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When Quantum Meets AI: Quantum Methods for Machine Learning and Machine Learning Methods for Quantum Systems
Tak Hur
Intelligence
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Summary
This thesis explores the intersection of quantum computing and artificial intelligence in two directions: using quantum methods for machine learning (Quantum for AI) and using machine learning for quantum systems (AI for Quantum). Key contributions include Neural Quantum Embedding (NQE) to improve classification on noisy hardware, a margin-based generalization analysis for quantum neural networks, a Mamba-based neural decoder for surface codes that reduces inference cost, and an interpretation of Stochastic Reconfiguration as tangent-space ridge regression with multi-shift variants.
Entities (9)
Relation Signals (7)
Mamba Decoder → appliedto → Surface Codes
confidence 97% · a Mamba-based neural decoder for surface codes
Mamba Decoder → reduces → inference-cost scaling
confidence 96% · reducing inference-cost scaling from quartic to quadratic in code distance.
Neural Quantum Embedding → improves → Classification
confidence 95% · Neural Quantum Embedding learns data representations that increase the trace distance... improving classification on noisy quantum hardware.
Neural Quantum Embedding → increases → Trace Distance
confidence 95% · Neural Quantum Embedding learns data representations that increase the trace distance between embedded class ensembles
Stochastic Reconfiguration → interpretedas → tangent-space ridge regression
confidence 94% · stochastic reconfiguration is interpreted as tangent-space ridge regression
Neural Quantum Embedding → extendsto → DQC1
confidence 93% · A training objective based on the Hilbert-Schmidt inner product extends this approach to deterministic quantum computation with one qubit (DQC1)
Margin Theory → predicts → Generalization
confidence 92% · margin distributions predict generalization more reliably than parameter-count metrics.
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Abstract
Abstract:This thesis studies the intersection of quantum computing and artificial intelligence in two directions: quantum methods for machine learning and machine learning methods for quantum systems. For quantum machine learning, Neural Quantum Embedding learns data representations that increase the trace distance between embedded class ensembles, lowering an embedding-dependent bound on empirical risk and improving classification on noisy quantum hardware. A training objective based on the Hilbert-Schmidt inner product extends this approach to deterministic quantum computation with one qubit (DQC1) and is demonstrated on an NMR quantum processor. A margin-based generalization analysis then connects quantum neural network performance to quantum state discrimination. In the studied benchmarks, margin distributions predict generalization more reliably than parameter-count metrics. For quantum systems, a Mamba-based neural decoder for surface codes matches a reproduced Transformer baseline in memory experiments while reducing inference-cost scaling from quartic to quadratic in code distance. Under an explicit decoder-induced-noise model, it achieves lower logical error rates and a higher effective threshold. For neural quantum states, stochastic reconfiguration is interpreted as tangent-space ridge regression, with its diagonal shift controlling the bias-variance trade-off under finite Monte Carlo sampling. Multi-shift stochastic reconfiguration reduces checkpoint-local validation residuals and update variance relative to fixed-shift SR, at additional computational cost. Together, these contributions show how learned representations, statistical control, and hardware constraints shape the exchange between quantum computing and machine learning.
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- Source: https://arxiv.org/abs/2609.25641v1
- Canonical: https://arxiv.org/abs/2609.25641v1
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When Quantum Meets AI: Quantum Methods for Machine Learning and Machine Learning Methods for Quantum Systems Tak Hur Department of Statistics and Data Science Graduate School Yonsei University arXiv:2609.25641v1 [quant-ph] 22 Sep 2026 When Quantum Meets AI: Quantum Methods for Machine Learning and Machine Learning Methods for Quantum Systems Advisor : Prof. Daniel K. Park A Dissertation Submitted to the Department of Statistics and Data Science and the Committee of the Graduate School of Yonsei University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Tak Hur August 2026 ACKNOWLEDGEMENTS I sincerely thank all the friends, colleagues, collaborators, and mentors whom I met dur- ing my PhD journey. Looking back, what remains most valuable to me is not the results or achievements, but the memories of struggle and joy I shared with the people around me. I would especially like to thank my advisor, Daniel K. Park, who not only guided me through the re- search, but also helped me grow to be a better person. I feel fortunate to have first met him when I was still an undergraduate. I have learned so much from him in the years since, and I can only hope to one day pass on to others what he has given me. I also thank my family, who have supported me throughout my life. Although I don’t express it often, knowing that you are proud of what I do means more to me than I can say. I would not be who I am today without your endless love and support. Finally, I thank my wife and best friend, Eun Hee. You have been by my side throughout my academic journey, from London to Seoul to Paris. The ups and downs we have shared along the way are among my most precious memories. I am excited for our next chapter together, wherever it may take us. If this thesis had a coauthor, it would be you. TABLE OF CONTENTS List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiv I Quantum for AI1 Chapter 1: Introduction to Quantum Machine Learning . . . . . . . . . . . . .2 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Introduction to Quantum Computing . . . . . . . . . . . . . . . . . . . 2 1.1.2 Introduction to Machine Learning . . . . . . . . . . . . . . . . . . . . 6 1.2 Quantum Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Quantum Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Quantum Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 i Chapter 2: Neural Quantum Embedding . . . . . . . . . . . . . . . . . . . . . .23 2.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.1 Empirical Risk and Trace Distance . . . . . . . . . . . . . . . . . . . . 24 2.1.2 Limitations of Deterministic Embeddings . . . . . . . . . . . . . . . . 26 2.2 Neural Quantum Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.1 The NQE Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.2 Training via Implicit Fidelity Loss . . . . . . . . . . . . . . . . . . . . 31 2.2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.4 Generalization, Expressibility, and Trainability . . . . . . . . . . . . . 37 2.3 Optimization via DQC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.1 The DQC1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.2 NQE Loss Function via Hilbert–Schmidt Inner Product . . . . . . . . . 43 2.3.3 Experimental Demonstration . . . . . . . . . . . . . . . . . . . . . . . 45 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Chapter 3: Generalization in Quantum Machine Learning . . . . . . . . . . . .51 3.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1.1 Generalization Bounds for Finite Hypothesis Classes . . . . . . . . . . 52 3.1.2 Complexity Measures for Infinite Hypothesis Classes . . . . . . . . . . 54 3.1.3 Margin Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Generalization Bounds for Quantum Models . . . . . . . . . . . . . . . . . . . 59 i 3.2.1 Uniform Generalization Bounds . . . . . . . . . . . . . . . . . . . . . 59 3.2.2 Limitations of Uniform Bounds . . . . . . . . . . . . . . . . . . . . . 60 3.3 Margin-Based Generalization for Quantum Neural Networks . . . . . . . . . . 61 3.3.1 Multiclass Classification with Quantum Neural Networks . . . . . . . . 61 3.3.2 Margin Generalization Bound . . . . . . . . . . . . . . . . . . . . . . 62 3.3.3 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.4 Connection to Quantum State Discrimination . . . . . . . . . . . . . . 70 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 I AI for Quantum74 Chapter 4: Neural Decoders for Quantum Error Correction . . . . . . . . . . .75 4.1 Introduction to Quantum Error Correction . . . . . . . . . . . . . . . . . . . . 76 4.1.1 Surface Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.1.2 The Decoding Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Neural Decoders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.1 Overview of Neural Decoder Literature . . . . . . . . . . . . . . . . . 82 4.2.2 AlphaQubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3 Mamba Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3.1 Mamba and State Space Models . . . . . . . . . . . . . . . . . . . . . 85 4.3.2 Architecture and Training . . . . . . . . . . . . . . . . . . . . . . . . 88 i 4.3.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Chapter 5: Neural Quantum States . . . . . . . . . . . . . . . . . . . . . . . . .98 5.1 Neural Quantum States and Stochastic Reconfiguration . . . . . . . . . . . . . 99 5.2 SR as Statistical Spectral Filtering . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Evidence Across System Scales . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.4 Multi-Shift Stochastic Reconfiguration . . . . . . . . . . . . . . . . . . . . . . 108 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Chapter 6: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 Appendix A: Supplementary Technical Details for Main Contributions . . . . . .136 A.1 Additional Details for Neural Quantum Embedding . . . . . . . . . . . . . . . 136 A.1.1 Implicit Fidelity Loss and Trace Distance . . . . . . . . . . . . . . . . 136 A.2 Additional Details for Margin Generalization . . . . . . . . . . . . . . . . . . 141 A.2.1 Proof Structure and Mixed-State Extension . . . . . . . . . . . . . . . 141 A.2.2 Experimental Protocol Notes . . . . . . . . . . . . . . . . . . . . . . . 143 A.3 Additional Details for Stochastic Reconfiguration as Spectral Filtering . . . . . 144 A.3.1 Fixed-Checkpoint Regression Identities . . . . . . . . . . . . . . . . . 144 iv Appendix B: Related Non-First-Author Contributions . . . . . . . . . . . . . . .148 B.1 Quantum Embeddings for Ligand-Based Virtual Screening . . . . . . . . . . . 149 B.2 Multi-Channel Convolutional Neural Quantum Embedding . . . . . . . . . . . 151 v LIST OF FIGURES 1.1 Schematicsofthetwodominantparadigmsinnear-termquantummachinelearn- ing. (a) A quantum neural network uses a parameterized quantum circuit푈(휃) trained via a classical optimizer. (b) A quantum kernel method uses a fixed quantum feature map to compute kernel entries, which are then processed by a classical algorithm such as an SVM. . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Schematic of a quantum convolutional neural network (QCNN). The circuit al- ternates between convolutional layers (two-qubit unitaries with shared parame- ters across neighboring pairs) and pooling layers, until a small number of qubits remain for the final measurement. . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Trace distance between class-averaged data ensembles퐷 tr (휌 − , 휌 + )for the con- ventional Z feature map and Neural Quantum Embedding on the balanced MNIST binary classification task (digits 0 vs. 1, 4 qubits). The blue dashed ref- erence line indicates the trace distance obtained by the conventional Z feature map without NQE. Results obtained on IBM quantum hardware (ibmq_toronto). 29 2.2 Overview of the NQE training procedure. A classical neural network푔(x,w) transforms input data into rotation angles for the quantum embedding circuit푉. The resulting quantum state|x⟩ = 푉 (푔(x,w)) |0⟩ ⊗푛 is used to compute the im- plicit fidelity loss via an overlap circuit푉 † (푔(x 푗 ,w)) 푉 (푔(x 푖 ,w)) |0⟩ ⊗푛 by mea- suring the probability of the all-zero outcome. The classical parametersware updated to maximize the distinguishability between classes. . . . . . . . . . . . 31 vi 2.3 QCNN training loss histories for balanced 4-qubit MNIST binary classification (digits 0 vs. 1). Blue solid, red dashed, and green dash-dotted lines represent the conventional Z feature map, PCA-NQE, and NQE, respectively. Thick lines indicate the theoretical lower bounds from Equation (2.11). Shaded re- gions represent one standard deviation over five (noiseless) or three (noisy) in- dependent trials.Left:noiseless simulation.Right:IBM quantum hardware (ibmq_jakarta, ibmq_toronto, ibmq_perth). . . . . . . . . . . . . . . . . . . . 34 2.4 Comparison of NQE against trainable unitary embeddings with one, two, and three trainable layers, using 8-qubit circuits on MNIST (top row) and Fashion- MNIST (bottom row) under noiseless (left column) and noisy (right column) simulation. The noisy simulations use the IBM Qiskit FakeGuadalupe environ- ment. Noiseless runs use1000iterations, learning rate0.01, and batches of128 samples per iteration. Noisy runs use200iterations, learning rate0.05, and batches of15samples per iteration. Classification accuracies are evaluated on held-out sets of2115(MNIST) and2000(Fashion-MNIST) samples, and the loss histories show the mean and one standard deviation over five independent trials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.5 Generalization diagnostics for embeddings with and without NQE.(a)Local ef- fective dimension of a four-qubit QNN with (solid green) and without (dashed purple) NQE, as a function of the number of data. NQE yields a smaller ef- fective model complexity, and hence a tighter generalization bound, across all dataset sizes (averaged over200experiments—10artificial datasets, each with 20random parameter initializations. Shaded regions denote one standard devi- ation).(b)Weight-norm complexity퐺 = ‖푊 ∗ ‖ 퐹 / √ 푁, which controls the data- dependent term of the quantum-kernel generalization bound Equation (2.25), as a function of the regularization weight휆. PCA-NQE (red circles) and NQE (green triangles) both lower퐺relative to the conventional Z feature map with- out NQE (blue squares) at every휆(mean and one standard deviation over five independent draws of1000MNIST samples). In both panels, lower values cor- respond to better expected generalization. . . . . . . . . . . . . . . . . . . . . 39 vii 2.6 Expressibility and trainability diagnostics for embeddings with and without NQE. (a) Deviation from a unitary2-design,휖 = √ Tr(퐴 † 퐴)from Equa- tion (2.26), on training (filled) and test (open) MNIST data. A larger deviation indicateslowerexpressibility. Both NQE variants are markedly less expressive than the conventional embedding. (b) Variance of the off-diagonal quantum- kernel elements, computed from1000MNIST samples (mean and one standard deviation over five iterations). The larger variance under NQE indicates that the kernel entries are not exponentially concentrated, improving the trainability of the quantum kernel method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.7 Experimental NQE-DQC1 circuit. A single probe qubit is acted upon by Hadamard gates, while the remaining register implements the controlled feature-map unitary and its Hermitian conjugate. Measurement of휎 푧 on the probe yields the Hilbert–Schmidt inner product required for the NQE-DQC1 loss. 42 2.8 NQE-DQC1 training on the NMR platform. (a) Schematic of the NMR DQC1 circuit: the probe qubit C1 controls the application of푉 † (푔(x 푗 )) 푉 (푔(x 푖 ))on qubits C2–C4. (b) Training loss versus iteration for NMR experiments (mark- ers) and numerical simulation (solid line). (c) Trace distance between class en- sembles for the training set (circles) and test set (triangles) across NQE training iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.9 Cross-platform training loss of the downstream PQC classifier with and without NQE. The classifier is optimized on the NQE-enhanced embedding (blue) and on the conventional Z feature map (red, “without NQE”). Dashed lines denote noiseless simulation, filled circles the NMR experiment, and asterisks the IBM superconducting hardware. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.10 Classification results with and without NQE. (a) Per-sample classification out- puts using the Z feature map alone (top) versus the NQE-enhanced embedding (bottom). (b) Scatter plot of predicted labels for all 500 MNIST test images (digits 0 vs. 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 viii 3.1 Tukey box-and-whisker plots of margin distributions for optimized 8-qubit QC- NNsontheQPRtask. ResultsareshownforQCNNswithone, five, andninelay- ers, under varying degrees of label randomization: 0% (left), 50% (middle), and 100% (right). Each legend entry reports the test accuracy with the correspond- ing generalization gap in parentheses. As randomization increases, the margin distributions shift to the left and the generalization gap widens, consistent with poorer generalization under the margin-sensitive bound in Equation (3.18). . . . 67 3.2 Comparison of the generalization gap, median margin (a margin-based metric), and effective parameters with threshold10 −2 (a parameter-based metric) as a function of: (a) the number of QCNN layers, (b) the percentage of randomized labels, and (c) the choice of variational ansatz. Since margins are inversely cor- related with the generalization gap (Equation (3.18)), the inverse median margin is plotted. The margin more closely tracks the generalization gap in these exper- iments, while effective parameters show inconsistent or opposite trends. . . . . 68 3.3 Mutual information (solid) and Kendall rank correlation coefficient (shaded) be- tween the generalization gap and various metrics. The first three columns rep- resent margin-based metrics (lower quartile, median, and mean of the margin distribution), while the last three represent parameter-based metrics (total pa- rameters, effective parameters at thresholds10 −1 and10 −2 ). Margin-based met- rics show stronger correlations with the generalization gap than parameter-based metrics in these experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 Tukey box-and-whisker plots of margin distributions for optimized 8-qubit QC- NNs on binary classification tasks using three quantum embedding schemes: fixed Z feature map (left), trainable quantum embedding (middle), and Neural Quantum Embedding (right). Results are shown for MNIST (bottom), Fashion- MNIST(middle), andKuzushiji-MNIST(top). Eachlegendentryreportsthetest accuracy with the corresponding generalization gap in parentheses. The black crossindicatesthemarginmean, andtheredcircleindicatestheunweightedtrace distance between class ensembles. Higher trace distances correspond to larger attainable margins and higher test accuracies—while the generalization gap re- mains small throughout—in these experiments, consistent with the connection established in Equation (3.23). . . . . . . . . . . . . . . . . . . . . . . . . . . 72 ix 4.1 Spacetime view of surface-code syndrome extraction for recurrent neural decod- ing. Each quantum error-correction (QEC) cycle produces stabilizer measure- ments and detection events, which are embedded as spatial features, processed recurrently over time, and decoded into a logical-error probability. . . . . . . . 79 4.2 Architecture of the Mamba decoder. (a) The Stabilizer Embedder converts raw measurements and detection events into푑 model -dimensional embeddings via lin- ear projections, positional encodings, and a ResNet. (b) The RNN Core pro- cesses each QEC cycle through퐿 = 3Syndrome Mixer layers with scaled skip connections. (c) Each Syndrome Mixer contains a Mamba-based mixer block, gated dense block, and dilated 2D convolutions. (d) The Readout Network maps the final hidden state to the logical-error probability푃 퐿 . . . . . . . . . . . . . . 89 4.3 Mamba decoder accuracy and inference speed . . . . . . . . . . . . . . . . . . 92 4.4 Real-time decoding performance.Main:Logical error per round (LER) for Mamba and Transformer decoders under simulated real-time conditions with decoder-induced noise proportional to computational complexity.Inset:LER without decoder-induced noise, showing comparable baseline accuracy. . . . . 94 4.5 Finite-size effective threshold under this latency model. Each panel shows logi- cal error per round versus physical error rate for code distances푑 = 3and푑 = 5. The effective threshold is identified where the푑 = 5curve crosses above the 푑 = 3curve. Under the assumed decoder-induced-noise model, the Transformer decoder yields푝 th ≈ 0.0097, while the Mamba decoder yields푝 th ≈ 0.0104. . . 95 5.1 Exact amplitude-regression diagnostics for the4 × 4Heisenberg graph of Equa- tion (5.22), with32nearest-neighbor and24diagonal bonds. At푁 푠 = 4096and 휆 = 10 −9 , matched noise exposes ViT overfitting at comparable amplitude-gap variance, while matched state shows lower gap variance and excess risk for the ViT. The plotted predictions and risks exclude the phase channel. . . . . . . . . 107 5.2 Large-scale fixed-checkpoint diagnostics for the퐿 = 100transverse-field Ising model family using a foundation NQS. The validation residual exhibits a U- shaped dependence on the SR diagonal shift, while the multi-batch variance de- creases with stronger regularization, as predicted by the noisy-ridge interpreta- tion in Equation (5.17). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 x 5.3 Checkpoint-local MS-SR ablations at퐾 = 4. Bars show raw validation residu- als and multi-batch variances averaged over fixed source checkpoints, with one standard error across checkpoints. Standard SR uses휆 train = 10 −4 . Bagging is shown at both the training shift and an oracle shift selected on the reporting data. All multi-shift methods use the same NTK-quantile grid. Values retain the doubled-target convention and, for퐽 1 –퐽 2 , Pauli Hamiltonian units. . . . . . . . 111 5.4 Online SR and MS-SR training on the8 × 8 퐽 1 –퐽 2 model. Five paired contin- uations start from a single shared SR checkpoint; energies per site are in Pauli units.(a)Trainingcurvesshowmeansoftracesbinnedinintervalsof20nominal SR-equivalent updates, with±1across-continuation sample standard deviation. Diamonds mark independent endpoint means. The horizontal axis counts one candidate per SR step and four per MS-SR step; the historical implementation’s extra solve and sampled batches are excluded from this nominal count.(b)Inde- pendent endpoint energies for the same five pairs. Bold marks the lower energy. Differences areΔ = (퐸 MS-SR − 퐸 SR )/64. Differences and individual±values are in units of10 −5 , with the latter combining the two Monte Carlo standard errors in quadrature. The across-pair confidence interval is given in the text. . . 113 A.1 NQE-DQC1 training results for the three alternative Hamiltonian-inspired feature maps (퐻 푋푌 ,퐻 푌 푍 ,퐻 푋푌 푍 ) on MNIST (left four columns) and Fashion- MNIST (right four columns). For each ansatz row, the left panel of each dataset shows the NQE loss퐿 NQE with the class trace distance during training in the inset; the right panel shows the downstream PQC loss퐿 PQC with (solid) and without (dashed) NQE. All ansatzes use circuit depth푀 = 4. The consistent decrease in퐿 NQE and improvement in퐿 PQC confirm that the NQE benefit is not specific to the푍(퐻 푋푍 ) feature map. . . . . . . . . . . . . . . . . . . . . . . 141 B.1 Trace-distance increase obtained by NQE training with the Z feature map on LIT-PCBA targets. The plot compares trace distances before and after NQE training for balanced (1:1) and imbalanced (1:6) activator/inactivator settings, separately for training and test sets. The log scale highlights that the untrained embeddingsproducenearlyindistinguishableclassensembles, whileNQEraises the class separation by several orders of magnitude for many targets. . . . . . . 150 xi B.2 Relationship between class trace distance and QCNN classification accuracy across CNQE configurations on CIFAR-10 and Tiny ImageNet binary tasks. Each point corresponds to a choice of dataset, classical-to-quantum interface, loss function, and embedding circuit. The positive trend supports the use of trace distance as an embedding-quality diagnostic for multi-channel NQE. . . . 152 xii LIST OF TABLES 2.1 SummaryofNQEresults forbalanced4-qubitMNISTbinaryclassification(dig- its 0 vs. 1). Noiseless accuracy is from statevector simulation, while noisy ac- curacy is from the IBM quantum hardware runs. The lower bound on empirical risk is computed as퐿 푆 ≥ (1 − 퐷 tr )/2. . . . . . . . . . . . . . . . . . . . . . . 34 4.1 Decoder hyperparameters. Top: model-specific parameters for Transformer and Mamba variants in Sycamore memory experiments (Syc.) and real-time decod- ing simulations (RT). Bottom: shared training and architecture parameters. . . . 90 5.1 Steady-state update-construction cost on the TFIM foundation NQS using one H100 80GB GPU. Values are medians over three paired device/seed settings, each with two warm-ups and five synchronized timed repetitions. Bagged SR uses the training shift and excludes the cost of shift tuning. . . . . . . . . . . . 114 A.1 Selected reproducibility details for the NQE experiments in Hur et al. [48]. . . . 138 A.2 Selected reproducibility details for the margin-generalization experiments. . . . 143 A.3 Selected protocol details for the SR spectral-filtering experiments. . . . . . . . 147 B.1 Summary of non-first-author contributions related to the main thesis. . . . . . . 148 xiii ABSTRACT When Quantum Meets AI: Quantum Methods for Machine Learning and Machine Learning Methods for Quantum Systems Tak Hur Department of Statistics and Data Science The Graduate School, Yonsei University This thesis studies the intersection of quantum computing and artificial intelligence in two directions. The first direction,Quantum for AI, asks how quantum models can be used for machine learning. The second direction,AI for Quantum, asks how machine learning can help solve problems that arise in quantum error correction and quantum many-body physics. The Quantum for AI part begins with background on quantum computing, supervised learning, quan- tum neural networks, and quantum kernels. It then presents Neural Quantum Embedding, a method for learning the data embedding used before quantum classification. The trace distance between embedded class ensembles sets a floor on the empirical risk achievable by the downstream classifier. And learning the embedding substantially raises this distinguishability, markedly improving classification accuracy on noisy quantum hardware. A DQC1-compatible training objective based on the Hilbert–Schmidt inner product extends the method to ensemble quantum systems and is demonstrated on an NMR quantum processor. The next contribution turns to generalization in quantum machine learning. It establishes a margin-based generalization bound for quantum neural networks, shows empirically that margin distribu- tions predict generalization more reliably than parameter-count metrics, and links achievable margins to the trace distance between class ensembles, connecting generalization to quantum state discrimination. The AI for Quantum part applies neural methods to two quantum problems. For quantum error cor- rection, the thesis develops a Mamba-based neural decoder for the surface code whose inference cost scales quadratically rather than quartically with code distance. The decoder matches the accuracy of a xiv Transformer baseline in memory experiments on simulated and real hardware data. In real-time decoding under an explicit decoder-induced-noise model, the Mamba decoder outperforms the Transformer base- line, achieving lower logical error rates and a higher effective error threshold. For neural quantum states, the thesis analyzes stochastic reconfiguration, a standard optimization method in variational Monte Carlo. It shows that the diagonal shift acts as a statistical spectral filter that trades bias against sampling variance under finite Monte Carlo sampling. A multi-shift variant combines independent regularized SR solves to reduce checkpoint-local validation residuals and update variance relative to standard SR at the training shift, at additional computational cost. Together, these contributions position the intersection of quantum computing and machine learning not as a one-way application of techniques, but as a bidirectional exchange in which each field supplies principled tools for the other’s hardest problems. Keywords: Quantum AI, Quantum Machine Learning, Quantum Error Correction, Neural Quantum States, Stochastic Reconfiguration, Generalization, Quantum Embeddings xv Part I Quantum for AI 1 CHAPTER 1 INTRODUCTION TO QUANTUM MACHINE LEARNING 1.1Background This chapter develops the technical background needed for theQuantum for AIdirection, while also introducing notation and machine learning concepts that reappear in Part I. Readers al- ready familiar with quantum computing may skip Section 1.1.1, and those with a background in machine learning may skip Section 1.1.2. 1.1.1Introduction to Quantum Computing Quantum computing exploits the principles of quantum mechanics—superposition, entangle- ment, and interference—to process information in ways that differ fundamentally from classical computation. This section reviews the essential building blocks of quantum computation used in the subsequent chapters. For a comprehensive treatment, we refer the reader to Nielsen and Chuang [1]. Qubits and Quantum States The fundamental unit of quantum information is thequbit, a two-level quantum system. Unlike a classical bit, which takes a definite value0or1, a qubit can exist in asuperpositionof both 2 computational basis states: |휓⟩ = 훼|0⟩ + 훽|1⟩,훼, 훽 ∈ ℂ,|훼| 2 + |훽| 2 = 1.(1.1) The state|휓⟩is a unit vector in a two-dimensional complex Hilbert spaceℋ ≅ ℂ 2 , and훼,훽 are calledprobability amplitudes. A system of푛qubits lives in the tensor product spaceℋ ⊗푛 ≅ ℂ 2 푛 . A general푛-qubit state can be written as |휓⟩ =∑ 푥∈0,1 푛 푐 푥 |푥⟩,∑ 푥 |푐 푥 | 2 = 1,(1.2) where|푥⟩denotes a computational basis state. The dimension of the state space grows exponen- tially with푛, meaning that describing a general quantum state requires an exponentially large number of complex amplitudes—a feature that lies at the heart of the potential power of quan- tum computation. An important consequence of this tensor product structure isentanglement: multi-qubit states that cannot be written as a product of individual qubit states. For example, the Bell state |Φ + ⟩ = 1 √ 2 (|00⟩ + |11⟩)exhibits correlations between qubits that have no classical explanation and serve as a key resource in many quantum algorithms and communication protocols. Quantum Gates and Circuits The evolution of a closed quantum system is described byunitary operators. A unitary operator 푈satisfies푈 † 푈 = 푈 † = 퐼, which ensures that quantum states remain normalized under evo- lution. In the circuit model of quantum computation, algorithms are constructed by composing elementary unitary operations calledquantum gates. 3 Common single-qubit gates include the Pauli operators and the Hadamard gate: 푋 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ,푌 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 −푖 푖0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ,푍 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 10 0 −1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ,퐻 = 1 √ 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 11 1 −1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ .(1.3) Other important single-qubit gates include parameterized rotation gates푅 푃 (휃) = 푒 −푖휃푃/2 for 푃 ∈ 푋, 푌 , 푍. Multi-qubit entangling gates, such as the controlled-NOT (CNOT) gate, are essential for creating entanglement: CNOT= |0⟩⟨0| ⊗ 퐼 + |1⟩⟨1| ⊗ 푋.(1.4) A fundamental result in quantum computing is that any unitary operation on푛qubits can be decomposed into a sequence of single-qubit gates and CNOT gates, forming auniversal gate set[1]. Aquantum circuitis a sequence of quantum gates applied to a register of qubits, typically initialized in the state|0⟩ ⊗푛 . The circuit model provides a convenient framework for designing andanalyzingquantumalgorithms, anditisthecomputationalmodelusedthroughoutthisthesis. Measurement Quantum measurement extracts classical information from a quantum system. In the standard computational basis measurement, measuring an푛-qubit state|휓⟩ = ∑ 푥 푐 푥 |푥⟩yields outcome푥 with probability 푝(푥) = |푐 푥 | 2 ,(1.5) as dictated by theBorn rule. After the measurement, the state collapses to the observed basis state|푥⟩, destroying the superposition. This probabilistic and irreversible nature of measurement 4 means that quantum algorithms must be carefully designed so that the desired answer is obtained with high probability, often by exploiting constructive and destructive interference among the amplitudes. More generally, measurements can be described by a set of measurement operators푀 푚 satisfying the completeness relation∑ 푚 푀 † 푚 푀 푚 = 퐼, where outcome푚occurs with probability 푝(푚) = ⟨휓|푀 † 푚 푀 푚 |휓⟩. Inthisthesis, weprimarilyworkwithcomputationalbasismeasurements and expectation values of observables푂, computed as⟨푂⟩ = ⟨휓|푂|휓⟩. The Promise of Quantum Computation The exponentially large state space of푛qubits does not, by itself, guarantee computational advantage—after all, measurement collapses the state and yields only a single classical outcome. The power of quantum computing lies in the ability to orchestrate interference and entanglement so that useful answers emerge with high probability. This principle has led to quantum algorithms that provably outperform the best known clas- sical algorithms for specific problems. Shor’s algorithm [2] factors integers in polynomial time, an exponential speedup over known classical methods. Grover’s algorithm [3] searches an un- structured database of푁items in푂( √ 푁 )queries, a quadratic improvement over the classical 푂(푁 )lower bound. These results demonstrate that quantum computers can, in principle, solve certain problems that are intractable for classical machines. In the near term, fully fault-tolerant quantum computers remain out of reach. Current de- vices, known asnoisy intermediate-scale quantum(NISQ) devices [4], contain tens to hun- dreds of qubits with limited coherence times and noisy gate operations. This has motivated the development of hybrid quantum-classical approaches that use shallow parameterized circuits amenable to near-term hardware, which we discuss in Section 1.2. 5 1.1.2Introduction to Machine Learning Machinelearning, broadlydefined, isthestudyofalgorithmsthatimprovetheirperformanceona task through experience [5]. While the field encompasses a wide range of paradigms—including unsupervised, reinforcement, and self-supervised learning—this thesis focuses primarily onsu- pervised learning, where a model learns a mapping from inputs to outputs given a labeled training dataset. This section traces the key developments in machine learning that motivate the quantum extensions discussed in the remainder of this thesis. From Classical Methods to Deep Learning Early machine learning research focused on models with strong theoretical foundations but lim- ited representational capacity. Linear classifiers, decision trees, and nearest-neighbor methods formed the initial toolkit. A major advance came withsupport vector machines(SVMs) [6], which use thekernel trickto implicitly map data into high-dimensional feature spaces where linear separation becomes possible. SVMs offered strong generalization guarantees grounded in statistical learning theory [7] and dominated many benchmarks through the 2000s. Kernels and the margin theory of SVMs appear twice in this thesis: first in Section 1.2.3, where we introduce their quantum generalization, and later in Chapter 3, where we leverage margin-based arguments from SVM theory to derive generalization bounds for quantum neural networks. The deep learning revolution was catalyzed byconvolutional neural networks(CNNs). Al- though the core ideas date back to the neocognitron [8] and LeNet [9], the breakthrough came in 2012 when AlexNet [10] won the ImageNet Large Scale Visual Recognition Challenge by a decisive margin, demonstrating that deep CNNs trained on GPUs could dramatically outperform hand-engineered feature pipelines. The key architectural principles—local connectivity, weight sharing, and hierarchical feature extraction—enabled CNNs to scale to high-dimensional im- 6 age data while keeping the parameter count manageable. Subsequent architectures such as VG- GNet[11], GoogLeNet[12], andResNet[13]continuedtopushperformancebyincreasingdepth and introducing structural innovations like skip connections. Beyondcomputervision,recurrent neural networks(RNNs)andtheirgatedvariants—Long Short-Term Memory (LSTM) [14] and Gated Recurrent Units (GRU) [15]—became the standard for sequential data, powering advances in machine translation, speech recognition, and natural language understanding. The Transformer Era The introduction of theTransformerarchitecture [16] marked a major shift across machine learning. By replacing recurrence with aself-attentionmechanism that computes pairwise in- teractions among all elements of a sequence in parallel, Transformers resolved the sequential bottleneck of RNNs and enabled efficient training on massive datasets. This architecture gave rise to large language models such as GPT [17] and BERT [18], and has since been adopted wellbeyondnaturallanguageprocessing—includingcomputervision(VisionTransformer[19]), protein structure prediction (AlphaFold [20]), and scientific computing. Transformer-based ar- chitectures now form an important part of the modern machine learning toolkit, especially for sequence modeling and large-scale representation learning. The Transformer paradigm also plays a central role in Part I of this thesis. In Chapter 4, we build on the AlphaQubit architecture, a Transformer-based neural decoder for quantum error correction, and propose a Mamba-based decoder that replaces the attention mechanism with a state-space model for improved scalability. In Chapter 5, transformer and foundation neural quantum states provide the parameter-rich setting in which we analyze stochastic reconfiguration as a statistical spectral filter for variational Monte Carlo. A recurring theme in these developments is thatarchitectural design—the choice of hy- 7 pothesis class—plays a decisive role in model performance. CNNs succeeded because their in- ductive biases match the structure of image data; Transformers succeeded because self-attention captures long-range dependencies that recurrent models struggle with. This observation moti- vates a central question of this thesis: can quantum computing offer new model classes with inductive biases that are advantageous for certain learning problems? Sources of Prediction Error To reason about the performance of any learning algorithm, it is useful to decompose the pre- diction error into distinct components [21]. Consider a supervised learning problem where we seek a function that maps inputs푥 ∈ 풳to outputs푦 ∈ 풴. Letℛ(푓 ) ∶= 피 (푥,푦)∼풟 [ℓ(푓 (푥), 푦)] denote therisk(expected loss) of a function푓with respect to the true data distribution풟, and let ̂ ℛ(푓 ) ∶= 1 푛 ∑ 푛 푖=1 ℓ(푓 (푥 푖 ), 푦 푖 )denote theempirical riskcomputed over푛training examples. In practice, we do not have access toℛ. Instead, we can only minimize ̂ ℛover a chosen hypothesis classℱ. Let푓 ∗ denote a population-risk minimizer over the unrestricted target class, let 푓 ∗ ℱ ∈ argmin 푓 ∈ℱ ℛ(푓 )denote the best-in-class predictor for the population risk, let ̂ 푓 ℱ ∈ argmin 푓 ∈ℱ ̂ ℛ(푓 )denote an empirical-risk minimizer, and let ̂ 푓denote the function returned by the actual training algorithm. The excess risk of ̂ 푓can then be decomposed as: ℛ( ̂ 푓 ) − ℛ(푓 ∗ ) = ℛ(푓 ∗ ℱ ) − ℛ(푓 ∗ ) ⏟ approximation + ℛ( ̂ 푓 ℱ ) − ℛ(푓 ∗ ℱ ) ⏟ estimation / generalization + ℛ( ̂ 푓 ) − ℛ( ̂ 푓 ℱ ) ⏟ optimization .(1.6) Each term captures a distinct source of error: •Approximation error: residual risk from the limitedexpressivityofℱ. •Estimation error: the cost of learning from finite data (generalization). •Optimization error: suboptimality from non-convex training. 8 This decomposition provides a useful organizing framework for Part I of the thesis. Specifically, the following chapters study how quantum embeddings set a representation-induced floor on achievable training loss and how margins affect generalization in quantum machine learning. 1.2Quantum Machine Learning 1.2.1History Quantum machine learning has been an important area of quantum computing research for over a decade. An influential early result in this direction was theHHL algorithm[22], proposed by Harrow, Hassidim, and Lloyd in 2009, which solves a system of linear equations퐴x=bin time logarithmic in the dimension under assumptions on sparsity, conditioning, and quantum state preparation. This result suggested that quantum computers could accelerate core linear algebra subroutines underlying many machine learning methods. Building on HHL, a wave of quantum machine learning algorithms emerged in the early 2010s. Rebentrost et al. [23] proposed a quantum support vector machine (QSVM) that performs classification in푂(log푁 )time by leveraging quantum linear algebra to solve the least-squares formulation of SVMs. Similarly, Lloyd et al. [24] introduced quantum principal component analysis (QPCA), which extracts eigenvalues and eigenvectors of a density matrix exponentially faster than classical PCA. These results generated significant optimism that quantum comput- ers could accelerate machine learning workloads under strong data-access and state-preparation assumptions. However, these algorithms share a critical assumption: they require efficientquantum ac- cesstoclassicaldata, typicallythroughaquantumrandomaccessmemory(QRAM)thatcanpre- pare arbitrary quantum states encoding the input data in superposition. Constructing a practical QRAM remains a major open challenge—the hardware overhead required to maintain coherent 9 access to푁classical data entries may negate the very speedup that the algorithms promise [25]. Without QRAM, the cost of loading classical data into quantum states can dominate the total runtime, reducing the practical advantage to at best polynomial. A more fundamental challenge to these early speedup claims came from a series ofde- quantizationresults by Tang [26], who demonstrated a classical algorithm for recommendation systems that matches the quantum algorithm’s performance up to polynomial factors, assuming only classicalsample and queryaccess to the input data. This work was followed by the de- quantization of quantum PCA [27], which showed that much of the claimed exponential speedup was an artifact of the strong input model (quantum state preparation) rather than a genuine com- putational advantage of quantum mechanics. Together, these results established that quantum speedups built on the HHL framework are far more fragile than initially believed. The QRAM bottleneck and the dequantization barrier shifted the focus of the field from fault-tolerant quantum linear-algebra subroutines toward models that can be run on noisy intermediate-scale quantum (NISQ) devices [4]. This shift did not remove the need for rigorous advantage claims. Rather, it changed the practical emphasis toward trainability, representation, finite samples, and hardware noise. Within this NISQ-era setting, two supervised-learning paradigms are central for this thesis: quantum neural networks and quantum kernel methods. They are illustrated in Figure 1.1 and discussed in the following subsections. 1.2.2Quantum Neural Networks The termparameterized quantum circuitcovers a broad family of trainable circuits used in variational quantum algorithms, including VQE for energy minimization [28] and QAOA for combinatorial optimization [29]. In this thesis, the termquantum neural network(QNN) refers more narrowly to supervised-learning trainable quantum models built from such parameterized circuits [30]. In a typical supervised QNN, a data encoding circuit푉 (x)first embeds classi- 10 Quantum Supervised Learning 1. Quantum Neural Networks (x 1 ,y 1 ),(x 2 ,y 2 ),...,(x N ,y N ) iid ∼D Given N labelled sample classical data, Find prediction function , f(x) min피 X,Y∼D | f(X)−Y | 2. Quantum Kernel Methods | 0⟩ | 0⟩ | 0⟩ | 0⟩ argmin θ L(θ) V(x) Data Embedding Variational Circuit U(θ) | 0⟩ | 0⟩ | 0⟩ | 0⟩ V(x) V † (x′ ) k(x,x′ )= | ⟨x | x′ ⟩ | 2 f(x;θ)=⟨x | U(θ) † OU(θ) | x⟩ Prediction function: Kernel function: V(x) | 0⟩ ⊗n = | x⟩ *Note Quantum Embedding: (a) Quantum Neural Network (QNN) Quantum Supervised Learning 1. Quantum Neural Networks (x 1 ,y 1 ),(x 2 ,y 2 ),...,(x N ,y N ) iid ∼D Given N labelled sample classical data, Find prediction function , f(x) min피 X,Y∼D | f(X)−Y | 2. Quantum Kernel Methods | 0⟩ | 0⟩ | 0⟩ | 0⟩ argmin θ L(θ) V(x) Data Embedding Variational Circuit U(θ) | 0⟩ | 0⟩ | 0⟩ | 0⟩ V(x) V † (x′ ) k(x,x′ )= | ⟨x | x′ ⟩ | 2 f(x;θ)=⟨x | U(θ) † OU(θ) | x⟩ Prediction function: Kernel function: V(x) | 0⟩ ⊗n = | x⟩ *Note Quantum Embedding: (b) Quantum Kernel Method (QKM) Figure 1.1: Schematics of the two dominant paradigms in near-term quantum machine learning. (a) A quantum neural network uses a parameterized quantum circuit푈(휃)trained via a classical optimizer. (b) A quantum kernel method uses a fixed quantum feature map to compute kernel entries, which are then processed by a classical algorithm such as an SVM. cal inputxinto a quantum state, followed by a trainable circuit푈(휃)with parameters휃. The prediction is obtained by measuring an observable푂: 푓 (x; 휃) = ⟨0|푉 † (x) 푈 † (휃) 푂 푈(휃) 푉 (x)|0⟩.(1.7) When the input is quantum data (e.g., a quantum state휌from a physical system), the encoding circuit푉 (x)is replaced by the quantum state itself, and the QNN acts directly on the given state. A classical optimizer iteratively updates휃to minimize a loss function computed from the measurement outcomes. One example of the trainable circuit푈(휃)is thehardware-efficient ansatz(HEA) [31], consisting of퐿layers of single-qubit rotation gates followed by entangling two-qubit gates: 푈(휃) = 퐿 ∏ ℓ=1 푊 ent ⋅ 푛 ⨂ 푖=1 푅(휃 (ℓ) 푖 ),(1.8) where푅(휃) = 푅 푍 (휃 1 )푅 푌 (휃 2 )푅 푍 (휃 3 )represents a general single-qubit rotation (up to global phase), and푊 ent denotes a fixed entangling layer such as a ladder of CNOT gates matching the hardware topology. Beyond generic hardware-efficient circuits, QNNs also include structured ansatz families that build architectural bias into the circuit. 11 Structured QNNs: Quantum Convolutional Neural Networks A widely studied structured QNN architecture is thequantum convolutional neural network (QCNN), introduced by Cong et al. [32] for classifying quantum phases of matter. QCNNs draw direct inspiration from classical CNNs by combining local receptive fields, shared parameters, and hierarchical pooling. The architecture alternates between convolutional and pooling layers, progressively reducing the system size until only a few qubits remain for measurement: 푈 QCNN = 푈 (퐿) conv ∘Pool (퐿−1) ∘ ⋯ ∘ 푈 (2) conv ∘Pool (1) ∘ 푈 (1) conv ,(1.9) where퐿denotes the number of convolutional layers. As illustrated in Figure 1.2, each convolu- tional layer applies a translationally invariant two-qubit parameterized unitary푈(휃) 푖,푖+1 across all neighboring qubit pairs, with shared parameters analogous to the weight-sharing mechanism in classical CNNs. Each pooling layer then halves the number of active qubits by measuring a subset and applying conditioned unitaries on the remaining qubits. After log 2 (푛)pooling oper- ations, only풪(1)qubits remain for the final classification measurement. Although QCNNs were originally designed for quantum data, prior work also applied them to classical data classification by prepending a data encoding circuit [33], where they serve as compact, parameter-sharing QNN classifiers. QCNNs enjoy several favorable properties as QNN architectures. Their hierarchical struc- ture and local operations provably avoid barren plateaus [34], enabling efficient training even for large system sizes. On the other hand, several local, hierarchical QCNN architectures admit efficient classical simulation or tensor-network contraction under additional structural assump- tions [35]. 12 3 Convolution Data Encoding Pooling | 0⟩ | 0⟩ | 0⟩ | 0⟩ | 0⟩ | 0⟩ | 0⟩ | 0⟩ Classical Computer argmin θ C(θ) ×l 1 ×l 2 ×l 3 FIG. 1. A schematic of the QCNN algorithm for an example of 8 input qubits. The quantum circuit consists of three parts: quantum data encoding (green rectangle), convolutional filters (blue rounded rectangle), and pooling (red circle). The quantum data encoding is fixed in a given structure of QCNN, while the convolutional filter and pooling use parameterized quantum gates. There are three layers in this example, and in each layer, multiple convolutional filters can be applied. The number of filters forith layer is denoted byl i . In each layer, the convolutional filter applies the same two-qubit ansatz to nearest neighbour qubits in a translationally-invariant way. Similarly, pooling operations within the layer use the same ansatz. In this example, the pooling operation is represented as a controlled unitary transformation, which is activated when the control qubit is 1. However, general controlled operations can also be considered. The measurement outcome of the quantum circuit is used to calculate the user-defined cost function. The classical computer is used to compute the new set of parameters based on the gradient, and the quantum circuit parameters are updated accordingly for the subsequent round. ing qubits with the two-qubit parameterized gates in the translationally invariant way. The optimization of the gate parameters can be car- ried out by iteratively updating the parameters based on the gradient of the cost function until some condi- tion for the termination is reached. The cost function gradient can be calculated classically or by using a quan- tum computer viaparameter-shift rule[31–33]. When the parameter-shift rule is used, QCNN requires an exponen- tially smaller number of quantum circuit executions com- pared to the general hierarchical structures inspired by tensor networks (e.g. tree tensor network) in Ref. [28]. While the latter usesO(n) runs, the former only uses O(log(n)) runs. C. Quantum data encoding Many machine learning techniques transform input dataXinto a di↵erent space to make it easier to work with. This transformation:X!X 0 is often called a feature map. In quantum computing, the same anal- ogy can be applied to perform aquantum feature map, which acts as:X!Hwhere the vector spaceHis a Hilbert space [34]. In fact, such feature mapping is mandatory when one uses quantum machine learning on classical data, since classical data must be encoded as a quantum state [9, 11, 35–38]. The quantum feature map x2X!|(x)i2His equivalent to applying a unitary transformationU (x) to the initial state|0i ⌦n to produce U (x)|0i ⌦n =|(x)i,wherenis the number of qubits. This refers to the green rectangle in Fig. 1. There exist numerous structures ofU (x)toencodethe classical input dataxinto a quantum state. In this work, we benchmark the performance of the QCNN algorithm with several di↵erent quantum data encoding techniques. These techniques are explained in detail in this section. 1. Amplitude encoding One of the most general approaches to encode classi- cal data as a quantum state is to associate normalized in- put data with probability amplitudes of a quantum state. This encoding scheme is known as the amplitude encod- ing (AE). The amplitude encoding represents input data ofx=(x 1 ,...,x N ) T of dimensionN=2 n as amplitudes of ann-qubit quantum state|(x)ias U (x):x2R N !|(x)i= 1 kxk N X i=1 x i |i,(2) where|iiis theith computational basis state. Clearly, with amplitude encoding, a quantum computer can rep- resent exponentially many classical data. This can be of great advantage in QCNN algorithms. Since the number of parameters subject to optimization scales asO(log(n)) (see Sec. I B), the amplitude encoding reduces the num- ber of parameters doubly-exponentially with the size (i.e. dimension) of the classical data. However, the quantum circuit depth for amplitude encoding usually grows as O(poly(N)), although there exists a method to reduce Figure 1.2: Schematic of a quantum convolutional neural network (QCNN). The circuit alter- nates between convolutional layers (two-qubit unitaries with shared parameters across neighbor- ing pairs) and pooling layers, until a small number of qubits remain for the final measurement. Gradient Computation: The Parameter-Shift Rule Training QNNs requires computing gradients of the cost function with respect to the circuit pa- rameters. Unlike classical neural networks, where backpropagation provides an efficient route to exactgradients, quantumcircuits donot admitdirectdifferentiation—measurements collapsethe quantum state and prevent the propagation of gradient information. Recent work has explored quantum analogues of backpropagation [36], showing that achieving backpropagation-like scal- ing requires access to multiple copies of the quantum state via shadow tomography. Akeyinsightenablinggradient-basedoptimizationofQNNsistheparameter-shift rule[37, 38]. For gates of the form푅 푃 (휃) = 푒 −푖휃푃/2 where푃 2 = 퐼(such as Pauli rotation gates), the partial derivative of the expectation value can be computed exactly using only two circuit evaluations: 휕 휕휃 푓 (휃) = 1 2 [푓 (휃 + 휋 2 ) − 푓 (휃 − 휋 2 )] , (1.10) where푓 (휃) =Tr[휌 푈 † (휃) 푂 푈(휃)]for an input state휌. This formula provides the exact an- 13 alytical gradient (not a finite-difference approximation) using only shifted circuit evaluations, making it naturally compatible with quantum hardware where only expectation values can be measured. The parameter-shift rule forms the foundation for gradient-based training of QNNs, enabling the use of optimizers such as gradient descent, Adam, and their variants. The Barren Plateau Problem AmajorobstacletotrainingQNNsisthebarren plateauphenomenon[39], whereinthevariance of the cost function gradient vanishes exponentially with the number of qubits푛: Var 휃 [ 휕퐶 휕휃 푘 ] ≤ 퐹(푛) ⋅ 2 −푛 ,(1.11) where퐹(푛)is a polynomial factor. When the gradient variance is exponentially small, an expo- nentialnumberofmeasurementshotsisrequiredtodistinguishthegradientfromstatisticalnoise, negating any potential quantum advantage. Barren plateaus can arise from various sources, in- cluding excessive circuit depth, global cost functions, noise, and high entanglement. Recent work has provided a unified understanding of barren plateaus through the lens of thedynami- cal Lie algebra(DLA) of the circuit generators [40, 41]. These results show that the gradient variance scales inversely with the dimension of the DLA, establishing that circuits with a poly- nomially scaling DLA can escape barren plateaus, while those with an exponentially scaling DLA cannot. We refer the reader to Larocca et al. [42] for a comprehensive review. 1.2.3Quantum Kernels Kernel methods offer an alternative paradigm for quantum machine learning, one that trades the parameterized circuits of QNNs for a fixed quantum feature map and classical post-processing. 14 This approach provides strong theoretical guarantees, and in constructed supervised-learning settings it has yielded rigorous separations under explicit computational assumptions. We begin with a brief review of classical kernel methods before introducing their quantum generalization. Classical Kernel Methods In classical machine learning, kernel methods [6, 7] address the fundamental challenge of learn- ing nonlinear decision boundaries by implicitly mapping data into a high-dimensionalfeature space. Given a feature map휙 ∶ 풳 → ℱthat embeds inputsx∈ 풳into a (possibly infinite- dimensional) Hilbert spaceℱ, a kernel function computes inner products in this feature space without explicitly constructing the embedding: 퐾(x,x ′ ) = ⟨휙(x), 휙(x ′ )⟩ ℱ .(1.12) Thiskernel trickenables algorithms like support vector machines to operate efficiently in feature spaces of exponential or even infinite dimension. Therepresenter theoremfurther guarantees that the optimal predictor in a kernel-regularized learning problem can be expressed as a linear combination of kernel evaluations on the training data: 푓 ∗ (x) = 푁 ∑ 푖=1 훼 푖 퐾(x,x 푖 ).(1.13) This elegant structure motivates the question: can quantum computers realize feature maps that are computationally intractable for classical machines, while remaining efficiently estimable? 15 Quantum Feature Maps Aquantum feature mapencodes classical dataxinto a quantum state by applying a unitary transformation to an initial state [43]: |휙(x)⟩ = 푉 (x)|0⟩ ⊗푛 .(1.14) The quantum states|휙(x)⟩live in the2 푛 -dimensional Hilbert space of푛qubits, and thequantum kernelis defined as the squared overlap between embedded states: 퐾(x,x ′ ) = |⟨휙(x)|휙(x ′ )⟩| 2 = |⟨0| ⊗푛 푉 † (x)푉 (x ′ )|0⟩ ⊗푛 | 2 .(1.15) This kernel can be estimated on a quantum computer by preparing the state푉 † (x)푉 (x ′ )|0⟩ ⊗푛 and measuring the probability of obtaining the all-zeros outcome. The key insight is that if 푉 (x)generates states with classically intractable correlations, the resulting kernel may be hard to compute classically, yet remain efficiently estimable on quantum hardware. The Z Feature Map A concrete realization of this idea is theZZ feature mapintroduced by Havlicek et al. [43]. This feature map applies퐷repetitions of a data-encoding unitary: 푉 (x) = ⎡ ⎢ ⎣ exp ⎛ ⎜ ⎝ 푖 ∑ 푖<푗 (휋 − 푥 푖 )(휋 − 푥 푗 )푍 푖 푍 푗 ⎞ ⎟ ⎠ exp ⎛ ⎜ ⎝ 푖 ∑ 푖 푥 푖 푍 푖 ⎞ ⎟ ⎠ 퐻 ⊗푛 ⎤ ⎥ ⎦ 퐷 ,(1.16) where퐻 ⊗푛 denotes Hadamard gates on all qubits,푍 푖 is the Pauli-푍operator on qubit푖, and x= (푥 1 , ... , 푥 푛 )is the input vector. The two-qubit푍interactions create entanglement that depends on the input data, embedding classical features into quantum correlations. 16 The structure of the Z feature map is designed so that the resulting kernel involves multi- qubitcorrelationsthatarebelievedtobeclassicallyhardtocompute. Whilecomputingthekernel exactly incurs a cost that scales exponentially in the number of qubits classically, a quantum computer can estimate it in polynomial time using the overlap measurement circuit. Constructed Quantum Advantage Results The question of whether quantum kernels can provide a computational advantage over classical methods was rigorously addressed by Liu et al. [44] in a constructed problem setting. Their work constructs a learning problem whose labels are defined through thediscrete logarithm problem, an instance of the abelian hidden subgroup problem that Shor’s algorithm solves in polynomial time but for which no efficient classical algorithm is known. The key result is the existence of a data distribution풟such that: 1.A quantum kernel classifier can learn to classify samples from풟with high accuracy. 2.No polynomial-time classical learner can classify significantly better than random guess- ing, assuming the discrete logarithm problem is classically intractable; the barrier is com- putational rather than a lack of training data. This provides a rigorous separation between quantum and classical learning under the stated distributional and computational assumptions. Importantly, the separation is shown to survive the finite-sampling (shot) noise incurred when the kernel entries are estimated from a finite number of measurements, addressing earlier concerns about the fragility of quantum speedup claims. This establishes that a provable quantum advantage with quantum kernels is possible in principle. However, whether such an advantage extends to natural, real-world datasets remains an open question. 17 Relationship to Quantum Neural Networks A fundamental connection between quantum kernels and QNNs was established by Schuld and Killoran [45]. They showed that QNNs with fixed data encodings are equivalent to linear models in the quantum feature space. Specifically, for a QNN with output 푓 (x) = ⟨휙(x)|푂|휙(x)⟩ =Tr[휌(x) ⋅ 푂],(1.17) where휌(x) = |휙(x)⟩⟨휙(x)|is the density matrix of the embedded state and푂is the observable, the prediction islinearin휌(x). This places QNNs within the framework of kernel methods in a reproducing kernel Hilbert space (RKHS). The representer theorem implies that the optimal observable for minimizing the training error can be written as: 푂 ∗ = 푁 ∑ 푖=1 훼 푖 |휙(x 푖 )⟩⟨휙(x 푖 )|,(1.18) where the coefficients훼 푖 are determined by the training data. This yields an important insight: under the corresponding loss and regularization setting, the kernel formulation can search over a larger linear hypothesis class associated with the same encoding than a fixed parameterized observable family. However, this does not imply that kernel methods are universally superior. The larger hy- pothesis class of kernel methods may lead to poorergeneralization—the model may overfit the training data. QNNs, by constraining the observable to have a specific parameterized form, ef- fectively regularize the model. This trade-off between expressivity and generalization is a central theme in Chapter 3. 18 Data Re-uploading and Model Expressivity The equivalence between QNNs and kernel methods breaks down when data isre-uploaded multiple times during the circuit [46]. In adata re-uploadingcircuit, input dataxis encoded into multiple layers: 푈(x, 휃) = 푈 퐷 (휃 퐷 )푉 (x) ⋯ 푈 1 (휃 1 )푉 (x),(1.19) where푉 (x)encodes the data and푈 ℓ (휃 ℓ )are trainable layers. Pérez-Salinas et al. [46] showed that data re-uploading circuits can act as universal function approximators, expressing arbitrary continuous functions of the input. The relationship between re-uploading circuits and kernel methods was clarified by Jerbi et al. [47]. They proved that mapping a퐷-layer data re-uploading circuit to an equivalent ker- nel model requiresΩ(퐷)additional ancilla qubits. This result has important implications for the design of quantum machine learning models: while single-encoding QNNs are equivalent to kernel methods and their advantage must come from the kernel itself, re-uploading circuits can leverage their depth to access richer function classes. Understanding when and how this additional expressivity translates to practical advantages remains an active area of research. 1.3Summary This chapter has laid the foundations for the quantum machine learning investigations that follow in Part I. We began with the essential building blocks of quantum computation—qubits, quantum gates, and measurement—and reviewed the key developments in classical machine learning that motivatequantumextensions, fromkernelmethodsandsupportvectormachinestodeeplearning and the transformer revolution. A central organizing principle introduced in this chapter is theerror decomposition(Equa- 19 tion (1.6)), which partitions prediction error into three components: •Approximation error: determined by the expressivity of the hypothesis class. •Optimization error: the gap between the achieved and optimal training loss. •Generalization error: the discrepancy between training and test performance. This framework provides a unified lens through which to analyze and improve quantum machine learning models, and it guides the structure of the subsequent chapters. We traced the evolution of quantum machine learning from early HHL-based algorithms, whose speedup claims depend on strong input-access assumptions, to NISQ-era approaches based on parameterized circuits and quantum feature maps. Variational quantum algorithms such as VQE and QAOA use parameterized quantum circuits for optimization problems in quan- tum chemistry, many-body physics, and combinatorial optimization. The supervised-learning focus of this thesis uses related circuit technology in two main forms: 1.Quantum Neural Networks (QNNs): Supervised-learning models that combine data en- coding, trainable parameterized circuits, and measurement-based prediction. Key chal- lenges include the barren plateau phenomenon, in which the gradient variance vanishes exponentially in deep or highly expressive circuits. 2.Quantum Kernel Methods (QKMs): Fixed quantum feature maps combined with classi- cal kernel algorithms. Constructed QKM problems provide rigorous supervised-learning separations under explicit assumptions, while the kernel framework also clarifies when QNNs behave as linear models in quantum feature space. The relationship between these approaches reveals a trade-off between expressivity, opti- mization, and generalization: QKMs use a fixed quantum feature map with classical kernel 20 learning, while QNNs restrict the trainable observable or circuit family and can thereby impose useful inductive bias. Data re-uploading circuits bridge these paradigms by adding expressivity beyond the simplest fixed-kernel picture. Concepts introduced in Chapter 1 and where they are used later. ConceptLater use in the dissertation Quantum embedding and feature maps Trace-distance empirical-risk bounds and Neural Quantum Embedding in Chapter 2. QCNNBackground architecture for downstream NQE classifiers and QPR experiments. Barren plateaus and trainability Motivation for structured circuits and embedding-aware QNN design in Chapters 2 and 3. Quantum kernel methodsAlternative supervised-QML paradigm and kernel interpretation of embeddings in Chapter 2. Sequence models and self-attention Hardware-aware QEC decoding and the Mamba/Transformer comparison in Chapter 4; the attention-based ViT ansatz for neural quantum states in Chapter 5. Roadmap for Part I.The concepts introduced in this chapter set the stage for Part I of this thesis, which explores how quantum computing can enhance machine learning: •Chapter 2: Neural Quantum Embeddingaddresses the embedding-induced training- loss limit in quantum supervised learning. Rather than using fixed feature maps, we train neural networks to learn embeddings that increase the distinguishability of quantum states from different classes. •Chapter3: GeneralizationinQuantumMachineLearningaddressesthegeneralization 21 side by developing margin-based bounds for quantum neural networks. Margins provide a statistical-control lens for finite training data and connect the trace-distance representation view to state discrimination. Roadmap for Part I.Part I of this thesis reverses the direction, exploring how artificial in- telligence techniques can address challenges in quantum computing: •Chapter 4: Neural Decoders for Quantum Error Correctiondevelops neural network- based decoders for quantum error correcting codes and studies their accuracy–latency trade-off in real-time decoding benchmarks. •Chapter 5: Neural Quantum Statesexplores the use of neural networks to represent quantum many-body states and analyzes the optimizer that makes these representations practical. We recast stochastic reconfiguration as tangent-space ridge regression, identify the expressivity gap as finite-sample residual noise, and develop multi-shift stochastic reconfiguration as a richer spectral filter. 22 CHAPTER 2 NEURAL QUANTUM EMBEDDING The results in this chapter are based onNeural Quantum Embedding: Pushing the Lim- its of Quantum Supervised Learning[48] andNeural Quantum Embedding via De- terministic Quantum Computation with One Qubit[49]. Code is available athttps: //github.com/takh04/neural-quantum-embedding. In the previous chapter, we saw that the performance of quantum machine learning clas- sifiers depends critically on how classical data is encoded into quantum states. In this chap- ter, we formalize this intuition by showing that, for binary classification under the linear loss (ℓ = 1 2 (1 − 푦푓 (x))), the empirical risk is lower bounded by the trace distance between embedded data ensembles, a quantity determined by the choice of data embedding. This trace-distance limit is not the entire optimization problem, but it identifies a representation-induced floor on the training loss that no downstream classifier circuit can overcome. We then show that con- ventional deterministic embedding schemes—amplitude encoding, angle encoding, and the Z feature map—do not guarantee that this trace distance will be large for a given dataset. This mo- tivatesNeural Quantum Embedding(NQE), a data-driven approach that uses classical neural networks to learn an embedding that increases the distinguishability of quantum states repre- senting different classes. In practice, NQE is trained using fidelity and Hilbert–Schmidt losses, which serve as tractable surrogates for the trace distance. 23 2.1Theoretical Background 2.1.1Empirical Risk and Trace Distance We consider a quantum binary classification task. Given a labeled dataset 푆 = (x − 푖 , −1) 푁 − 푖=1 ∪ (x + 푖 , +1) 푁 + 푖=1 ,(2.1) with푁 = 푁 − + 푁 + samples, a quantum embedding circuit푉maps each classical inputx∈ ℝ 푚 to a quantum state: |x⟩ = 푉 (x)|0⟩ ⊗푛 .(2.2) A parameterized quantum circuit푈(휃)is then applied, followed by measurement of an observ- able푂, yielding a prediction function: 푓 (x; 휃) = ⟨x|푈 † (휃) 푂 푈(휃)|x⟩.(2.3) This prediction process can be recast as a quantum state discrimination problem [50]. Let 푂be a Hermitian observable with eigenvalues±1(equivalently,푂 2 = 퐼), which is the case for the Pauli observables used below. We define two positive operator-valued measure (POVM) elements parameterized by휃: 퐸 ± (휃) = 퐼 ± 푈 † (휃) 푂 푈(휃) 2 ,(2.4) which satisfy퐸 + (휃) + 퐸 − (휃) = 퐼and퐸 ± (휃) ≥ 0. The probability of obtaining outcome±1 given inputxis: 푃(퐸 ± (휃) |x) = ⟨x|퐸 ± (휃)|x⟩.(2.5) 24 The natural loss function for classification is then the misclassification probability: ℓ(푓 (x; 휃), 푦) = 푃(퐸 ̄푦 (휃) |x),(2.6) wherē푦denotes the complement of푦, i.e., if푦 = +1then̄푦 = −1and vice versa. Let푓 (x) = ⟨x|푈 † (휃) 푂 푈(휃)|x⟩ ∈ [−1, 1]be the prediction. The loss then takes the equivalentlinearform ℓ(푓 (x; 휃), 푦) = 1 2 (1 − 푦 푓 ( x)) .(2.7) Lower bound of empirical risk.The empirical risk over the training set푆can be written as: 퐿 푆 = 1 푁 ⎡ ⎢ ⎣ 푁 − ∑ 푖=1 푃(퐸 + (휃) |x − 푖 ) + 푁 + ∑ 푖=1 푃(퐸 − (휃) |x + 푖 ) ⎤ ⎥ ⎦ .(2.8) Minimizing퐿 푆 is therefore equivalent to minimizing the probability of misclassification in a quantum state discrimination problem, for which the minimum achievable error probability is known exactly: the Helstrom bound [50]. Applying this bound, the empirical risk is lower bounded by: 퐿 푆 ≥ 1 2 − 퐷 tr (푝 − 휌 − , 푝 + 휌 + ), (2.9) where 휌 ± = 1 푁 ± 푁 ± ∑ 푖=1 |x ± 푖 ⟩⟨x ± 푖 |(2.10) are the class-averaged density matrices (data ensembles),푝 ± = 푁 ± /푁are the class priors, and 퐷 tr (휌, 휎) = 1 2 ‖휌 − 휎‖ 1 denotes the trace distance. For balanced binary datasets, where푝 + = 푝 − = 1/2, this bound can equivalently be written as 퐿 푆 ≥ 1 2 (1 − 퐷 tr (휌 − , 휌 + )) .(2.11) 25 This unweighted form is the one used for the balanced experiments and figures below. This result has a direct implication: the best possible misclassification probability over all binary measurements is controlled by the trace distance between the two data ensembles푝 − 휌 − and푝 + 휌 + . This quantity is fixed once the embedded states have been prepared and is not im- proved by the subsequent parameterized unitary푈(휃). Thus, even an expressive trainable circuit cannot overcome poor class distinguishability created at the embedding stage. Furthermore, the minimum loss is achieved when the POVM퐸 − (휃), 퐸 + (휃)forms aHel- strom measurement—the measurement that optimally discriminates between the two quantum state ensembles. The training of a quantum neural network can therefore be viewed as a process of finding this optimal Helstrom measurement. Contractive property of trace distance.The importance of the embedding is further under- scored by the contractive property of the trace distance. For quantum channels, which are com- pletely positive and trace preserving (CPTP), trace distance is contractive. More generally, the same contractivity holds for positive trace-preserving (PTP) maps on Hermitian operators [51, 52]: 퐷 tr (Λ(휌), Λ(휎)) ≤ 퐷 tr (휌, 휎).(2.12) Subsequent quantum operations after the embedding, including the parameterized circuit푈(휃) and quantum noise channels, are described by CPTP maps. This is particularly relevant for NISQ devices, where hardware noise can further reduce the trace distance between quantum states. 2.1.2Limitations of Deterministic Embeddings We now examine commonly used deterministic quantum embedding schemes and explain why they do not guarantee a large trace distance for a given dataset. 26 Amplitude embeddingencodes a normalized classical vectorx∈ ℝ 2 푛 directly into the am- plitudes of an푛-qubit state: |x⟩ = 2 푛 −1 ∑ 푖=0 푥 푖 |푖⟩.(2.13) Thisismaximallyqubit-efficient(2 푛 featureswith푛qubits)butrequirescircuitdepth풪(2 푛 )[53]. Thefidelityreducestotheclassicalinnerproduct|⟨x|x ′ ⟩| 2 = |x⋅x ′ | 2 , sothequantumfeaturespace inherits the geometry of the input space without additional structure. Angle embeddingencodes each feature as a rotation angle on a dedicated qubit: |x⟩ = 푛 ⨂ 푗=1 푅 푃 (푥 푗 )|0⟩,(2.14) where푅 푃 (휃) = 푒 −푖휃푃/2 for푃 ∈ 푋, 푌 , 푍. This produces only product states with no entangle- ment. Z feature mapintroduces entanglement through two-qubit푍interactions: 푉 (휙(x)) = ⎡ ⎢ ⎣ exp ⎛ ⎜ ⎝ 푖 ∑ 푘 휙 푘 (x) 푍 푘 + 푖 ∑ 푘<푙 휙 푘,푙 (x) 푍 푘 푍 푙 ⎞ ⎟ ⎠ 퐻 ⊗푛 ⎤ ⎥ ⎦ 퐿 ,(2.15) where퐿 ≥ 1is the number of repetitions [43]. The standard choices휙 푘 (x) = 푥 푘 and휙 푘,푙 (x) = (휋 − 푥 푘 )(휋 − 푥 푙 )/2are made without data-dependent justification. Although Suzuki et al. [54] show that the choice of휙significantly impacts performance, no guidelines exist for selecting it for a given problem. The limitation of fixed embeddings.All three schemes aredata-agnostic: their structure is fixed independently of the training data, and there is no guarantee that they produce a large trace distance between data ensembles for a given classification task. A natural idea to address this 27 is to introduce trainable parameters within the quantum circuit. In atrainable unitary embed- ding[55], a parameterized unitary푈 tra (휃)is applied before the data encoding: |x; 휃⟩ = 푉 (x) 푈 tra (휃)|0⟩ ⊗푛 .(2.16) However, themaximumachievabletracedistanceisupperboundedbythediamonddistance[48]: max 휃 퐷 tr (푝 + 휌 + (휃), 푝 − 휌 − (휃)) ≤ 퐷 ⋄ (푝 + ℰ + , 푝 − ℰ − ) , (2.17) whereℰ ± are quantum channels with Kraus operators퐾 ± 푖 = 푉 (x ± 푖 )/ √ 푁 ± , determined entirely by the embedding circuit푉. The trainable unitary푈 tra (휃)does not improve this upper bound. A related restriction applies todata re-uploading[46], since such circuits can be transformed into a form where the embedding component is separated from the trainable parameters by in- troducing ancilla qubits [47]. Thus, trainable quantum layers alone do not remove the need for an embedding that separates the classes well. Experimental evidence.Figure 2.1 compares the trace distance achieved by the conventional Z feature map against a data-driven embedding on the balanced MNIST binary classification task (digits 0 vs. 1, 4 qubits); the embedding itself is developed in Section 2.2 and we defer its details until then. The Z feature map achieves a trace distance of only퐷 tr ≈ 0.27, which by Equation (2.11) implies a lower bound on the empirical risk of퐿 푆 ≥ 0.36. The data-driven embedding, by contrast, achieves퐷 tr ≈ 0.84, reducing the lower bound to퐿 푆 ≥ 0.08. The same pattern holds on Fashion-MNIST across different qubit counts (4, 8, 12), and becomes even more pronounced under hardware noise, where the contractive property further degrades already-low trace distances. These results motivate adata-drivenapproach that increases the trace distancebeforethe 28 5 V(g(x i ,w ⋆ )) | 0⟩ | 0⟩ | 0⟩ | 0⟩ argmin θ L(θ) QCNN (a) (b) (c) (d) (%) (%) FIG. 2. (a) Schematic representation of the quantum circuit used in the experiments. The green rectangle indicates the Neural Quantum Embedding (NQE), which transforms classical data푥 푖 into a quantum state|푥 푖 ⟩. The blue rectangles represent two-qubit parameterized quantum gates of the Quantum Convolutional Neural Network (QCNN), designed for binary classification tasks. (b) Plot depicting the evolution of the trace distance between two ensembles of quantum states embedded by the NQE models during training on the ibmqtoronto device, compared to the trace distance from conventional quantum embedding without NQE. (c) Noiseless QCNN simulation results. (d) The results from QCNN experiments conducted on IBM quantum devices. In (c) and (d), the blue solid, red dashed, and green dash-dotted lines represent the mean training loss histories for conventional Z feature embedding, PCA-NQE, and NQE, respectively. The shaded regions in the figure represent one standard deviation from the mean. These values are acquired from five repetitions of each QCNN training with random initialization of parameters. The thicker versions of these lines indicate the theoretical lower bounds for each method. This presents a significant limitation sinceE ± are entirely predetermined by the choice of quantum embedding circuit 푈 emb (·), without any guarantee that the diamond distance will be large. The trainable unitary푈 tra (휃)does not contribute to improving the upper bound of the maximum trace distance. Alternatively, one may consider the data re-uploading tech- nique in which the trainable unitary and quantum embedding circuit are repeatedly applied multiple times: Φ(푥;휃):|0⟩ ⊗푛 → |푥;휃⟩= 퐿 Ö 푙=1 푈 emb (푥)푈 tra (휃 푙 )|0⟩ ⊗푛 .(8) However, Ref [51] demonstrates that data-reuploading quan- tum embedding can be exactly transformed into a form where all the trainable unitaries follow the quantum embedding cir- cuits by introducing ancilla qubits. Upon such transformation, the embedding can be expressed as 푈 ′ tra (θ)푈 ′ emb (푥)|0⟩ ⊗푛+푛 ′ ,(9) whereθ=[휃 1 ,...,휃 퐿 ]and푛 ′ ∈푂(퐿log ( 퐿 ) ). Importantly, the embedding circuit is independent of the parameters of the trainable unitary gates. Consequently, the maximum trace distance is once again determined by the choice of quantum embedding circuit, which, in turn constrains the data distin- guishability. Furthermore, employing multiple layers of train- able unitary and quantum embedding circuit increases the total circuit depth significantly, making it not only unsuitable for NISQ applications but also susceptible to the barren plateaus problem. Figure 3 displays a comparison of the NQE and trainable unitary embedding under noiseless and noisy environments employing eight qubits. The PCA-NQE and NQE are imple- mented with adjustments made in both PCA and the classical neural network to accommodate the use of eight qubits (see Appendix A 1). The QCNN circuits are trained utilizing ei- ther trainable unitary embedding or NQE techniques. For the trainable unitary embedding we used following parameterized quantum circuit, 퐿 Ö 푙=1 " 푉(휙(푥))exp 푖 ∑︁ 푖 휃 푙 푖 푌 푖 +푖 ∑︁ 푖,푗 휃 푙 푖,푗 푌 푖 푌 푗 !# |0⟩ ⊗푛 .(10) Here,푉(휙(푥))is the Z feature map explained in Eq. (4) and 휃 푙 푖 (휃 푙 푖,푗 )are the trainable parameters of the embedding. Figure 2.1: Trace distance between class-averaged data ensembles퐷 tr (휌 − , 휌 + )for the conven- tional Z feature map and Neural Quantum Embedding on the balanced MNIST binary classifi- cation task (digits 0 vs. 1, 4 qubits). The blue dashed reference line indicates the trace distance obtained by the conventional Z feature map without NQE. Results obtained on IBM quantum hardware (ibmq_toronto). data become quantum states, thereby sidestepping the PTP contractive constraint. We develop this next as Neural Quantum Embedding. 2.2Neural Quantum Embeddings The preceding analysis reveals a bottleneck: the trace distance between data ensembles is fixed by the embedding circuit, and subsequent quantum processing cannot increase it. Neural Quan- tum Embedding (NQE) [48] addresses this limitation by operatingbeforethe quantum domain, using a classical neural network to learn a data-dependent preprocessing that increases the trace distance. 29 2.2.1The NQE Framework The core idea of NQE is to place a classical neural network푔 ∶ ℝ 푚 × ℝ 푟 → ℝ 푚 ′ in front of the quantum embedding, so that the data are mapped to the rotation angles that yield the most discriminative feature map. The NQE mapping is |x⟩ = 푉 (푔(x,w)) |0⟩ ⊗푛 ,(2.18) where푉is a fixed quantum embedding circuit (e.g., the Z feature map of Equation (2.15)), 푔is a classical neural network with trainable parametersw∈ ℝ 푟 , and푚 ′ ≤ 푚allows for dimensionality reduction. Because푔transformstheclassicalinputparametersbeforeanyquantumstateexists, NQEes- capes the contractive bound that constrains the trainable-unitary embedding of Equation (2.16). There, the trainable unitary acts on a fixed data-dependent channel ensemble and remains sub- ject to the diamond-distance bound of Equation (2.17). Here, the achievable trace distance is determined by the chosen feature-map family, the Hilbert-space dimension, the class priors, and the capacity, optimization, and training data used to fit푔. FortheZZfeaturemapspecifically, NQEsuppliestheguidancethatEquation(2.15)lacksfor choosing휙: the fixed angle functions휙 푘 and휙 푘,푙 are replaced by learned functions푔 푘 (x,w)and 푔 푘,푙 (x,w). The NQE framework is not tied to this choice. It applies to any embedding circuit푉, including amplitude and angle encoding. Whenever푔can represent the original preprocessing map, the fixed embedding is recovered as a special case, and any improvement beyond it reflects the added expressivity of the learned map. 30 4 V(g(x i ,w)) | 0⟩ | 0⟩ | 0⟩ | 0⟩ x g(x,w) V † (g(x j ,w)) w′ =argmin w ∑ ij [ | ⟨x i (w) | x j (w)⟩ | 2 − 1 2 ( 1+y i y j ) ] 2 Update parameters: w←w′ Post-process data ⋮ ⋮ Pr ( | 0⟩ ⊗n ) | x i (w)⟩=V(g(x i ,w)) | 0⟩ ⊗n FIG. 1. Overview of the NQE training. The unitary transformation that maps푥 푖 to the quantum feature space is determined by the output of a classical neural network denoted by푔(푥 푖 ,푤), where푤represents trainable parameters. The resulting quantum state is|푥 푖 (푤)⟩= 푉(푔(푥 푖 ,푤))|0⟩ ⊗푛 . The goal of the training is to produce mapping functions that can separate the two classes of data into two orthogonal subspaces. Efficient calculation of the fidelity between the two quantum states produced by the feature map is performed using a quantum computer. feature embedding, PCA-NQE, and NQE, respectively. The thicker versions of these lines indicate the theoretical lower bounds for each method. The mean values are calculated from five repetitions of each QCNN training with random initial- ization of parameters. The shaded regions in the figure illus- trate one standard deviation from the mean. The empirical risks for all models converge toward their respective theo- retical minima, affirming that the trained QCNN adequately approximates optimal Helstrom measurements. Classifica- tion accuracies achieved on the test dataset are shown in the table within the figure. NQE methods led to significant en- hancements, as demonstrated by reduced empirical risks and improved classification accuracies. Improvements in classi- fication accuracies are expected as NQE embeds the training data into a specific subspace within the given Hilbert space such that the state distinguishability is maximized. The lo- calization of embedded data facilitates the use of ML models with reduced complexity, thereby implying an enhancement in generalization capability. Numerical results supporting this intuition are presented in Sections I D, I E, and Appendix F. Note that in some instances, the training loss falls below the theoretical lower bound of empirical risk. This occurs be- cause the training loss is computed from a randomly sampled mini-batch of data in each iteration. Figure 2(d) presents the mean QCNN training loss histo- ries obtained using IBM quantum devices. The mean values are calculated from three independent trials on ibmq jakarta, ibmqperth, and ibmqtoronto devices with random initializa- tion of parameters. The shaded regions in the figure illus- trate one standard deviation from the mean. The presence of noise and imperfections in the quantum devices prevents the empirical risks from reaching their theoretical lower bounds. Nonetheless, the training performance is significantly im- proved by NQE. Notably, for both NQE methods, the empirical risk rapidly approaches or even falls below the theoretical limit of the conventional method. This result underscores that, even on the current noisy quantum devices, our method can outper- form the theoretical optimum of the Z feature map without any additional quantum error mitigation. Moreover, a sub- stantial improvement in classification accuracy was recorded, reaching 96% (PCA-NQE) and 90% (NQE), whereas the con- ventional method achieved only 52%. These findings demon- strate that NQE enhances the noise resilience of QML models, improving the utility of NISQ devices. Comprehensive details of these experiments are provided in Appendix A 2. 2. NQE versus Trainable Unitary Embedding We also conduct numerical comparisons between NQE and trainable unitary embedding to further investigate the advan- tage of NQE. Trainable unitary embedding utilizes trainable unitary푈 tra (휃)to find the quantum embedding that separates the data well. More specifically, one can implement a trainable unitary embeddingΦ(푥;휃)as Φ(푥;휃):|0⟩ ⊗푛 → |푥;휃⟩=푈 emb (푥)푈 tra (휃)|0⟩ ⊗푛 .(5) In this case, the data embedded ensembles are expressed as, 휌 ± (휃)= 1 푁 ± ∑︁ 푖 푈 emb (푥 ± 푖 )푈 tra (휃)|0⟩⟨0| ⊗푛 푈 † tra (휃)푈 † emb (푥 ± 푖 ) =E ± (|휓(휃)⟩⟨휓(휃)|),(6) where|휓(휃)⟩=푈 tra (휃)|0⟩ ⊗푛 andE ± (·)are quantum channels that maps휌→ Í 푖 퐾 ± 푖 휌퐾 ±† 푖 with퐾 ± 푖 =푈 emb (푥 ± 푖 )/ √ 푁 ± . Now the maximum trace distance between two data ensembles is upper bounded by the diamond distance, max 휃 퐷 tr (푝 + 휌 + (휃), 푝 − 휌 − (휃))= max 휃 ∥푝 + E + (|휓(휃)⟩⟨휓(휃)|)−푝 − E − (|휓(휃)⟩⟨휓(휃)|)∥ 1 ≤퐷 ⋄ (푝 + E + , 푝 − E − ).(7) Figure 2.2: Overview of the NQE training procedure. A classical neural network푔(x,w)trans- forms input data into rotation angles for the quantum embedding circuit푉. The resulting quan- tumstate|x⟩ = 푉 (푔(x,w)) |0⟩ ⊗푛 isusedtocomputetheimplicitfidelitylossviaanoverlapcircuit 푉 † (푔(x 푗 ,w)) 푉 (푔(x 푖 ,w)) |0⟩ ⊗푛 by measuring the probability of the all-zero outcome. The clas- sical parametersware updated to maximize the distinguishability between classes. 2.2.2Training via Implicit Fidelity Loss The ideal training objective for NQE would directly maximize퐷 tr (푝 − 휌 − , 푝 + 휌 + ),since by Equa- tion (2.9) this directly lowers the empirical risk bound. However, computing the trace distance requires full tomography of the class-averaged density matrices, which is computationally pro- hibitive. Instead, NQE employs animplicit fidelity lossderived from pairwise state fidelities. For a pair of training samples(x 푖 , 푦 푖 )and(x 푗 , 푦 푗 ), the implicit fidelity loss is defined as: ℓ fid (( x 푖 , 푦 푖 ), (x 푗 , 푦 푗 )) = [∣⟨x 푖 |x 푗 ⟩∣ 2 − 1 2 (1 + 푦 푖 푦 푗 )] 2 ,(2.19) where|⟨x 푖 |x 푗 ⟩| 2 is the fidelity between the embedded quantum states. The target value 1 2 (1+푦 푖 푦 푗 ) equals1when the labels agree (푦 푖 = 푦 푗 ) and0when they disagree (푦 푖 ≠ 푦 푗 ). The NQE training 31 objective is therefore: w ∗ =argmin w ∑ 푖,푗 ℓ fid (( x 푖 , 푦 푖 ), (x 푗 , 푦 푗 )) .(2.20) Train/test separation.The NQE preprocessing network is trained only from labeled pairs drawn from the training split. After this embedding is fixed, the downstream quantum clas- sifier is trained on the training split using the learned embedding. Held-out test data are used only for final evaluation and for post hoc diagnostics such as test-set trace distance. The connection between this pairwise fidelity loss and the trace distance operates through two complementary mechanisms: Same-class pairs (푦 푖 = 푦 푗 ).The loss drives the fidelity|⟨x 푖 |x 푗 ⟩| 2 → 1for pairs within the same class. This directly increases the purity of the class-averaged density matrices: Tr[(휌 ± ) 2 ] = 1 (푁 ± ) 2 푁 ± ∑ 푖,푗=1 ∣⟨x ± 푖 |x ± 푗 ⟩∣ 2 .(2.21) When Tr[(휌 ± ) 2 ] = 1, the density matrix휌 ± is a pure state, meaning all same-class data points are mapped to the same quantum state. Higher purity corresponds to tighter clustering within each class in the quantum feature space. Cross-class pairs (푦 푖 ≠ 푦 푗 ).The loss drives the fidelity|⟨x + 푖 |x − 푗 ⟩| 2 → 0for pairs from different classes. For a balanced dataset with paired examples, convexity of the trace distance gives 퐷 tr (휌 − , 휌 + ) ≤ 1 푁 푁 ∑ 푖=1 √ 1 − |⟨x + 푖 |x − 푖 ⟩| 2 ,(2.22) so reducing the cross-class fidelities relaxes this upper bound and, together with the same-class clustering that drives each ensemble toward a pure representative, makes a large trace distance 32 attainable. Together, these two effects push the embedded quantum states toward a configuration where the two classes are represented by nearly pure states that are nearly orthogonal—precisely the condition that maximizes the trace distance and thus minimizes the lower bound on empirical risk. The full derivation is given in Section A.1.1. Practical computation.The fidelity|⟨x 푖 |x 푗 ⟩| 2 between two embedded states can be measured on a quantum computer by preparing the overlap circuit 푉 † (푔(x 푗 ,w)) 푉 (푔(x 푖 ,w)) |0⟩ ⊗푛 (2.23) and measuring the probability of the all-zero outcome Pr(|0⟩ ⊗푛 ), which equals the fidelity. This pure-state fidelity can also be computed via the swap test [56]. In Section 2.3, we introduce a DQC1-compatible variant in which the training loss is written in terms of Hilbert–Schmidt inner products between feature-map unitaries. 2.2.3Experimental Results We now present experimental results demonstrating the effectiveness of NQE across multiple metrics. All experiments use binary classification of MNIST digits (0 vs. 1) with a 4-qubit QCNN (reviewed in Section 1.2.2), unless otherwise noted. We consider two NQE architectures that differ in how they handle high-dimensional inputs. InPCA-NQE, PCA first reduces the input to푛features, which are then passed through a fully connected neural network (two hidden layers, ReLUactivations)thatoutputs2푛rotationanglesfortheZZfeaturemap. InNQE(direct), a 2D convolutional neural network processes the full-resolution input (e.g.,28 × 28images) with max pooling layers, bypassing PCA to learn nonlinear feature extraction end-to-end. 33 5 V(g(x i ,w ⋆ )) | 0⟩ | 0⟩ | 0⟩ | 0⟩ argmin θ L(θ) QCNN (a) (b) (c) (d) (%) (%) FIG. 2. (a) Schematic representation of the quantum circuit used in the experiments. The green rectangle indicates the Neural Quantum Embedding (NQE), which transforms classical data퐿 퐿 into a quantum state|퐿 퐿 →. The blue rectangles represent two-qubit parameterized quantum gates of the Quantum Convolutional Neural Network (QCNN), designed for binary classification tasks. (b) Plot depicting the evolution of the trace distance between two ensembles of quantum states embedded by the NQE models during training on the ibmqtoronto device, compared to the trace distance from conventional quantum embedding without NQE. (c) Noiseless QCNN simulation results. (d) The results from QCNN experiments conducted on IBM quantum devices. In (c) and (d), the blue solid, red dashed, and green dash-dotted lines represent the mean training loss histories for conventional Z feature embedding, PCA-NQE, and NQE, respectively. The shaded regions in the figure represent one standard deviation from the mean. These values are acquired from five repetitions of each QCNN training with random initialization of parameters. The thicker versions of these lines indicate the theoretical lower bounds for each method. This presents a significant limitation sinceE ± are entirely predetermined by the choice of quantum embedding circuit 푀 emb (·), without any guarantee that the diamond distance will be large. The trainable unitary푀 tra (푁)does not contribute to improving the upper bound of the maximum trace distance. Alternatively, one may consider the data re-uploading tech- nique in which the trainable unitary and quantum embedding circuit are repeatedly applied multiple times: ω(퐿;푁):|0→ ↑푀 ↓|퐿;푁→= 푁 ! 푂=1 푀 emb (퐿)푀 tra (푁 푂 )|0→ ↑푀 .(8) However, Ref [51] demonstrates that data-reuploading quan- tum embedding can be exactly transformed into a form where all the trainable unitaries follow the quantum embedding cir- cuits by introducing ancilla qubits. Upon such transformation, the embedding can be expressed as 푀 ↔ tra (ω)푀 ↔ emb (퐿)|0→ ↑푀+푀 ↔ ,(9) whereω=[푁 1 ,...,푁 푁 ]and푂 ↔ ↗푃(푄log ( 푄 ) ). Importantly, the embedding circuit is independent of the parameters of the trainable unitary gates. Consequently, the maximum trace distance is once again determined by the choice of quantum embedding circuit, which, in turn constrains the data distin- guishability. Furthermore, employing multiple layers of train- able unitary and quantum embedding circuit increases the total circuit depth significantly, making it not only unsuitable for NISQ applications but also susceptible to the barren plateaus problem. Figure 3 displays a comparison of the NQE and trainable unitary embedding under noiseless and noisy environments employing eight qubits. The PCA-NQE and NQE are imple- mented with adjustments made in both PCA and the classical neural network to accommodate the use of eight qubits (see Appendix A 1). The QCNN circuits are trained utilizing ei- ther trainable unitary embedding or NQE techniques. For the trainable unitary embedding we used following parameterized quantum circuit, 푁 ! 푂=1 " 푅(푆(퐿))exp # 푇 $ 퐿 푁 푂 퐿 푈 퐿 +푇 $ 퐿,푃 푁 푂 퐿,푃 푈 퐿 푈 푃 %& |0→ ↑푀 .(10) Here,푅(푆(퐿))is the Z feature map explained in Eq. (4) and 푁 푂 퐿 (푁 푂 퐿,푃 )are the trainable parameters of the embedding. Figure 2.3: QCNN training loss histories for balanced 4-qubit MNIST binary classification (dig- its0vs.1). Bluesolid, reddashed, andgreendash-dottedlinesrepresenttheconventionalZZfea- ture map, PCA-NQE, and NQE, respectively. Thick lines indicate the theoretical lower bounds from Equation (2.11). Shaded regions represent one standard deviation over five (noiseless) or three (noisy) independent trials.Left:noiseless simulation.Right:IBM quantum hardware (ibmq_jakarta, ibmq_toronto, ibmq_perth). Table 2.1: Summary of NQE results for balanced 4-qubit MNIST binary classification (digits 0 vs. 1). Noiseless accuracy is from statevector simulation, while noisy accuracy is from the IBM quantum hardware runs. The lower bound on empirical risk is computed as퐿 푆 ≥ (1 − 퐷 tr )/2. Method퐷 tr Lower bound Noiseless acc. (%) Noisy acc. (%) Z (no NQE) 0.2730.36484.5 ± 8.052.7 ± 3.3 PCA-NQE 0.8400.08098.8 ± 0.096.1 ± 0.5 NQE 0.7920.104 99.9 ± 0.0 90.1 ± 4.5 Convergence to theoretical bounds.Figure 2.3(a) shows the noiseless QCNN training loss histories for the conventional Z feature map, PCA-NQE, and NQE. For all three embedding methods, the QCNN training loss approaches the respective theoretical lower bounds predicted by Equation (2.9), indicating that the trained QCNN approximates the corresponding Helstrom measurement in these experiments. The critical difference lies in the lower bounds themselves: NQE methods achieve substantially lower theoretical limits, translating into higher classification accuracy. Table 2.1 summarizes the key results: the conventional Z feature map achieves84.5± 8.0%noiseless accuracy, while PCA-NQE and NQE achieve98.8 ± 0.0%and99.9 ± 0.0%, respectively. 34 NQE vs. trainable unitary embedding.We compare NQE against trainable unitary embed- dings with퐿 = 1, 2, 3layers on both MNIST and Fashion-MNIST datasets using 8-qubit circuits (Figure 2.4). In both noiseless and noisy environments, NQE consistently achieves lower train- ing loss and higher classification accuracy than all trainable unitary variants. Trainable unitary embedding adds parameterized quantum gates that increase the overall circuit depth, making the model more susceptible to hardware noise and barren plateaus [39]. Under noisy simula- tion (IBM FakeGuadalupe), the advantage of NQE becomes even more pronounced: the train- able unitary embedding’s deeper circuits suffer greater noise-induced degradation, while NQE’s classical preprocessing adds no quantum circuit depth. Robustness on real quantum hardware.Figure 2.3(b) presents the QCNN training loss his- tories obtained on IBM quantum hardware (ibmq_jakarta, ibmq_toronto, and ibmq_perth). The contractive property (Equation (2.12)) provides a direct explanation for the dramatic difference in noise robustness between conventional and NQE-enhanced embeddings. Hardware noise con- stitutes a PTP map that can onlyreducetrace distance: •Small퐷 tr (Z≈ 0.27):Even modest noise pushes the already-close quantum states past the decision boundary, collapsing classification accuracy to52.7 ± 3.3%—barely above random guessing. •Large퐷 tr (PCA-NQE≈ 0.84):Noise reduces the trace distance, but the states remain well-separated and classification remains robust, achieving96.1 ± 0.5%(PCA-NQE) and 90.1 ± 4.5%(NQE) on real hardware. In these experiments, the noisy empirical risk achieved by NQE on real quantum hardware falls belowthe noiseless theoretical lower bound of the conventional Z feature map. Thus, NQE with real hardware noise achieves a lower empirical risk than the best possible Helstrom limit 35 6 (a) (b) (c) (d) Classification accuracy Without NQE47.1±1.2 PCA-NQE84.4±0.5 NQE50.8±0.3 Classification accuracy Without NQE67.4±3.7 PCA-NQE98.5±0.0 NQE99.2±0.4 Classification accuracy Without NQE68.7±2.3 PCA-NQE98.8±0.2 NQE99.5±0.41 Classification accuracy Layer=1100±0.0 Layer=2100±0.0 Layer=399.68±0.2 PCA-NQE98.1±0.1 NQE99.5±0.0 Classification accuracy Layer=1100±0.0 Layer=299.1±1.8 Layer=399.9±0.0 PCA-NQE93.02±0.2 NQE87.3±2.1 Classification accuracy Layer=156.9±5.6 Layer=256.1±1.2 Layer=366.6±3.1 PCA-NQE98.5±0.0 NQE99.2±0.4 Classification accuracy Layer=192.3±0.8 Layer=292.1±0.4 Layer=393.0±0.4 PCA-NQE94.6±0.2 NQE94.6±0.5 Classification accuracy Layer=163.2±1.7 PCA-NQE95.9±0.9 2 Classification accuracy Layer=166.9±4.1 PCA-NQE93.1±1.6 3 Classification accuracy Without NQE47.1±1.2 PCA-NQE84.4±0.5 NQE50.8±0.3 Classification accuracy Without NQE67.4±3.7 PCA-NQE98.5±0.0 NQE99.2±0.4 Classification accuracy Without NQE68.7±2.3 PCA-NQE98.8±0.2 NQE99.5±0.41 Classification accuracy Layer=1100±0.0 Layer=2100±0.0 Layer=399.68±0.2 PCA-NQE98.1±0.1 NQE99.5±0.0 Classification accuracy Layer=1100±0.0 Layer=299.1±1.8 Layer=399.9±0.0 PCA-NQE93.02±0.2 NQE87.3±2.1 Classification accuracy Layer=156.9±5.6 Layer=256.1±1.2 Layer=366.6±3.1 PCA-NQE98.5±0.0 NQE99.2±0.4 Classification accuracy Layer=192.3±0.8 Layer=292.1±0.4 Layer=393.0±0.4 PCA-NQE94.6±0.2 NQE94.6±0.5 Classification accuracy Layer=163.2±1.7 PCA-NQE95.9±0.9 2 Classification accuracy Without NQE47.1±1.2 PCA-NQE84.4±0.5 NQE50.8±0.3 Classification accuracy Without NQE67.4±3.7 PCA-NQE98.5±0.0 NQE99.2±0.4 Classification accuracy Without NQE68.7±2.3 PCA-NQE98.8±0.2 NQE99.5±0.41 Classification accuracy Layer=1100±0.0 Layer=2100±0.0 Layer=399.68±0.2 PCA-NQE98.1±0.1 NQE99.5±0.0 Classification accuracy Layer=1100±0.0 Layer=299.1±1.8 Layer=399.9±0.0 PCA-NQE93.02±0.2 NQE87.3±2.1 Classification accuracy Layer=197.0±0.9 Layer=297.0±1.1 Layer=392.5±4.6 PCA-NQE98.5±0.0 NQE99.2±0.4 Classification accuracy Layer=192.3±0.8 Layer=292.1±0.4 Layer=393.0±0.4 PCA-NQE94.6±0.2 NQE94.6±0.5 Classification accuracy Layer=163.2±1.7 PCA-NQE95.9±0.9 2 FIG. 3. Comparative analysis between Neural Quantum Embeddings and Trainable Unitary Embeddings with one, two and three trainable layers. The numerical simulations were conducted under noiseless (left) and noisy (right) environments, utilizing MNIST (top) and Fashion- MNIST (bottom) datasets. For noiseless simulations, we used 1000 iterations, a learning rate of 0.01 learning rate, and batches of 128 data points per iteration. For noisy simulations, we used 200 iterations, a learning rate of 0.05, and batches of 15 data point per iteration. The noisy model simulations utilized the IBM Qiskit FakeGuadalupe environment. The classification accuracies were evaluated using a sample size of 2115 and 2000 data points for the MNIST and Fashion-MNIST datasets, respectively. The mean and one standard deviation from five independent iterations are shown for the loss history. Figure 3 (a) and (c) depict the training loss history and classification accuracies in a noiseless environment for the MNIST (0,1) and Fashion-MNIST (T-shirt/Top,Trouser) datasets [52], respectively. In this experiment, we explored the impact of one, two, and three trainable unitary layers in comparison to PCA-NQE and NQE. The results indicate that the NQE methods achieves a notably lower training loss than the trainable unitary embedding, suggesting superior e!cacy of NQE in improving data separability (by increasing the trace distance). Additionally, the NQE methods resulted in higher classification accuracies. Figure 3, panels (b) and (d), depict the training loss history and classification accuracies in a noisy environment for the MNIST and Fashion-MNIST datasets, respectively. A single- layer trainable unitary embedding was evaluated against PCA- NQE. The choice of a single layer was based on its minimal circuit depth, rendering it less susceptible to noise interference. The results indicate NQE are more e”ective in enhancing data separability under presence of noise. Additionally, PCA-NQE yields considerably higher classification accuracies. These advantages stem from the larger initial trace distance and the reduced circuit depths associated with PCA-NQE. We em- ployed a noisy simulation using the IBM FakeGuadalupe de- vice. This simulator mimics the essential characteristics of the ibmqguadalupe device, including its basis gates, qubit con- nectivity, qubit relaxation (T 1 ) and rephrasing (T 2 ) times, and readout error rates. D. E!ective Dimension As demonstrated thus far, NQE has proven to be an e!cient method for reducing the lower bound of empirical risk. While this advancement enhances the ability to learn from data, an- other essential measure of successful machine learning is the generalization performance, that is, the ability to make ac- curate predictions on unseen data based on what has been learned. The simulation and experimental results presented in the previous section indicate an improvement in prediction accuracy on test data with the implementation of NQE. In this section, we provide additional evidence of improved gen- eralization performance by analyzing the e”ective dimension (ED) [15, 53, 54] of a QML model constructed with and with- out NQE. Intuitively, ED quantifies the number of parameters that active in the sense that they influence the outcome of its statistical model. To investigate this further, we focus on the local e”ective di- mension (LED) introduced in Ref. [54], as it takes into account the data and learning algorithm dependencies and is compu- tationally more convenient. Importantly, the LED exhibits a Figure 2.4: Comparison of NQE against trainable unitary embeddings with one, two, and three trainable layers, using 8-qubit circuits on MNIST (top row) and Fashion-MNIST (bottom row) under noiseless (left column) and noisy (right column) simulation. The noisy simulations use the IBM Qiskit FakeGuadalupe environment. Noiseless runs use1000iterations, learning rate 0.01, and batches of128samples per iteration. Noisy runs use200iterations, learning rate0.05, and batches of15samples per iteration. Classification accuracies are evaluated on held-out sets of2115(MNIST) and2000(Fashion-MNIST) samples, and the loss histories show the mean and one standard deviation over five independent trials. associated with the unoptimized Z feature map in a noiseless setting. This result, summarized in Table 2.1, shows that improving the embedding can matter more than reducing moderate hardware noise for this task. Beyond these training-loss improvements, NQE also affects two properties that are not vis- ible in the training loss alone: thegeneralizationof the trained model to unseen data and its trainability. We examine both in the following subsection. 36 2.2.4Generalization, Expressibility, and Trainability The trace-distance analysis of Section 2.1.1 concerns only thetrainingloss. A low training loss does not by itself guarantee good performance on unseen inputs, nor does it indicate how readily the model can be trained. In this subsection we present numerical evidence that NQE improves two further,distinctproperties of the resulting models. The first isgeneralization—the ability to predict well on unseen data—which we quantify through the local effective dimension of the quantum neural network and the weight-norm complexity of the quantum kernel model. The second istrainability, which is governed by theexpressibilityof the embedding and which we probe through the deviation from a unitary2-design and the variance of the quantum kernel entries. Here, a high expressibility is detrimental, as it induces barren plateaus and kernel con- centration that impede optimization. These two axes are addressed separately below. Unless noted otherwise, the experiments use the same PCA-NQE and NQE networks introduced above (optimized onibmq_toronto; see Section 2.2.3) and binary MNIST (digits 0 vs. 1). Effective dimension of the quantum neural network.Theeffective dimensionmeasures how many parameters of a model are “active” in the sense that they meaningfully influence its out- put, and it acts as a capacity measure that upper bounds the generalization error of a statistical model [57]. We use thelocal effective dimension(LED), which is better matched to a specific learned model than the global effective dimension. The LED is positively correlated with the generalization error, so a smaller LED indicates better expected generalization. Figure 2.5(a) reports the LED of a four-qubit QNN as a function of the number of data, averaged over200 instances (10artificial datasets× 20random parameter initializations), with and without NQE. Acrosstheentirerangeofdatasetsizes, theNQE-enhanced modelhasasubstantiallylowereffec- tive dimension (rising from≈ 10.4to≈ 13.5) than the conventional embedding (which plateaus 37 near≈ 18.5), and the reduction holds in all200instances. Because the effective dimension can also be read as the volume of solution space the model class occupies, the smaller value under NQE indicates a more constrained hypothesis class, consistent with the improved test accuracy reported above. Generalization in the quantum kernel method.The benefit of NQE is not confined to quan- tum neural networks; it also tightens a generalization bound for thequantum kernel method (QKM). Given the embedding, the quantum kernel is푘 푄 (x 푖 ,x 푗 ) = |⟨x 푖 |x 푗 ⟩| 2 , and a kernel model predicts푓 (x; 푊 ) =Tr[푊 |x⟩⟨x|], with푊obtained by minimizing a regularized least-squares cost, 푊 ∗ =argmin 푊 ∈ℂ 2 푛 ×2 푛 1 푁 푁 ∑ 푖=1 (푓 (x 푖 ; 푊 ) − ℎ(x 푖 )) 2 + 휆‖푊 ‖ 2 퐹 ,(2.24) whereℎis the target function,‖ ⋅ ‖ 퐹 is the Frobenius norm, and휆is a regularization weight that trades training error for generalization. For this estimator, the generalization gap obeys [58] ∣푅(푊 ) − 푅 푁 (푊 )∣ ≤ 풪 ⎛ ⎜ ⎝ ‖푊 ∗ ‖ 퐹 √ 푁 + √ log(1/훿) 푁 ⎞ ⎟ ⎠ (2.25) with probability at least1 − 훿, where푅and푅 푁 denote the true and empirical risks. The data- dependent term is governed by the weight-norm complexity퐺 = ‖푊 ∗ ‖ 퐹 / √ 푁. Because NQE changes both the kernel matrix퐾 푄 and the embedded states|x⟩, it directly affects퐺. Fig- ure 2.5(b) plots퐺for binary MNIST (푁 = 1000) across regularization weights휆 ∈ [0.1, 0.9]. Both PCA-NQE and NQE yield a markedly smaller퐺than the conventional embedding at ev- ery휆—that is, a tighter upper bound on the generalization error—with PCA-NQE giving the smallest values. 38 7 FIG. 4. The local e!ective dimension for the circuit with (solid green) and without (dashed blue) NQE. These results are based on ten sets of experiments, each on a distinct artificial dataset with 20 repetitions with random initialization of parameters. The reported values represent the average across all 200 experiments. positive correlation with the generalization error, allowing for straightforward interpretation of the results: a smaller LED corresponds to a smaller generalization error, and vice versa. Our numerical investigation employs a four-qubit QNN (see Appendix A 3), and the results are presented in Fig. 4. In the figure, the green solid and purple dashed lines represent the mean local e!ective dimension of the tested QML model with and without NQE, respectively. The mean values are com- puted from 200 trials, where each trial consists of ten artificial datasets, and for each dataset, the experiment is repeated 20 times with random initialization of parameters. The shaded areas in the figure represent one standard deviation. The sim- ulation results unequivocally demonstrate that NQE consis- tently reduces the LED in all 200 instances tested and across a wide range of training data sizes, signifying an improvement in generalization performance. The e!ective dimension can also be interpreted as a mea- sure of the volume of the solution space that a specific model class can encompass. A smaller e!ective dimension in a QML model implies a reduced volume of the solution space. This observation is particularly relevant as it suggests that mod- els with smaller e!ective dimensions are less prone to en- countering barren plateaus [55]. Consequently, the simulation results further indicate that NQE not only enhances general- ization performance but also improves the trainability of the model. Further evidence substantiating this improvement for both QNN and QKM will be presented in a later section. E. Generalization in Quantum Kernel Method Up to this point, the investigation of NQE has primarily fo- cused on its application within the context of quantum neural networks. In this section, we extend the analysis to demon- strate that NQE also enhances the performance of the quantum kernel method. Given a quantum embedding, the kernel func- tion can be defined as 퐿 퐿 (푀 푀 ,푀 푁 )= ! ! " 푀 푀 ! ! 푀 푁 # ! ! 2 ,(11) which can be computed e”ciently on a quantum computer. The quantum kernel method refers to an approach that uses the kernel matrix푁 퐿 , of which each entry is the kernel of the corresponding data points, in a method like a classical support vector machine [17, 32]. The potential quantum advantage of such approach is based on the hardness to compute certain quantum kernel functions classically [17, 18, 56]. The quantum kernel method attempts to determine the func- tion푂(푀;푃)=Tr(푃|푀→↑푀|)to predict the true underlying function푄(푀)for unseen data푀. The optimal parameters푃 ↓ are obtained by minimizing the cost function 푃 ↓ =arg min 푂↔C 2 퐿 ↗2 퐿 1 푅 푃 $ 푀=1 ( 푂(푀 푀 ;푃)↘푄(푀 푀 ) ) 2 ↘푆||푃|| 2 F ,(12) where||·|| F is the Frobenius norm. The second term is the regularization term with a hyperparameter푆. The purpose of including the regularization term is to reduce the generaliza- tion error at the expense of the training error. Specifically, considering the true error푇(푃)=E 푄 |푂(푀;푃)↘푄(푀)|, and the training error푇 푃 (푃)= % 푃 푀=1 |푂(푀 푀 ;푃)↘푄(푀 푀 )|/푅, the generalization error is upper-bounded as | 푇(푃)↘푇 푃 (푃) | ≃O & ||푃|| F ⇐ 푅 + ' log ( 1/푈 ) 푅 ( ,(13) with probability at least 1↘푈(see Ref [32] Supplementary Information Section 4.C). The optimal푃 ↓ can be expressed as푃 ↓ = % 푃 푀=1 % 푃 푁=1 푄(푀 푀 )(푁 퐿 +푆푉) ↘1 푀,푁 |푀 푁 →↑푀 푁 |. Here, the em- ployment of NQE a!ects푃 ↓ as both푁 퐿 and|푀 푁 →vary with the quantum embedding. We performed empirical evaluations to assess the e!ective- ness of NQE in reducing푊=||푃 ↓ || F / ⇐ 푅and thereby improv- ing the upper bound of the generalization error. The analysis proceeded in three steps: loading the dataset, computing the quantum kernel matrix with and without NQE, and calculat- ing the generalization error bound with and without NQE. The experiments tested both PCA-NQE and NQE, with neural net- work parameters optimized on the ibmqtoronto processor, as detailed in Appendix A 1. During the dataset loading phase, binary datasets containing푅=1,000 samples from classes 0,1were constructed from the MNIST dataset. As outlined in Appendix A 1, both PCA-NQE and the conventional quan- tum embedding without NQE were preceded by PCA to reduce the number of features to four. In contrast, the NQE utilized the original 28x28 image datasets. Subsequently, three quan- tum kernel matrices were constructed: one without NQE, one with PCA-NQE, and one with NQE. The(푋,푌)th entry of the quantum kernel corresponds to퐿 퐿 (푀 푀 ,푀 푁 ), the fidelity overlap between pairs of data-embedded quantum states. This fidelity overlap was computed using Pennylane [57] numerical simu- lation. Finally, for each of the quantum kernel matrices, the corresponding upper bound of the generalization error푊= 8 FIG. 5. A comparative analysis of the generalization error bound 퐿with varying regularization weights푀. This plot illustrates the performance enhancement – lower generalization error bound – when employing NQE (green triangles) and PCA-NQE (red circles) over conventional methods without NQE (blue squares) in quantum kernel methods. PCA-NQE and NQE were optimized on the ibmqtoronto quantum hardware. The error bound퐿was determined based on five independent numerical simulations, presenting both the mean and one standard deviation of퐿. ! ||푁 → || 2 F /푂was calculated. Here,||푁 → || 2 F can be expressed as||푁 → || 2 F = " 퐿 " 푀 # (푃 푁 +푀푄) ↑1 푃 푁 (푃 푁 +푀푄) ↑1 $ 퐿,푀 푅 퐿 푅 푀 . The experimental procedure was repeated five times, each time with di!erent sets of 1,000 samples of data. Figure 5 presents the mean and one standard deviation of the gener- alization error bound퐿. These values were obtained under the Z feature embedding, both with and without NQE, and were examined across various regularization parameters푀. The results clearly illustrate that NQE significantly decreases the upper bound of the generalization error in quantum kernel methods. F. Expressibility and Trainability In both QNN and QKM, there exists a trade-o!between ex- pressibility and trainability. In the QNN framework, highly ex- pressive quantum circuits often leads to barren plateaus, char- acterized by exponentially vanishing gradients, which severely hinders the trainability of the model [27, 55]. In the QKM framework, highly expressive embedding induces a quantum kernel matrix whose elements exhibit an exponential concen- tration [21]. Specifically, the concentration of quantum kernel element푃 푁 퐿,푀 =푆 푁 (푇 퐿 ,푇 푀 )can be expressed by Chebyshev’s inequality, Pr % & & & 푃 푁 퐿,푀 ↑E % 푃 푁 퐿,푀 ' & & & ↓푈 ' ↔ Var % 푃 푁 퐿,푀 ' 푈 2 (14) for any푈>0. The quantum kernel element푃 푁 퐿,푀 arising from highly expressive quantum embedding displays an ex- ponentially reduction in variance as the number of qubits increases. Consequently, an exponentially large number of quantum circuit executions is necessary to accurately approx- (a)(b) FIG. 6. (a) A comparative analysis of expressibility with and without NQE models. The deviation from unitary 2-design is depicted, where a smaller deviation indicates higher expressibility. The deviation is derived from 12,665 (2,115) MNIST training (test) data results. (b) A comparative analysis of the variance of quantum kernel elements with and without NQE models. The variance was computed from the o!-diagonal elements of the quantum kernel matrix푃 푁 , constructed from 1,000 samples of MNIST datasets. The mean and one standard deviation from five independent iterations are shown. For both (a) and (b), NQE models are optimized using the ibmqtoronto quantum device. imate the quantum kernel matrix푃 푁 . This poses a significant challenge in the e”cient implementation of QKM. NQE addresses this challenge by constraining quantum em- bedding to ensure large distinguishability. NQE strategi- cally limits the expressibility of the embedded quantum states, thereby enhancing the trainability of QML models. This im- provement is achieved by exploiting the prior knowledge that a quantum embedding with large distinguishability can e!ec- tively approximate the true underlying function. Figure 6(a) illustrates how the expressibility varies as we apply NQE models. Here, we investigated the Hilbert-Schmidt norm of the deviation from unitary 2-design [55, 58], as a measure of expressibility. More specifically, the deviation is given as, 푉= ( Haar (|푊↗↘푊|) ≃2 푋푊↑ ( E (|푌↗↘푌|) ≃2 푋푌,(15) where the first integral is taken over the Haar measure, and the second integral is taken over ensemble of data embed- ded quantum states,E. We then define the deviation norm, 푍= ) Tr(푉 † 푉), where a small푍indicates a highly expres- sive quantum embedding and vice versa. In this experiment, we employed NQE and PCA-NQE, which had been previ- ously optimized with ibmqtoronto hardware as detailed in Section I C 1. The value of푍was numerically computed for ensembles of training and test datasets of classes0,1from MNIST data, utilizing 12,665 instances for training data and 2,115 for test data. We observe that for both training and test data, both NQE methods lead to reduction in the expressibility, consequently enhancing trainability. Figure 6(b) demonstrates how the variance of the quantum kernel elements varies as we apply NQE models. To determine the variance, we initially computed quantum kernel matrix 푃 푁 using PCA-NQE, NQE, and quantum embedding with- out NQE, following the procedures outlined in Section I E. (a)(b) Figure 2.5: Generalization diagnostics for embeddings with and without NQE.(a)Local effec- tive dimension of a four-qubit QNN with (solid green) and without (dashed purple) NQE, as a function of the number of data. NQE yields a smaller effective model complexity, and hence a tighter generalization bound, across all dataset sizes (averaged over200experiments—10artifi- cial datasets, each with20random parameter initializations. Shaded regions denote one standard deviation).(b)Weight-norm complexity퐺 = ‖푊 ∗ ‖ 퐹 / √ 푁, which controls the data-dependent term of the quantum-kernel generalization bound Equation (2.25), as a function of the regular- ization weight휆. PCA-NQE (red circles) and NQE (green triangles) both lower퐺relative to the conventional Z feature map without NQE (blue squares) at every휆(mean and one standard deviation over five independent draws of1000MNIST samples). In both panels, lower values correspond to better expected generalization. Expressibility and trainability.Both the QNN and QKM frameworks face a trade-off be- tween expressibility and trainability: highly expressive circuits tend to exhibit barren plateaus— exponentially vanishing gradients that obstruct training [39, 42]—and, in the kernel setting, they induce an exponential concentration of the kernel entries, so that exponentially many measure- ments are needed to resolve퐾 푄 [59]. NQE mitigates both effects by deliberatelylimitingthe expressibility of the embedding, exploiting the prior that an embedding with large class distin- guishability already suffices to approximate the target function. We quantify expressibility in two complementary ways. First, Figure 2.6(a) reports the 39 Hilbert–Schmidt norm휖 = √ Tr(퐴 † 퐴)of the deviation from a unitary2-design, 퐴 = ∫ Haar (|휓⟩⟨휓|) ⊗2 푑휓 − ∫ ℰ (|휙⟩⟨휙|) ⊗2 푑휙,(2.26) where the first integral is over the Haar measure and the second is over the ensembleℰof data- embedded states, evaluated on12,665training and2,115test MNIST samples. A small휖marks a highly expressive embedding (close to a2-design). Both NQE variants have a substantially larger deviation (휖 ≈ 0.37) than the conventional embedding (휖 ≈ 0.13), on both training and test data, confirming that NQE produces alessexpressive embedding. Second, Figure 2.6(b) shows the variance of the off-diagonal quantum-kernel entries (from1000MNIST samples): the NQE variants exhibit far larger kernel variance (≈ 0.19for PCA-NQE,≈ 0.13for NQE) than the conventional embedding (≈ 0.02). A larger variance means the kernel entries arenot exponentially concentrated, so the kernel matrix can be estimated reliably with far fewer circuit executions. Together, the two panels show that NQE trades a controlled reduction in expressibil- ity for improved trainability in both frameworks. These diagnostics indicate that NQE improves not only the achievable training loss but also the generalization and trainability of the resulting models. 2.3Optimization via DQC1 The NQE training procedure described in Section 2.2.2 relies on measuring the fidelity|⟨x 푖 |x 푗 ⟩| 2 between pairs of embedded quantum states, which requires preparing all푛qubits in the pure state |0⟩ ⊗푛 . While this is straightforward on gate-based quantum computers such as superconducting and trapped-ion platforms, it is not naturally matched toensemble quantum systems—most notably nuclear magnetic resonance (NMR) quantum processors [60]—where the thermal equi- librium state is highly mixed and preparing a global pure state across all qubits is impractical. In 40 8 FIG. 5. A comparative analysis of the generalization error bound 퐿with varying regularization weights푀. This plot illustrates the performance enhancement – lower generalization error bound – when employing NQE (green triangles) and PCA-NQE (red circles) over conventional methods without NQE (blue squares) in quantum kernel methods. PCA-NQE and NQE were optimized on the ibmqtoronto quantum hardware. The error bound퐿was determined based on five independent numerical simulations, presenting both the mean and one standard deviation of퐿. ! ||푁 → || 2 F /푂was calculated. Here,||푁 → || 2 F can be expressed as||푁 → || 2 F = " 퐿 " 푀 # (푃 푁 +푀푄) ↑1 푃 푁 (푃 푁 +푀푄) ↑1 $ 퐿,푀 푅 퐿 푅 푀 . The experimental procedure was repeated five times, each time with di!erent sets of 1,000 samples of data. Figure 5 presents the mean and one standard deviation of the gener- alization error bound퐿. These values were obtained under the Z feature embedding, both with and without NQE, and were examined across various regularization parameters푀. The results clearly illustrate that NQE significantly decreases the upper bound of the generalization error in quantum kernel methods. F. Expressibility and Trainability In both QNN and QKM, there exists a trade-o!between ex- pressibility and trainability. In the QNN framework, highly ex- pressive quantum circuits often leads to barren plateaus, char- acterized by exponentially vanishing gradients, which severely hinders the trainability of the model [27, 55]. In the QKM framework, highly expressive embedding induces a quantum kernel matrix whose elements exhibit an exponential concen- tration [21]. Specifically, the concentration of quantum kernel element푃 푁 퐿,푀 =푆 푁 (푇 퐿 ,푇 푀 )can be expressed by Chebyshev’s inequality, Pr % & & & 푃 푁 퐿,푀 ↑E % 푃 푁 퐿,푀 ' & & & ↓푈 ' ↔ Var % 푃 푁 퐿,푀 ' 푈 2 (14) for any푈>0. The quantum kernel element푃 푁 퐿,푀 arising from highly expressive quantum embedding displays an ex- ponentially reduction in variance as the number of qubits increases. Consequently, an exponentially large number of quantum circuit executions is necessary to accurately approx- (a)(b) FIG. 6. (a) A comparative analysis of expressibility with and without NQE models. The deviation from unitary 2-design is depicted, where a smaller deviation indicates higher expressibility. The deviation is derived from 12,665 (2,115) MNIST training (test) data results. (b) A comparative analysis of the variance of quantum kernel elements with and without NQE models. The variance was computed from the o!-diagonal elements of the quantum kernel matrix푃 푁 , constructed from 1,000 samples of MNIST datasets. The mean and one standard deviation from five independent iterations are shown. For both (a) and (b), NQE models are optimized using the ibmqtoronto quantum device. imate the quantum kernel matrix푃 푁 . This poses a significant challenge in the e”cient implementation of QKM. NQE addresses this challenge by constraining quantum em- bedding to ensure large distinguishability. NQE strategi- cally limits the expressibility of the embedded quantum states, thereby enhancing the trainability of QML models. This im- provement is achieved by exploiting the prior knowledge that a quantum embedding with large distinguishability can e!ec- tively approximate the true underlying function. Figure 6(a) illustrates how the expressibility varies as we apply NQE models. Here, we investigated the Hilbert-Schmidt norm of the deviation from unitary 2-design [55, 58], as a measure of expressibility. More specifically, the deviation is given as, 푉= ( Haar (|푊↗↘푊|) ≃2 푋푊↑ ( E (|푌↗↘푌|) ≃2 푋푌,(15) where the first integral is taken over the Haar measure, and the second integral is taken over ensemble of data embed- ded quantum states,E. We then define the deviation norm, 푍= ) Tr(푉 † 푉), where a small푍indicates a highly expres- sive quantum embedding and vice versa. In this experiment, we employed NQE and PCA-NQE, which had been previ- ously optimized with ibmqtoronto hardware as detailed in Section I C 1. The value of푍was numerically computed for ensembles of training and test datasets of classes0,1from MNIST data, utilizing 12,665 instances for training data and 2,115 for test data. We observe that for both training and test data, both NQE methods lead to reduction in the expressibility, consequently enhancing trainability. Figure 6(b) demonstrates how the variance of the quantum kernel elements varies as we apply NQE models. To determine the variance, we initially computed quantum kernel matrix 푃 푁 using PCA-NQE, NQE, and quantum embedding with- out NQE, following the procedures outlined in Section I E. Figure 2.6: Expressibility and trainability diagnostics for embeddings with and without NQE. (a) Deviation from a unitary2-design,휖 = √ Tr(퐴 † 퐴)from Equation (2.26), on training (filled) and test (open) MNIST data. A larger deviation indicateslowerexpressibility. Both NQE variants are markedly less expressive than the conventional embedding. (b) Variance of the off-diagonal quantum-kernel elements, computed from1000MNIST samples (mean and one standard deviation over five iterations). The larger variance under NQE indicates that the ker- nel entries are not exponentially concentrated, improving the trainability of the quantum kernel method. this section, we show that NQE can be reformulated using theHilbert–Schmidt inner product in place of fidelity, and that this quantity can be computed efficiently using thedeterministic quantum computation with one qubit(DQC1) model [61]. This reformulation extends NQE to ensemble quantum platforms and demonstrates its experimental viability on an NMR quantum processor. 2.3.1The DQC1 Model The DQC1 model, introduced by Knill and Laflamme [61], is a restricted model of quantum computation that uses one probe qubit with nonzero polarization alongside푛qubits in the max- 41 2 V(g(x i ,w)) V † (g(x j ,w)) H I ⊗n 2 n ⋮ ⋮ I+ασ z 2 x g(x,w) Update w FIG. 1. Schematic of the neural quantum embedding via DQC1, illustrating the process for identifying the optimal quantum feature map for classification tasks. Data points with different labels are represented as squares and triangles. Protocol overview—Let the class label of the input data x 푖 be푦 푖 ∈ +1,−1. The quantum data embedding circuit must be constructed to maximize the fidelity between quan- tum states corresponding to input data points with the same label (푦 푖 =푦 푗 ) and minimize it for those with different labels. This can be formulated as maximizing the distinguishability between휌 ± = Í 푚 ± 푖 |x 푖 ⟩⟨x i |/푚 ± , which denotes the quantum state representation of data samples labeled±1 [27]. Here,푚 ± indicates the number of data points in class±1. For quantum data embedding, this is equivalent to finding a quantum feature map that maximizes the trace distance,∥휌 + −휌 − ∥ 1 /2 [14]. Our method utilizes a classical neural network to optimize quantum data embedding according to these criteria. The objective function subject to minimization is 퐿 NQE = ∑︁ 푖푗 1 2 푛 Tr 푉(푔 x 푖 )푉 † (푔 x 푗 ) − 1+푦 푖 푦 푗 2 2 ,(1) where푔 x 푖 =푔(x 푖 ,w)is defined as the output of the neural network,wis the set of trainable parameters, and푉is the quantum feature map that encodes푉(푔 x )|휓⟩=|x⟩. Then |x⟩is provided as an input to a QML algorithm, such as the quantum neural network [7, 28–34] or the quantum kernel method [6, 35, 36], to construct a classifier. The first term in Eq. (B1) is the HS inner product between two푛-qubit unitary operators, which quantifies the similarity between two data points. Finding the solution to minimizing Eq. (B1) corresponds to learningwsuch that the HS inner product is large when the two data points belong to the same class and small otherwise. It is also related to the distance defined by the Frobenius norm,∥퐴∥ 퐹 = √︁ Tr(퐴 † 퐴), since ∥푉(푔 x 푖 )−푉(푔 x 푗 )∥ 2 퐹 =2−2Re h Tr 푉(푔 x 푖 )푉(푔 x 푗 ) † i . Thus, the task of minimizing Eq. (B1) reduces to efficiently computing the HS inner product within the objective function. As previously noted, DQC1 provides an efficient means for estimating this metric [20]. Additionally,푉can be optimized by employing a classical neural network and deep learning techniques. This approach overcomes the limitation imposed by completely positive and trace-preserving (CPTP) maps, which cannot increase the trace distance between quantum states. Specifically, any subsequent CPTP mapΛsatisfies the inequality∥Λ(휌 + ) −Λ(휌 − )∥ 1 ≤ ∥휌 + −휌 − ∥ 1 [37]. Further details are provided in the Supplementary Material [38]. The schematic of NQE-DQC1 is depicted in Fig. 1 using the task of embedding a subset of the MNIST database of hand- written digits [39] as an example. In the following, we demon- strate a proof-of-principle implementation of NQE-DQC1 on a nuclear magnatic resonance (NMR) quantum processor. Experimental scheme—We experimentally demonstrate the NQE-DQC1 protocol on an NMR platform.The experi- ments are conducted on a Bruker 300 MHz spectrometer at room temperature. The sample used is 13 C-labeled trans- crotonic acid dissolved in푑 6 -acetone, with the four 13 C nuclear spins forming a four-qubit quantum processor [40– GzGzGz H (a) (b) (c) 51015 Itera�on 051015 Itera�on 0 0 0.5 1.0 Trace distance 051015 Itera�on theo exp 0 0.2 0.4 0.6 Test setTraining set FIG. 2. (a) Experimental quantum circuit for NQE-DQC1 training. The experimental setup uses a four-qubit NMR quantum processor comprising four carbon spins. The operations within the red dashed box initialize the system to|0⟩⟨0| ⊗퐼/2 푛 with푛=3. The gray bar represents a 1 ms푧-gradient pulse, which is applied to eliminate coherence. The unitary operators푉and푉 † are determined by the output of the neural network푔, where푥 푖 serves as the image data input, andwrepresents the trainable parameters.퐻denotes the Hadamard gate, and measurements are performed solely on the probe qubit. (b) Training loss histories are presented, with the solid line and diamonds representing the simulated and experimental mean values of the training loss across 10 pairs of images, respectively. (c) Trace distance is used as a metric to evaluate the distinguishability of images from different categories. Twenty pairs of images are randomly selected from both the training and test datasets. Solid dots indicate the mean trace distance, while error bars denote the standard deviation. Figure 2.7: Experimental NQE-DQC1 circuit. A single probe qubit is acted upon by Hadamard gates, while the remaining register implements the controlled feature-map unitary and its Her- mitian conjugate. Measurement of휎 푧 on the probe yields the Hilbert–Schmidt inner product required for the NQE-DQC1 loss. imally mixed state. In DQC1 framework, the initial state of the(푛 + 1)-qubit system is: 휌 0 = |0⟩⟨0| ⊗ 퐼 2 푛 ,(2.27) where the first qubit (theprobe) is prepared as a clean qubit and the remaining푛qubits are in the maximally mixed state퐼/2 푛 . The DQC1 circuit proceeds as follows: a Hadamard gate퐻is applied to the probe qubit, followed by a controlled-푈gate (with the probe as control and the푛mixed qubits as targets), and finally another Hadamard gate on the probe. Measuring the Pauli휎 푧 observable on the probe qubit yields: ⟨휎 푧 ⟩ = ReTr(푈) 2 푛 . (2.28) By replacing the final Hadamard with a phase gate푆 † 퐻, one can similarly extract ImTr(푈)/2 푛 , thereby obtaining the full normalized trace Tr(푈)/2 푛 . 42 Despite its limited initial resources, the DQC1 model is believed to offer computational power beyond classical simulation. Shor and Jordan [62] showed that estimating the Jones polynomial—a problem related to knot invariants—is complete for the DQC1 complexity class. Moreover, Poulinetal.[63]demonstratedanexponentialquantumspeedupforestimatingfidelity decay in quantum chaos using DQC1. The quantum correlations responsible for this computa- tional power arise not from entanglement in the conventional sense, but from quantum discord between the probe and the mixed register [64]. Crucially for our purposes, the DQC1 model is a natural fit for NMR quantum processors. In NMR, the quantum state at thermal equilibrium is close to the maximally mixed state, with only a small polarization on each spin. Through spatial averaging techniques, one can prepare an effective DQC1 state with a polarized probe and a maximally mixed encoding register, which realizes the structure idealized in Equation (2.27). 2.3.2NQE Loss Function via Hilbert–Schmidt Inner Product We now reformulate the NQE training objective in terms of a quantity that DQC1 can directly measure. TheHilbert–Schmidt (HS) inner productbetween two unitary operators푈 1 and푈 2 is defined as: ⟨푈 1 , 푈 2 ⟩ HS = Tr(푈 † 1 푈 2 ) 2 푛 .(2.29) This quantity is directly related to the normalizedFrobenius distancebetween the two unitaries: 1 2 푛 ‖푉 (푔(x 푖 ,w)) − 푉 (푔(x 푗 ,w))‖ 2 퐹 = 2 − 2ReTr[푉 (푔(x 푖 ,w)) 푉 † (푔(x 푗 ,w))] /2 푛 ,(2.30) where푉 (푔(x,w))is the quantum embedding circuit parameterized by the neural network output, as in Equation (2.18). Minimizing this normalized Frobenius distance for same-class pairs and 43 increasing it for cross-class pairs is analogous to the fidelity-based objective of Equation (2.19), but now expressed entirely in terms of the HS inner product. The NQE-DQC1 loss function is defined as: 퐿 NQE = ∑ 푖,푗 [ 1 2 푛 ReTr[푉 (푔(x 푖 ,w)) 푉 † (푔(x 푗 ,w))] − 1 + 푦 푖 푦 푗 2 ] 2 .(2.31) The target value(1+ 푦 푖 푦 푗 )/2is identical to the fidelity-based loss: it equals1for same-class pairs and0for different-class pairs. The key difference is that the fidelity|⟨x 푖 |x 푗 ⟩| 2 has been replaced by the measured real part of the HS inner product Tr[푉 (푔(x 푖 ,w)) 푉 † (푔(x 푗 ,w))] 2 푛 .(2.32) The crucial observation is that this HS inner product is precisely what DQC1 measures. Setting 푈 = 푉 (푔(x 푖 ,w)) 푉 † (푔(x 푗 ,w))(2.33) in Equation (2.28), the DQC1 measurement on the probe qubit yields: ⟨휎 푧 ⟩ = Re Tr [푉 (푔( x 푖 ,w)) 푉 † (푔(x 푗 ,w))] 2 푛 ,(2.34) which directly provides the real part of the HS inner product needed for the loss function (Equa- tion (2.31)). The loss drives same-class pairs toward an HS inner product of1(similar unitaries) and different-class pairs toward low HS overlap, providing an HS-compatible class-separation surrogate. The classical neural network푔(x,w)is optimized via standard gradient descent. Each loss evaluation is estimated from repeated DQC1 probe-qubit measurements. As in the original NQE 44 framework, the embedding circuit푉can be any fixed quantum circuit, such as the Z feature map (Equation (2.15)) with휙(x)replaced by the learned function푔(x,w). 2.3.3Experimental Demonstration NMR platform and initialization.The NQE-DQC1 protocol was experimentally demon- strated on a Bruker 300 MHz NMR spectrometer using 13 C-labeledtrans-crotonic acid dissolved in d6-acetone [49]. The molecule provides four carbon nuclear spins: C1 serves as the probe qubit, while C2–C4 serve as the three encoding qubits (푛 = 3). The initial state is prepared via spatial averaging as an effective DQC1 state휌 0 = |0⟩⟨0| ⊗ 퐼/2 3 , matching the structure of Equation (2.27). The quantum embedding circuit uses the Z feature map (Equation (2.15)) with푀 = 1repetition, and the classical input data is preprocessed via PCA to reduce the dimensionality to five principal components before being fed to the neural network 푔. NQE-DQC1 training results.The training was performed on 500 MNIST images (bi- nary classification of digits 0 vs. 1), with 15 training iterations and 10 randomly sampled data pairs per iteration. For each pair(x 푖 ,x 푗 ), the DQC1 circuit implements the unitary 푈 = 푉 † (푔(x 푗 ,w)) 푉 (푔(x 푖 ,w))controlled by the probe qubit to measure the HS inner product via Equation (2.34). Figure 2.8(b) shows the training loss as a function of iteration: the loss con- verges to approximately zero by iteration 10, and the NMR experimental results closely match the numerical simulations, confirming the practical feasibility of the DQC1-based approach. Figure 2.8(c) demonstrates that the trace distance between class ensembles increases for both the training and test sets across iterations, consistent with the HS surrogate being aligned with improved distinguishability in this experiment. 45 2 V(g(x i ,w)) V † (g(x j ,w)) H I ⊗n 2 n ⋮ ⋮ I+ασ z 2 x g(x,w) Update w FIG. 1. Schematic of the neural quantum embedding via DQC1, illustrating the process for identifying the optimal quantum feature map for classification tasks. Data points with different labels are represented as squares and triangles. Protocol overview—Let the class label of the input data x 푖 be푦 푖 ∈ +1,−1. The quantum data embedding circuit must be constructed to maximize the fidelity between quan- tum states corresponding to input data points with the same label (푦 푖 =푦 푗 ) and minimize it for those with different labels. This can be formulated as maximizing the distinguishability between휌 ± = Í 푚 ± 푖 |x 푖 ⟩⟨x i |/푚 ± , which denotes the quantum state representation of data samples labeled±1 [27]. Here,푚 ± indicates the number of data points in class±1. For quantum data embedding, this is equivalent to finding a quantum feature map that maximizes the trace distance,∥휌 + −휌 − ∥ 1 /2 [14]. Our method utilizes a classical neural network to optimize quantum data embedding according to these criteria. The objective function subject to minimization is 퐿 NQE = ∑︁ 푖푗 1 2 푛 Tr 푉(푔 x 푖 )푉 † (푔 x 푗 ) − 1+푦 푖 푦 푗 2 2 ,(1) where푔 x 푖 =푔(x 푖 ,w)is defined as the output of the neural network,wis the set of trainable parameters, and푉is the quantum feature map that encodes푉(푔 x )|휓⟩=|x⟩. Then |x⟩is provided as an input to a QML algorithm, such as the quantum neural network [7, 28–34] or the quantum kernel method [6, 35, 36], to construct a classifier. The first term in Eq. (B1) is the HS inner product between two푛-qubit unitary operators, which quantifies the similarity between two data points. Finding the solution to minimizing Eq. (B1) corresponds to learningwsuch that the HS inner product is large when the two data points belong to the same class and small otherwise. It is also related to the distance defined by the Frobenius norm,∥퐴∥ 퐹 = √︁ Tr(퐴 † 퐴), since ∥푉(푔 x 푖 )−푉(푔 x 푗 )∥ 2 퐹 =2−2Re h Tr 푉(푔 x 푖 )푉(푔 x 푗 ) † i . Thus, the task of minimizing Eq. (B1) reduces to efficiently computing the HS inner product within the objective function. As previously noted, DQC1 provides an efficient means for estimating this metric [20]. Additionally,푉can be optimized by employing a classical neural network and deep learning techniques. This approach overcomes the limitation imposed by completely positive and trace-preserving (CPTP) maps, which cannot increase the trace distance between quantum states. Specifically, any subsequent CPTP mapΛsatisfies the inequality∥Λ(휌 + ) −Λ(휌 − )∥ 1 ≤ ∥휌 + −휌 − ∥ 1 [37]. Further details are provided in the Supplementary Material [38]. The schematic of NQE-DQC1 is depicted in Fig. 1 using the task of embedding a subset of the MNIST database of hand- written digits [39] as an example. In the following, we demon- strate a proof-of-principle implementation of NQE-DQC1 on a nuclear magnatic resonance (NMR) quantum processor. Experimental scheme—We experimentally demonstrate the NQE-DQC1 protocol on an NMR platform.The experi- ments are conducted on a Bruker 300 MHz spectrometer at room temperature. The sample used is 13 C-labeled trans- crotonic acid dissolved in푑 6 -acetone, with the four 13 C nuclear spins forming a four-qubit quantum processor [40– GzGzGz H (a) (b) (c) 51015 Itera�on 051015 Itera�on 0 0 0.5 1.0 Trace distance 051015 Itera�on theo exp 0 0.2 0.4 0.6 Test setTraining set FIG. 2. (a) Experimental quantum circuit for NQE-DQC1 training. The experimental setup uses a four-qubit NMR quantum processor comprising four carbon spins. The operations within the red dashed box initialize the system to|0⟩⟨0| ⊗퐼/2 푛 with푛=3. The gray bar represents a 1 ms푧-gradient pulse, which is applied to eliminate coherence. The unitary operators푉and푉 † are determined by the output of the neural network푔, where푥 푖 serves as the image data input, andwrepresents the trainable parameters.퐻denotes the Hadamard gate, and measurements are performed solely on the probe qubit. (b) Training loss histories are presented, with the solid line and diamonds representing the simulated and experimental mean values of the training loss across 10 pairs of images, respectively. (c) Trace distance is used as a metric to evaluate the distinguishability of images from different categories. Twenty pairs of images are randomly selected from both the training and test datasets. Solid dots indicate the mean trace distance, while error bars denote the standard deviation. Figure 2.8: NQE-DQC1 training on the NMR platform. (a) Schematic of the NMR DQC1 circuit: the probe qubit C1 controls the application of푉 † (푔(x 푗 )) 푉 (푔(x 푖 ))on qubits C2–C4. (b) Training loss versus iteration for NMR experiments (markers) and numerical simulation (solid line). (c) Trace distance between class ensembles for the training set (circles) and test set (triangles) across NQE training iterations. 46 3 43]. The molecular structure of the sample is shown in Fig.2(a). The internal Hamiltonian governing this system isH NMR =→ ! 퐿 퐿 퐿 푀 퐿 푀 /2+ ! 퐿,푁 푁푂 퐿푁 푀 퐿 푀 푀 푁 푀 /2, where퐿 퐿 /2푁is the Larmor frequency of the푃th spin, and푂 퐿푁 represents the scalar coupling between the푃th and푄th spins. The specific parameters are provided in Supplemental Material [38]. In the NMR experimental setup, single-qubit rotations are imple- mented through transverse radio-frequency pulses, while two- qubit gates are realized via free evolutions of the interactions inH NMR . The experimental control accuracy is further en- hanced using gradient-based optimization to generate shaped pulses [44]. The experimental NQE-DQC1 circuit is depicted in Fig.2(a), where C 1 serves as the probe qubit, and the re- maining spins act as encoding qubits. The entire NQE-DQC1 experiment consists of three stages. (i) Initialization. The entire system is first prepared into |0000↑from thermal equilibrium using the spatial averaging method [45]. Subsequently, each of the encoding qubits under- goes a 2 ms푅 푂 (푁/2)pulse (푁/2 rotation about the푆-axis) and a 1 ms푇-gradient pulse. This operation initializes the system to푈 0 =|0↑↓0|↔푉/2 푃 , where푉is the 8↗8 identity matrix and 푊=3. (i) NQE training. The dataset used in this experiment comes from 500 images of the MNIST dataset [39], which depicts handwritten digits “0” and “1”. To construct the training dataset, ten pairs of images are randomly selected, with each pair comprising two images that may either belong to the same category or di!erent categories. The gray-scale values of the images are then extracted and subjected to principal component analysis (PCA). It is important to note that PCA preprocessing is not strictly necessary, as the NQE training at this stage is able to handle the “dimensionality reduction” [14]. However, to reduce the number of training parameters of the classical neural network, we preprocess the images using classical PCA before implementing the quantum circuit. After applying PCA, we obtain two 1↗5 vectors, e.g.,x 1 andx 2 , corresponding to the two images in each pair. Each vectorxis then non-linearly mapped to a quantum state on the encoding qubits via|x↑=푋(푌 x )|0↑. Here,푋denotes a 푍-feature map [6], expressed as 푋(ω)= " exp # 푃 $ 푄 ω 푄 푍 푄 +ω 푃+푄 푍 푄 푍 푄+1 % 푎 ↔푃 & 푅 ,(2) where푎is the Hadamard gate,푍 푄 is the Pauli-푇operator on the푏th spin,ω 푄 represents the푏th entry of the vectorx, and 푐is the number of layers, which we set푐=1. As required by the DQC1 protocol, the implementation of this feature map depends on the state of the probe qubit C 1 . Starting from 푈 0 , the following gate sequence,U=푎 1 V † 푆 (푌 x 2 )V 푆 (푌 x 1 )푎 1 , is applied (right to left). Here, the Hadamard gate푎 1 can create superposition (or vice versa) on C 1 , whileV 푆 (푌 x )= |0↑↓0|↔푉+|1↑↓1|↔푋(푌 x )represents a controlled feature map conditioned on the state of C 1 . The corresponding circuit is illustrated in Fig.2(a). (i) Measurement and optimization. Here, we only need to measure the probe qubit along its푇-direction, as↓푀 푀 ↑= Re ' Tr[푋(푌 x 1 )푋 † (푌 x 2 )] ( /2 푃 . The experimental result is then H H H ...... (a) (b) Z Z Embedding 2468101214 Itera�on 0.2 0.3 0.4 0.5 Without NQE theo exp (NMR) exp (IBM) With NQE PQC FIG. 3. (a) Quantum circuit for classification tasks. Classical images are first encoded into quantum states using NQE and then processed by the PQC classifier. The PQC consists of two layers, incorporating four푑parameters and two CNOT gates. Measurements of↓푀 푀 ↑ are performed on the third spin. (b) Optimization results of the PQC. The optimization results for the PQC are presented, where solid lines, dots, and dashed lines correspond to the loss푒 PQC from numerical simulations, NMR experiments, and IBM experiments, respectively. Red and blue denote results obtained under traditional푍-feature embedding (without NQE) and with NQE encoding, respectively. substituted into Eq. (B1) to compute the loss value푒 NQE . Subsequently, the neural network푌is optimized using the gradient↘푒 NQE , which is computed from the average loss value obtained across the ten pairs of image data. In the next iteration, ten new pairs of images are randomly selected, and the above steps are repeated to further optimize the neural network. NQE-DQC1 training results—In the experiment, we per- form 15 iterations of NQE training using the DQC1 circuit. The training loss푒 NQE decreases with each iteration, as shown in Fig.2(b). The triangle markers indicate the experimental re- sult from the NMR device, while the solid line represents num- berical simulations. We observe that푒 NQE converges close to zero around the 10th iteration. Furthermore, the experimental results closely match the simulation results, demonstrating the high control accuracy of our experiment. To evaluate the NQE performance, we examine how the trace distance varies over training iterations. Twenty pairs of images, each from the two categories, are randomly selected from both the training and test datasets. Ideally, we expect the trace distance for each pair to approach 1 after the NQE optimization. We compute the trace distance for the 20 pairs at each iter- ation. In Fig.2(c), the solid circles represent the mean trace distance across the 20 pairs, while the error bars indicate the standard deviation. For both the training and test datasets, the gradual increase in trace distance through successive iterations Figure 2.9: Cross-platform training loss of the downstream PQC classifier with and without NQE. The classifier is optimized on the NQE-enhanced embedding (blue) and on the conven- tional Z feature map (red, “without NQE”). Dashed lines denote noiseless simulation, filled circles the NMR experiment, and asterisks the IBM superconducting hardware. Classificationresults.After NQE training, a parameterized quantum circuit (PQC) classifier is deployed on the NQE-enhanced embedding. The classifier uses 2 circuit layers with 4 trainable rotation parameters and 2 CNOT gates, and classification is performed by measuring⟨휎 푧 ⟩on the third qubit. Without NQE (using the raw Z feature map), the classifier achieves 54% accuracy on the 500-sample dataset, which is close to random guessing and consistent with the low trace distance of the unoptimized embedding. With NQE preprocessing, the accuracy reaches 98%. 47 4 Training setTest set NQE 0.06-0.090.040.20-0.390.230.08-0.30 -0.490.38-0.500.430.590.46-0.47-0.44 0.01-0.010.040.11-0.16-0.10-0.13-0.09 0.640.79-0.430.68-0.410.37-0.40-0.35 0100200300400500 (a)(b) -feature Image index -1.0 -0.5 0 0.5 1.0 “1”“0” accuracy: 54.0% accuracy: 98.0% -1.0 -0.5 0 0.5 1.0 “1”“0” FIG. 4. Experimental classification results after processing through the PQC circuit for푍-feature and NQE encoding. (a) The top and bottom panels present the results for푍-feature embedding and NQE, respectively. The numerical values correspond to the experimental⟨휎 푧 ⟩, while the height of the color bars serves as a visual representation (red for negative values and green for positive). Additionally, NMR peaks provide visual indicators, with downward peaks representing “0” and upward peaks representing “1”. Handwritten images outlined in red indicate incorrect classifications. These results clearly highlight the substantial improvement in classification accuracy achieved through NQE encoding. (b) Classification results for all 500 images, with the dashed line representing the decision boundary. The classification accuracy improves from 54.0% without NQE (comparable to random guessing) to 98.0% with NQE encoding. shows that NQE significantly improves the separability of the embedded data. This substantial improvement confirms the ef- fectiveness of NQE-DQC1 in enhancing the distinguishability between categories in classification tasks. Classification results—We further substantiate the effec- tiveness of NQE in QML by training a parameterized quan- tum circuit (PQC) and implementing classification tasks. The sturcture of the PQC is shown in Fig. 3(a) and consists of two parts. The first part embeds classical data into quantum states, while the second part trains the PQC to efficiently classify the data. For the task of classifying handwritten digits “0” and “1”, we apply a double-layer PQC with four parameters of휃 and two CNOT gates, and measure⟨휎 푧 ⟩on the third qubit. The training of this PQC is carried out using the standard parameter-shift rule [46–49], and further experimental details can be found in Supplemental Material [38]. . We compare the classification performance of the traditional 푍-feature embedding (without NQE) [6] and our protocol. This performance is visualized by measuring the loss function of the PQC, defined as퐿 PQC = 1 푚 Í 푖 1 2 (1−푓 푖 ×푦 푖 ), at each iteration. Here,푓is the experimental output⟨휎 푧 ⟩on the third qubit, and푦is the label of the image (푦=−1 for digit “0” and 푦=+1 for digit “1”). We randomly select a batch of푚=10 images at each iteration to compute퐿 PQC . The experimental results are shown in Fig. 3(b). The blue and red curves represent the loss histories of퐿 PQC from nu- merical simulations with and without NQE, respectively. The solid circles represent the experimental results from the NMR device. We observe that퐿 PQC decreases significantly through iterative optimization when NQE is utilized. In contrast, the PQC training using the traditional푍-feature embedding with- out NQE shows only minimal improvement, highlighting the effectiveness of our NQE protocol. Moreover, by leveraging the neural network obtained from the NQE training, we can apply it to subsequent classifica- tion tasks on other physical systems. This highlights an addi- tional advantage of the NQE-DQC1 protocol: its extendability. DQC1 is specifically designed for ensemble systems, making platforms such as NMR well-suited for its implementation. However, once the neural network is trained, the subsequent tasks can also be executed on other quantum platforms. As a comparison, we train the PQC using IBM cloud superconduct- ing processors, and the corresponding results are depicted in Fig. 3(b). The red and blue stars represent IBM experimental results without and with NQE encoding, respectively. The experimental trends align well with numerical simulations, al- though they perform slightly worse than the NMR experiments. This further demonstrates the extendability of the NQE-DQC1 protocol in classification tasks. After training the PQC, we evaluate its prediction accu- racies. Eight images are randomly selected from both the training and test datasets, with each image encoded into quan- tum states via NQE before being processed through the trained PQC circuit. The expectation values⟨휎 푧 ⟩of the third spin are measured, and these values are used to determine the classifi- cation results, as shown in Fig. 4(a). Ideally, a negative⟨휎 푧 ⟩ (red bar), corresponding to a downward NMR peak, indicates the digit “0,” while a positive⟨휎 푧 ⟩(green bar) or an upward peak indicates the digit “1.” For both the training and test datasets, the classification accuracy is significantly improved through NQE encoding, yielding more precise⟨휎 푧 ⟩values and more distinct experimental spectral features. Without NQE Figure 2.10: Classification results with and without NQE. (a) Per-sample classification outputs using the Z feature map alone (top) versus the NQE-enhanced embedding (bottom). (b) Scatter plot of predicted labels for all 500 MNIST test images (digits 0 vs. 1). Cross-platform extendability.An important feature of the NQE-DQC1 approach is cross- platform transferability. The neural network푔(x,w)trained through the NMR-compatible DQC1 protocol was deployed on IBM cloud superconducting quantum processors without retraining, and the experimental trends agreed with the numerical simulations (Figure 2.9) [49]. This is possible because NQE training produces a set of classical neural network parameters w ∗ that define a data preprocessing function푔(x,w ∗ ), which is independent of the hardware used to evaluate the training loss. The DQC1 protocol is used during the NQE training phase to evaluate the loss function. Once training is complete, the optimized preprocessing can be paired with another platform that implements the same embedding circuit푉. Additional validation on Fashion-MNIST and satellite image datasets supports the applicability of the approach across the tested data domains [49]. Further supplementary analyses on feature-map ansatz robustness and multi-class extensions, together with reproducibility details, are collected in Section A.1. 48 2.4Summary In this chapter, we addressed a central question in quantum machine learning: how does the data embedding constrain the achievable performance of a quantum classifier? We established three main results. First, we showed that the empirical risk of the binary quantum classifier under the linear loss is lower bounded by the trace distance between embedded data ensembles (Equation (2.9)), a quantity determined by the data embedding. The contractive property of trace distance under quantum channels (Equation (2.12)) further implies that subsequent noisy quantum processing cannot increase this distinguishability. We demonstrated that conventional deterministic em- beddings (amplitude encoding, angle encoding, and the Z feature map) are data-agnostic and offer no guarantee of producing a large trace distance for a given classification task, and that trainable quantum embedding strategies remain constrained by the distinguishability allowed by the embedding construction. Second, we introduced Neural Quantum Embedding (NQE), which avoids the contractive constraint by inserting a classical neural network between the raw data and the quantum embed- ding circuit. Because the neural network operates on classical parameters rather than quantum states, it is not subject to the quantum-state contractive inequality at this preprocessing stage. The implicit fidelity loss (Equation (2.19)) provides a tractable training objective that simulta- neously clusters same-class states and separates cross-class states in the quantum feature space. In the reported IBM hardware experiments on binary MNIST, PCA-NQE raises the trace dis- tance from≈ 0.27to≈ 0.84, improving classification accuracy from52.7%to96.1%under real hardware noise and achieving a lower empirical risk than the noiseless Helstrom limit associated with the conventional Z feature map. Third, we showed that NQE can be extended to ensemble quantum systems by replacing the 49 fidelity-based loss with the Hilbert–Schmidt inner product, which is directly measurable via the DQC1 model using a single probe qubit and a mixed encoding register. This reformulation was experimentally validated on an NMR quantum processor, where it achieved 98% classification accuracy on MNIST compared to 54% without NQE. The deployment of the learned neural net- work parameters on superconducting quantum processors further shows that the preprocessing learned through the DQC1 protocol can be transferred across platforms that implement the same embedding circuit. Throughout this chapter, we focused on a representation-induced source of training error in quantum machine learning: the achievable training loss can be reduced only if the embedding makes the class ensembles sufficiently distinguishable. However, low training error does not automatically guarantee good performance on unseen data. In the next chapter, we turn to the complementary question ofgeneralization: under what conditions does a quantum classifier trained on a finite dataset maintain its performance on new, previously unseen inputs? We will show that the trace-distance perspective also connects naturally to margin-based generalization analysis. 50 CHAPTER 3 GENERALIZATION IN QUANTUM MACHINE LEARNING The results in this chapter are based onUnderstanding Generalization in Quantum Ma- chine Learning with Margins[65]. Code is available athttps://github.com/takh04/ Q-margin. In the previous chapter, we studied a representation-induced limit on quantum supervised learning. Neural Quantum Embedding increases the trace distance between embedded data en- sembles and thereby lowers the achievable training loss. However, achieving low training error is only part of the challenge: a model that memorizes its training data without learning the under- lying pattern will fail on new, unseen inputs. This chapter turns to the complementary question ofgeneralization—the ability of a model to maintain its performance beyond the training set. As discussed in Section 1.1.2, the generalization error is the discrepancy between the true risk and the empirical risk, arising because the model is trained on a finite sample rather than the full data distribution. Controlling this error requires understanding the complexity of the hypothesis class. In classical machine learning, uniform generalization bounds based on com- plexity measures such as VC dimension and Rademacher complexity have been the primary theoretical tools. However, recent work has shown that such bounds are often too vacuous to explain empirical comparisons between models that generalize and models that memorize data, 51 both in classical deep learning [66] and in quantum machine learning [67]. In this chapter, we develop amargin-basedgeneralization framework for quantum neural networksthatrespondstotheselimitations. Drawinginspirationfromtheclassicalmargintheory of Bartlett et al. [68], we establish a high-probability bound for multiclass QNN classifiers that remains a function-class bound, but is margin-sensitive through the empirical margin loss. We experimentally show that margin-based metrics are stronger predictors of generalization per- formance than parameter-based metrics in the studied QPR setting. Furthermore, we connect the margin to quantum state discrimination, showing that quantum embeddings with large trace distances—such as those produced by NQE (Chapter 2)—can support larger margins and thus tighter margin-based generalization bounds when the other bound parameters are comparable. 3.1Theoretical Background This section reviews the key concepts from classical generalization theory that form the foun- dation for our quantum margin bounds. We build the theory progressively: starting from the simplest case of finite hypothesis classes, moving to the tools needed for infinite classes, and finally arriving at margin theory. We assume a supervised learning setting as established in Sec- tion 1.1.2: given a hypothesis classℋand a training set푆 = (푥 푖 , 푦 푖 ) 푚 푖=1 drawn i.i.d. from an unknown distribution풟, we seek to bound the difference between the true risk푅(ℎ) = 피 (푥,푦)∼풟 [1(argmax 푗 ℎ(푥) 푗 ≠ 푦)]and the empirical risk ̂ 푅(ℎ) = 푚 −1 ∑ 푚 푖=1 1(argmax 푗 ℎ(푥 푖 ) 푗 ≠ 푦 푖 )for any hypothesisℎ ∈ ℋ. 3.1.1Generalization Bounds for Finite Hypothesis Classes We begin with the simplest setting: a hypothesis classℋcontaining finitely many hypotheses. Even this restricted case reveals the core tension in generalization theory—the gap between con- 52 trolling error for asingle fixedhypothesis and controlling error for thebest hypothesis selected from data. Concentration for a single hypothesis.Consider a single, fixed hypothesisℎ. Since the train- ing samples(푥 1 , 푦 1 ), ... , (푥 푚 , 푦 푚 )are drawn i.i.d., the empirical risk ̂ 푅(ℎ)is an average of푚in- dependent bounded random variables, each taking values in[0, 1]. Hoeffding’s inequality [21] gives a sharp concentration bound: Pr[|푅(ℎ) − ̂ 푅(ℎ)| > 휖] ≤ 2exp(−2푚휖 2 ).(3.1) Thus, for anyfixedhypothesis, the empirical risk converges exponentially fast to the true risk as the sample size푚grows. If we could specify which hypothesis to evaluate before seeing the data, generalization would be straightforward. The selection problem.In practice, however, we do not fixℎin advance. Instead, we use the training data to select the hypothesis with the lowest empirical risk: ̂ ℎ =argmin ℎ∈ℋ ̂ 푅(ℎ). This data-dependent selection invalidates the direct application of Equation (3.1), because the selected hypothesis is correlated with the training sample. A hypothesis that happens to perform well on the training set may do so by chance rather than by capturing the true pattern. Union bound overℋ.Whenℋis finite, we can resolve this by applying Hoeffding’s in- equality toeveryℎ ∈ ℋsimultaneously and taking a union bound. The probability thatany hypothesis deviates by more than휖is at most: Pr[∃ ℎ ∈ ℋ ∶ |푅(ℎ) − ̂ 푅(ℎ)| > 휖] ≤ ∑ ℎ∈ℋ 2exp(−2푚휖 2 ) = 2|ℋ|exp(−2푚휖 2 ).(3.2) 53 Setting this probability to훿and solving for휖, we obtain thefinite hypothesis class bound: with probability at least1 − 훿, for allℎ ∈ ℋsimultaneously, 푅(ℎ) ≤ ̂ 푅(ℎ) + √ ln|ℋ| +ln(2/훿) 2푚 .(3.3) This result has an appealing interpretation: the “price” of selecting the best hypothesis from |ℋ|candidates is only logarithmic in the class size. Even if|ℋ|is astronomically large, the generalization penalty grows slowly. Limitation.The finite hypothesis class bound breaks down when|ℋ| = ∞—as is the case for hypothesis classes parameterized by continuous parameters. Neural networks, support vector machines, and quantum circuits all define infinite hypothesis classes, rendering Equation (3.3) vacuous. This motivates the development of complexity measures that can handle infinite classes, which we turn to next. 3.1.2Complexity Measures for Infinite Hypothesis Classes When the hypothesis class is infinite, we can no longer enumerate its elements. Instead, we need complexity measures that capture the “effective size” ofℋ—how richly it can fit arbitrary patterns—without requiring finiteness. VC Dimension TheVapnik–Chervonenkis (VC) dimension[7] was the first such measure. A hypothesis class ℋshattersa set of푚points if, for every possible labeling of those points, there exists anℎ ∈ ℋ that realizes that labeling. The VC dimension푑 VC is the largest푚such that some set of푚points can be shattered byℋ. 54 The fundamental theorem of PAC learning [7, 21] establishes that푑 VC characterizes learn- ability: a binary hypothesis class is PAC-learnable if and only if its VC dimension is finite. Moreover, the VC generalization bound states that with high probability, 푅(ℎ) ≤ ̂ 푅(ℎ) + 푂 ⎛ ⎜ ⎝ √ 푑 VC 푚 ⎞ ⎟ ⎠ .(3.4) While the VC dimension elegantly extends the finite class bound to infinite classes (note the structural similarity between Equations (3.3) and (3.4), with ln|ℋ|replaced by푑 VC ), it has a significant limitation for modern models: the VC dimension of a neural network scales with its number of parameters, producing bounds that are vacuous for overparameterized models that nonetheless generalize well in practice. Rademacher Complexity Rademacher complexity[69] provides a data-dependent measure of hypothesis class complex- ity. For a sample푆 = 푧 푖 푚 푖=1 and a function classℱ, thesample Rademacher complexityis defined as: ℜ(ℱ| 푆 ) = 피 휎 ⎡ ⎢ ⎣ sup 푓 ∈ℱ 1 푚 푚 ∑ 푖=1 휎 푖 푓 (푧 푖 ) ⎤ ⎥ ⎦ ,(3.5) where휎 1 , ... , 휎 푚 are independent Rademacher random variables taking values±1with equal probability. Intuitively, the Rademacher complexity measures how well functions inℱcan correlate with random noise—a class that can fit random labels has high Rademacher complexity and is prone to overfitting. A fundamental result connects Rademacher complexity to generalization [69, 21]: for any 훿 > 0, with probability at least1 − 훿over the random draw of an i.i.d. sample푆of size푚, the 55 following holds for allℎ ∈ ℋ: 푅(ℎ) ≤ ̂ 푅(ℎ) + 2ℜ(ℋ| 푆 ) + 3 √ ln(2/훿) 2푚 .(3.6) This bound isuniform—it holds simultaneously for all hypotheses inℋ—and becomes tighter as the sample size푚increases or the Rademacher complexity decreases. Compared to the VC bound, the Rademacher bound has the advantage of beingdata-dependent: the sample Rademacher complexity depends on the specific data points, not just the abstract hypothesis class. Covering Numbers and Dudley’s Entropy Integral In practice, computing the Rademacher complexity directly can be challenging. A powerful technique for bounding it usescovering numbers, which bridge the gap between infinite and finite by discretizing function classes at progressively finer resolutions. The휖-covering number of a set퐴with respect to a norm‖ ⋅ ‖, denoted풩(퐴, 휖, ‖ ⋅ ‖), is the minimum number of balls of radius휖needed to cover퐴. The key idea is that even an infinite function class can be approximated to precision휖by a finite set of size풩(퐴, 휖, ‖⋅‖)—effectively reducing the problem to the finite case at each resolution scale. The connection between covering numbers and Rademacher complexity is established throughDudley’s entropy integral[21]: ℜ(퐴) ≤inf 훼≥0 ( 4훼 √ 푚 + 12 푚 ∫ √ 푚 훼 √ ln풩(퐴, 훽, ‖ ⋅ ‖ 2 ) 푑훽) .(3.7) This integral aggregates the covering numbers across all scales훽. Fine scales (small훽) capture local complexity, while coarse scales (large훽) capture global structure. Dudley’s entropy in- 56 tegral is the key technical tool we will use in Section 3.3.2 to derive our margin generalization bound for QNNs. 3.1.3Margin Theory The complexity measures discussed above—VC dimension, Rademacher complexity, covering numbers—all provideuniformbounds that depend on the hypothesis classℋas a whole.Mar- gin theoryintroduces a complementary,hypothesis-dependentperspective: the generalization bound depends on how confidently the specific learned hypothesis classifies the training data. The concept of margin has been central to generalization theory since the early days of support vector machines [6]. Classical Margin Bounds For binary classification, the margin of a classifier푓on a sample(푥, 푦)is defined as푦 ⋅ 푓 (푥), where푦 ∈ −1, +1. The sample is correctly classified with margin훾if푦 ⋅ 푓 (푥) ≥ 훾 > 0. For multiclass classification with푘classes, the margin is generalized through themargin operator: ℳ(푣, 푦) = 푣 푦 −max 푖≠푦 푣 푖 ,(3.8) where푣 = (푣 1 , ... , 푣 푘 )is the output vector of the classifier and푦is the true label. A positive marginℳ(푣, 푦) > 0indicates correct classification, and a larger margin corresponds to a more confident prediction. To incorporate margins into generalization bounds, one replaces the hard0-1loss with a Lipschitz surrogate such as theramp loss푙 훾 ∶ ℝ → ℝ + , parameterized by the margin threshold 57 훾 > 0: 푙 훾 (푥) = ⎧ ⎨ ⎩ 0,if푥 > 훾, 1 − 푥/훾,if0 ≤ 푥 ≤ 훾, 1,if푥 < 0. (3.9) The ramp loss upper bounds the0-1loss while remaining Lipschitz, which makes it suitable for Rademacher-complexity arguments. The theorem below is stated in terms of the empirical margin error, ̂ 푅 훾 (ℎ) = 1 푚 푚 ∑ 푖=1 1 (ℳ(ℎ(푥 푖 ), 푦 푖 ) ≤ 훾) ,(3.10) which counts both misclassified samples and correctly classified samples whose margin is at most훾. Margin Bounds for Deep Networks A landmark result by Bartlett et al. [68] established spectrally-normalized margin bounds for deepneuralnetworks. Theirkeyinsightwasthatthemargindistribution, normalizedbythespec- tral norms of the weight matrices, is strongly correlated with generalization performance. This represented a significant advance over classical uniform bounds, which depend on the number of parameters or the VC dimension and often provide vacuous estimates for overparameterized networks. Specifically, for a deep network with weight matrices푊 1 , ... , 푊 퐿 , the generalization bound scales as: 푅(ℎ) ≤ ̂ 푅 훾 (ℎ) + ̃ 푂 ⎛ ⎜ ⎜ ⎝ ∏ 퐿 ℓ=1 ‖푊 ℓ ‖ 휎 훾 √ 푚 ⎞ ⎟ ⎟ ⎠ ,(3.11) where‖푊 ℓ ‖ 휎 denotes the spectral norm of theℓ-th weight matrix. Unlike parameter-count bounds, this bound becomes tighter when the model achieves larger margins. Subsequent work 58 confirmedthatmargin-basedmetricsareamongthestrongestpredictorsofgeneralizationindeep learning [70, 71, 72, 73]. Our contribution in this chapter is to extend this margin-based framework from classical neural networks to quantum neural networks, adapting the covering number techniques to the quantum setting. 3.2Generalization Bounds for Quantum Models We now review the existing approaches to understanding generalization in quantum machine learning. These approaches predominantly rely onuniform bounds—bounds that hold for all hypotheses within the function class—and we discuss their strengths and limitations. 3.2.1Uniform Generalization Bounds A growing body of work has established uniform generalization bounds for quantum models using various complexity measures. Parameter-basedbounds.The most widely adopted approach is based on the number of train- able parameters. Caro et al. [74] proved that the generalization gap of a QNN with푇trainable gates, trained on푚samples, is bounded by: |푅(ℎ) − ̂ 푅(ℎ)| ≤ 푂 ⎛ ⎜ ⎝ √ 푇 푚 ⎞ ⎟ ⎠ ,(3.12) with high probability. When only a subset of parameters undergo substantial change during training, the bound improves to scale with the number ofeffective parametersrather than the total parameter count. This result has become a standard tool in the QML community, providing a simple and interpretable estimate of generalization. 59 Effective dimension.Abbas et al. [57] introduced theeffective dimensionof a QML model as a measure of its complexity, connecting it to the Fisher information matrix. Models with smaller effective dimension are expected to generalize better, as they occupy a smaller effective volume in the hypothesis space. Quantum Rademacher and statistical complexity.Bu et al. [75] derived bounds on the Rademacher complexity of quantum circuits, relating the generalization capability to properties of the quantum circuit architecture. Subsequent work [76, 77] extended this analysis to study the effects of quantum resources and noise on the statistical complexity of quantum models. Information-theoretic approaches.Banchi et al. [78] analyzed QML generalization from a quantum information standpoint, connecting the generalization gap to information-theoretic quantities such as the mutual information between the training data and the learned hypothesis. More recently, Caro et al. [79] established information-theoretic generalization bounds for learning from quantum data, providing a complementary perspective to the complexity-based approaches. 3.2.2Limitations of Uniform Bounds Despite the theoretical elegance of the bounds reviewed above, they share a limitation for the comparisons studied here: they areuniform boundsthat hold for all hypotheses in the function class simultaneously. In classical deep learning, a seminal study by Zhang et al. [66] demon- strated that modern neural networks can easily memorize datasets with completely randomized labels, achieving zero training error on random noise. Since uniform bounds must hold forall functions in the hypothesis class—including those that memorize random labels—they can be too vacuous to distinguish a model that has learned meaningful patterns from one that has merely 60 memorized the training data in many practical settings [80, 81]. Building on this observation, Gil-Fuster et al. [67] demonstrated that the same phenomenon occurs in QML: quantum neural networks, including QCNNs, can overfit randomized labels on quantum datasets. Despite the small number of qubits and parameters, QNNs are expressive enough to fit random labels, showing that uniform bounds can also be too vacuous for QML memorization regimes. These findings motivate a shift from uniform bounds towarddata-dependentand hypothesis-dependentgeneralization measures. In classical deep learning, this shift led to the discovery that margin-based metrics are among the strongest predictors of generaliza- tion [71, 72, 73]. In the next section, we extend this margin-based perspective to quantum neural networks. 3.3Margin-Based Generalization for Quantum Neural Networks We now present our main contribution, a margin-based generalization bound for multiclass clas- sification with quantum neural networks. We begin by formalizing the QNN classification setup, derive the margin bound, and validate it experimentally. 3.3.1Multiclass Classification with Quantum Neural Networks Consider a푘-class classification task where the input is an푛-qubit quantum state휌 ∈ ℂ 푁×푁 with푁 = 2 푛 , and the label푦 ∈ [푘] = 1, ... , 푘. A QNN performs classification using a parameterized unitary circuit푈(휃) ∈ 핌 QNN and a set of positive operator-valued measurements (POVMs)퐸 푖 푘 푖=1 satisfying∑ 푘 푖=1 퐸 푖 = 퐼and퐸 푖 ≥ 0. The QNN maps an input quantum state to 61 a푘-dimensional probability vector: ℎ 휃 (휌) = Tr(푈(휃)휌푈 † (휃)퐸 푖 ) 푘 푖=1 ,(3.13) where each componentℎ 휃 (휌) 푖 represents the probability of assigning the input to class푖. Given푚training samples푆 = (휌 푖 , 푦 푖 ) 푚 푖=1 drawn i.i.d. from an unknown distribution풟, the goal is to find optimal parameters휃 ∗ that minimize the true error: 푅(ℎ ∗ ) = 피 (휌,푦)∼풟 [1(argmax 푗 ℎ ∗ (휌) 푗 ≠ 푦)] .(3.14) The generalization gap푔(ℎ) = 푅(ℎ) − ̂ 푅(ℎ)measures the discrepancy between the true error and the empirical error on the training set. 3.3.2Margin Generalization Bound We now derive the margin-based generalization bound for QNNs. The key idea is to bound the Rademacher complexity of the margin loss function class in terms of the quantum channel components. For a margin threshold훾 > 0, the margin loss function class is: ℱ 훾 = (휌, 푦) ↦ 푙 훾 (ℳ(ℎ(휌), 푦)) ∶ ℎ ∈ ℋ ,(3.15) whereℳis the margin operator (Equation (3.8)) and푙 훾 is the ramp loss (Equation (3.9)). Ap- plying the Rademacher generalization bound (Equation (3.6)) toℱ 훾 , we obtain: Pr (휌,푦)∼풟 [argmax 푖 ℎ(휌) 푖 ≠ 푦] ≤ ̂ 푅 훾 (ℎ) + 2ℜ(ℱ 훾 | 푆 ) + 3 √ ln(2/훿) 2푚 ,(3.16) 62 for allℎ ∈ ℋ, with probability at least1 − 훿. The central technical contribution is an analytic bound on the Rademacher complexity ℜ(ℱ 훾 | 푆 )in terms of the quantum channel components. This derivation proceeds in three steps: Step 1: Lipschitz continuity.We first establish the Lipschitz properties of the function class. The margin loss푙 훾 (ℳ(⋅, 푦))is2/훾-Lipschitz with respect to the푙 푝 norm for any푝 ≥ 1. The quantummeasurement function푔(푥) = 푥 † 퐸 푖 푥 푘 푖=1 (forpure stateinputs휌 = |푥⟩⟨푥|with푥 ∈ ℂ 푁 ) is2퐸-Lipschitz, where퐸 = √ ∑ 푖 ‖퐸 푖 ‖ 2 휎 and‖ ⋅ ‖ 휎 denotes the spectral norm. This result follows from each measurement function푔 푖 being2‖퐸 푖 ‖ 휎 -Lipschitz for normalized quantum states. Step 2: Covering number bound.Using the Lipschitz continuity, we reduce the covering number ofℱ 훾 restricted to the sample푆to the covering number of the set푈푋 ∶ 푈 ∈ 핌 QNN , where푋 ∈ ℂ 푁 ×푚 is the data matrix whose columns are the quantum state vectors: ln풩((ℱ 훾 )| 푆 , 휖, ‖ ⋅ ‖ 2 ) ≤ ⌈ 32푚푏 2 퐸 2 휖 2 훾 2 ⌉ ln4푁 2 ,(3.17) where푏is a distance bound such that‖푈 − 푈 ref ‖ 2,1 ≤ 푏for all푈 ∈ 핌 QNN , with푈 ref serving as a reference unitary matrix. This bound is derived using matrix covering techniques origi- nally introduced by Bartlett et al. [68] and Zhang [82], adapted to the complex-valued setting of quantum circuits. Here‖퐴‖ 2,1 denotes the sum of Euclidean norms of the columns of퐴. The parameter 푏 measures the radius of the accessible unitary family around the reference unitary in this matrix norm. Step 3: Dudley’s entropy integral.Finally, we bound the Rademacher complexity using Dud- ley’s entropy integral (Equation (3.7)) with the covering number bound from Step 2. This yields our main result: 63 Theorem 3.1(Margin Generalization Bound for QNNs).Consider an푛-qubit QNN with uni- tary푈 ∈ 핌 QNN and POVMs퐸 푖 푘 푖=1 for푘-class classification. Let푏be a distance bound such that‖푈 − 푈 ref ‖ 2,1 ≤ 푏for any푈 ∈ 핌 QNN . Then, for any훿 > 0and훾 > 0, with proba- bility at least1 − 훿over the random draw of an i.i.d. sample푆of size푚, the following holds for allℎ ∈ ℋ: 푅(ℎ) ≤ ̂ 푅 훾 (ℎ) + ̃ 푂 ⎛ ⎜ ⎜ ⎝ 푏 훾 √ 푛 푚 푘 ∑ 푖=1 ‖퐸 푖 ‖ 2 휎 + √ ln(1/훿) 푚 ⎞ ⎟ ⎟ ⎠ ,(3.18) where ̃ 푂hides logarithmic factors in푛,푏,퐸,훾, and푚. Interpreting the quantities in the bound.The parameter푛is the number of qubits, so푁 = 2 푛 sets the Hilbert-space dimension entering the covering argument. The number of classes is푘, the sample size is푚, and훿is the failure probability of the high-probability statement. The margin threshold훾controls the trade-off between empirical margin loss and complexity: larger 훾is more demanding on the training margins but reduces sensitivity to small confidence gaps. The term푏 = ‖푈 − 푈 ref ‖ 2,1 is a radius of the reachable unitary family around a fixed reference unitary. Finally,∑ 푖 ‖퐸 푖 ‖ 2 휎 measures how the POVM can amplify perturbations of the quantum state into perturbations of class probabilities. This bound has several important properties that distinguish it from the uniform bounds discussed in Section 3.2: Dependence on margin.Unlike parameter-count or norm-only uniform bounds, the margin bound contains an empirical margin term evaluated on the learned classifier. The first term ̂ 푅 훾 (ℎ)measures the fraction of training samples with margin at most훾, and the second term scales as1/훾. A model that achieves large margins on the training data will have a small ̂ 푅 훾 (ℎ) for a reasonably large훾, resulting in a tighter bound. In the randomized-label experiments below, memorizing models are reflected by smaller margins and looser margin bounds, which is 64 precisely the behavior missing from parameter-count bounds. Role of measurement operators.The bound depends on the spectral norms of the POVMs ∑ 푖 ‖퐸 푖 ‖ 2 휎 , which quantifies the influence of the measurement choice on generalization. This is a uniquely quantum feature: the generalization performance depends not only on the circuit architecture but also on how information is extracted from the quantum state. Corollary for projective measurements. When the POVMs are projective measurements—a common choice in QML models—the bound simplifies significantly: Corollary 3.2.Under the conditions of Theorem 3.1, suppose the POVMs퐸 푖 푘 푖=1 are pro- jective measurements, i.e.,퐸 2 푖 = 퐸 푖 and퐸 푖 퐸 푗 = 0for all푖 ≠ 푗. Then: 푅(ℎ) ≤ ̂ 푅 훾 (ℎ) + ̃ 푂 ⎛ ⎜ ⎝ 푏 훾 √ 푛푘 푚 + √ ln(1/훿) 푚 ⎞ ⎟ ⎠ .(3.19) For projective measurements,퐸 2 푖 = 퐸 푖 implies‖퐸 푖 ‖ 휎 = 1, so∑ 푖 ‖퐸 푖 ‖ 2 휎 = 푘. The bound is therefore independent of the specific measurement operators, depending only on the number of classes푘. 3.3.3Experimental Validation We now validate our theoretical framework through extensive experiments on the quantum phase recognition(QPR)task—aquantumdataclassificationproblemofsignificantphysicalrelevance. 65 Quantum Phase Recognition Quantum phase recognition (QPR) [83, 84] is a classification task aimed at identifying quantum phases of matter, a problem of fundamental importance in condensed matter physics [85, 86, 87]. We consider the generalized cluster Hamiltonian: 퐻(퐽 1 , 퐽 2 ) = 푛 ∑ 푗=1 (푍 푗 − 퐽 1 푋 푗 푋 푗+1 − 퐽 2 푋 푗−1 푍 푗 푋 푗+1 ) ,(3.20) where푋 푗 and푍 푗 are Pauli operators acting on site푗, and퐽 1 ,퐽 2 are tunable parameters control- ling the interaction strengths. Depending on the values of퐽 1 and퐽 2 , the ground state falls into one of four distinct phases: (1) ferromagnetic, (2) antiferromagnetic, (3) symmetry-protected topological (SPT), or (4) trivial. Determining the phase of a given ground state with unknown interaction parameters constitutes a four-class classification problem. We use 8-qubit QCNNs [32, 33] for this task, training on 20 data points evenly split across the four classes and evaluating test accuracy on 1,000 held-out samples. The small training set is deliberately chosen to explore the overfitting regime, following the methodology of Gil-Fuster et al. [67]. To test the models’ ability to distinguish genuine patterns from noise, we introduce label randomization at varying levels: 0% (pure labels), 50% (half random), and 100% (fully random). The QCNNs are trained using Adam with learning rate0.001and full-batch updates, train for up to 5,000 iterations with early stopping based on convergence of interval-averaged losses, and average the margin-distribution plots over 15 repetitions with different training samples. Margin Distribution and Generalization Performance Figure 3.1 presents the margin distributions of optimized QCNNs using Tukey box-and-whisker plots, along with the corresponding test accuracies and generalization gaps, for models with one, 66 Understanding Generalization in Quantum Machine Learning with Margins Figure 1.A Tukey box-and-whisker plot depicting the margin distributions of optimized 8-qubit Quantum Convolutional Neural Networks (QCNNs). The results for QCNNs with one, five, and nine layers are displayed, along with their corresponding test accuracies and generalization gaps indicated in the legend. QCNNs were trained for 4-class classification task aimed at quantum phase recognition (QPR). The experiment was performed with varying degrees of label noise: QPR dataset with pure labels (left), half randomly labelled dataset (middle), and full randomly labelled datasets (right). As the noise (randomization) level increases, the margin distributions tend to exhibit a more pronounced skew towards the left, indicating that a greater proportion of samples are classified with smaller margins. Notably, the margin distribution exhibits a strong positive correlation with test accuracy across all scenarios. 4.2. Predicting Generalization Gap: Parameters vs Margins In our earlier analyses, we identified a strong correlation be- tween margin distributions and generalization performance. Here, we demonstrate the effectiveness of margin-based metrics in estimating the generalization gap, highlighting their advantages over traditional uniform bounds. Caro et al.(2022) showed that the generalization gap in QML models can be estimated based on the number of train- able parameters. Specifically, they proved that, in the worst case, the generalization gap scales with the square root of the parameter count. Furthermore, when only a subset of parameters undergoes substantial change during training, the generalization gap scales with the square root of the number of effective parameters that undergo significant up- dates. Despite being uniform bounds, the parameter-based generalization approach has become a standard method for understanding generalization behavior in QML models. To illustrate the effectiveness of margins in estimating the generalization gap, we compare three margin-based met- rics with three parameter-based metrics. For margin-based metrics, we analyze the lower quartile, median, and mean values of the margin distribution. For parameter-based met- rics, we examine both the total number of parameters and the count of effective parameters that undergo substantial change during optimization. Specifically, we define effec- tive parameters using two thresholds,10 →1 and10 →2 , which represent the minimum change in a parameter (the rotation angle of parameterized quantum gates) for it to be consid- ered effective. Figure2depicts how the generalization gap, median margin, and effective parameters (with a10 →2 threshold) change in response to variations in the number of QCNN layers, the percentage of randomized labels, and the choice of varia- tional ansatz. Since margins are inversely correlated with generalization gap (as shown in Theorem3.1), the inverse of the median margin is plotted instead. Additionally, fol- lowing the results ofCaro et al.(2022), we plot the square root of the number of effective parameters. Figure2(a) shows how the generalization gap, inverse me- dian margin, and the square root of effective parameters vary with the number of QCNN layers—1, 3, 5, 7, and 9. The generalization gap reaches its peak at five layers before slightly decreasing. While the effective parameters increase monotonically with the number of layers, the median margin effectively captures the peak at five layers. Similarly, in Figure2(b), the margin effectively captures the rising generalization gap as the percentage of randomized labels increases, while the number of effective parameters show an inverse trend. In Figure2(c), we analyze three variational ans ̈ atze: QCNN, QCNN with shared parameters (Cong et al.,2019;Hur et al., 2022), and Strongly Entangling Layers (Bergholm et al., 2020), arranged in decreasing order of expressibility. In this case, the QCNN with shared parameters constrains the local parameterized unitaries within the convolutional layers to share identical parameter values. The margin correctly reflects the rising generalization gap as expressibility de- 5 Figure 3.1: Tukey box-and-whisker plots of margin distributions for optimized 8-qubit QCNNs on the QPR task. Results are shown for QCNNs with one, five, and nine layers, under varying degrees of label randomization: 0% (left), 50% (middle), and 100% (right). Each legend entry reports the test accuracy with the corresponding generalization gap in parentheses. As random- ization increases, the margin distributions shift to the left and the generalization gap widens, consistent with poorer generalization under the margin-sensitive bound in Equation (3.18). five, and nine QCNN layers. The results reveal a clear and consistent pattern across all configurations: •A right-skewed margin distribution (larger margins) is associated with higher test accu- racy and a tighter generalization bound, as indicated by the smaller right-hand side of Equation (3.18). •Increasinglabelrandomizationshiftsthemargindistributionstotheleft, reflectingthatthe model assigns lower confidence to randomly labeled data. This leftward shift is accom- panied by a larger measured generalization gap (reported in the legend) and a larger gen- eralization bound, correctly capturing the deteriorating generalization under label noise. •Whenlabelsarenotfullyrandomized, deeperQCNNs(morelayers)tendtoachievehigher test accuracy and rightward-shifted margin distributions, suggesting that increased ex- pressibility helps the model find hypotheses closer to the optimal one. 67 (a)(b)(c) Figure 2: Illustration of how the generalization gap, median of the margin distribution (a margin- based metric), and effective parameters with a10 −2 threshold (a parameter-based metric) vary with the number of layers, percentage of randomized labels, and the choice of variational ans ̈ atze. rics, we examine both the total number of parameters and the count of effective parameters that undergo substantial change during optimization. Specifically, we define effective parameters using two thresholds,10 −1 and10 −2 , which represent the minimum change in a parameter (the rotation angle of parameterized quantum gates) for it to be considered effective. Figure 2 depicts how the generalization gap, median margin, and effective parameters (with a10 −2 threshold) change in response to variations in the number of QCNN layers, the percentage of ran- domized labels, and the choice of variational ansatz. Since margins are inversely correlated with generalization gap (as shown in Equation (5)), the inverse of the median margin is plotted instead. Additionally, following the results of Caro et al. (2022), we plot the square root of the number of effective parameters. Figure 2(a) shows how the generalization gap, inverse median margin, and the square root of effec- tive parameters vary with the number of QCNN layers—1, 3, 5, 7, and 9. The generalization gap reaches its peaks at five layers before slightly decreasing. While the effective parameters increase monotonically with the number of layers, the median margin effectively captures the peak at five layers. Similarly, in Figure 2(b), the margin effectively captures the rising generalization gap as the per- centage of randomized labels increases, while the effective parameters show an inverse trend. In Figure 2(c), we analyze three variational ans ̈ atze, QCNN, QCNN with shared parameters (Cong et al., 2019; Hur et al., 2022), and Strongly Entangling Layers (Bergholm et al., 2020), arranged in decreasing order of expressibility. In this case, the QCNN with shared parameters constrains the local parameterized unitaries within the convolutional layers to share identical parameter values. The margin correctly reflects the rising generalization gap as expressibility decreases, while the effective parameters show an inverse trend. In summary, the margin reliably captures variations in the generalization gap across different hy- perparameters, whereas effective parameters fails to do so and sometimes show the opposite trend. Note that the median margin is used instead of the lower quartile (Q1) to avoid infinite inverse mar- gins when Q1 is zero, a scenario that occasionally arises when the labels are fully randomized and the model employs an inexpressive variational ansatz. Effective parameters are shown instead of the total number of parameters, as the latter remains constant despite changes in randomized label percentages, unlike the generalization gap. Additional comparisons—margin mean and Q1 versus total and effective parameters—without label noise are provided in Appendix A.2, Figure 6. Thus far, we have compared how margins and the number of parameters capture generalization by varying one hyperparameter at a time while keeping the others fixed. Now, we present a more comprehensive comparison with all hyperparameters varied simultaneously. More specifically, for a given set of hyperparametersλ, we examine the correlation between the generalization gapg(λ)and the corresponding metricμ(λ). Here, as before, the metrics can be either margin-based or parameter- based. To assess correlation, we analyzed two different evaluation methods: mutual information and Kendall rank correlation coefficient. Both methods (or their variants) have been used to assess generalization capacity in classical deep learning (Jiang et al., 2019; 2020; Dziugaite et al., 2020). 6 Figure 3.2: Comparison of the generalization gap, median margin (a margin-based metric), and effective parameters with threshold10 −2 (a parameter-based metric) as a function of: (a) the number of QCNN layers, (b) the percentage of randomized labels, and (c) the choice of variational ansatz. Since margins are inversely correlated with the generalization gap (Equa- tion (3.18)), the inverse median margin is plotted. The margin more closely tracks the gen- eralization gap in these experiments, while effective parameters show inconsistent or opposite trends. These observations support the margin distribution as an informative indicator of generaliza- tion performance in the studied settings, even in scenarios where the model can overfit random labels—precisely the regime where uniform bounds fail. Predicting the Generalization Gap: Margins vs. Parameters Akeyquestioniswhethermargin-basedmetricsofferapracticaladvantageoverparameter-based metrics for predicting the generalization gap. To address this, we compare three margin-based metrics (lower quartile, median, and mean of the margin distribution) against three parameter- based metrics (total parameter count, and effective parameters at thresholds10 −1 and10 −2 ). Figure 3.2 compares these metrics across three experimental axes: (a) varying the number of QCNN layers, (b) varying the percentage of randomized labels, and (c) varying the variational ansatz (QCNN, QCNN with shared parameters [32, 33], and Strongly Entangling Layers [88]). The results show that: 68 •The margin reliably captures variations in the generalization gap across all three hyperpa- rameters. For example, the generalization gap peaks at five layers before slightly decreas- ing, and the inverse median margin accurately tracks this non-monotonic behavior. •Effective parameters fail to capture the generalization gap in several settings and some- times show the opposite trend. For instance, the effective parameters increase monoton- ically with the number of layers, failing to capture the peak in generalization gap at five layers. For a more comprehensive comparison, we compute both themutual informationand the Kendall rank correlation coefficientbetween each metric and the generalization gap across all hyperparameter configurations simultaneously. The mutual information퐼(푔; 휇)quantifies the reduction in uncertainty about the generaliza- tion gap푔given the metric휇. The Kendall rank correlation coefficient휏measures the ordinal association between the generalization gap and the metric—specifically, whether the ranking of models by the metric agrees with their ranking by generalization performance: 휏 퐺,푀 = 1 푛(푛 − 1) ∑ 푖<푗 [1 +sgn(푔 푖 − 푔 푗 )sgn(휇 푖 − 휇 푗 )] ,(3.21) where퐺 = [푔 1 , ... , 푔 푛 ]and푀 = [휇 1 , ... , 휇 푛 ]are lists of generalization gaps and corresponding metrics. This coefficient ranges from0to1, where휏 = 1indicates perfect agreement between the two rankings (all pairs concordant), and휏 = 0perfect disagreement (all pairs discordant). Figure 3.3 shows that margin-based metrics are more strongly correlated with the general- ization gap than parameter-based metrics in both evaluation methods. This supports the use of margin-based quantities not only as theoretically motivated terms in Theorem 3.1, but also as practical diagnostics for evaluating the generalization performance of QML models. 69 margin (Q1) margin (median) margin (mean) paramseff. params (10 1 ) eff. params (10 2 ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Correlation Values Mutual Information Kendall's Tau Figure 3: Comparative analysis of mutual information (solid) and Kendall rank correlation coeffi- cients (shaded) between the generalization gap and various metrics. The first three columns represent margin-based metrics, while the last three columns represent parameter-based metrics. In mutual information analysis, we treat the hyperparameterλas a random vector, withg(λ) andμ(λ)as functions of this vector. Mutual information, denoted byI(g;μ), is defined as I(g;μ) =E h log P(g,μ) P(g)P(μ) i ,which simplifies toH(g)−H(g|μ). Here,H(g)represents the entropy ofg, quantifying the uncertainty of the generalization gap, whileH(g|μ)is the conditional entropy ofggivenμ, indicating the remaining uncertainty about the generalization gap givenμ. Thus, higher mutual informationI(g;μ)suggests thatμprovides more information aboutg, reducing the uncertainty associated withg. In contrast, the Kendall rank correlation coefficientτmeasures the strength and direction of the as- sociation between two variables. For pairs of generalization gap and metric values,(g(λ 1 ),μ(λ 1 )) and(g(λ 2 ),μ(λ 2 )), we expect the metric to accurately capture the ranking of generalization perfor- mance. Specifically, ifg(λ 1 )< g(λ 2 ), thenμ(λ 1 )should also be less thanμ(λ 2 ). This implies that the metricμeffectively predicts relative generalization, with lowerμvalues corresponding to better generalization (lowerg). Consider lists ofngeneralization gaps and corresponding metrics, denoted asG= [g 1 ,...,g n ]andM= [μ 1 ,...,μ n ], where eachg i andμ i corresponds tog(λ i )andμ(λ i ), respectively. The Kendall rank correlation coefficient betweenGandMis given by τ(G,M) = 1 2n(n−1) X i<j [1 +sgn(g i −g j )sgn(μ i −μ j )].(7) This coefficient ranges from 0 to 1, whereτ= 1indicates perfect agreement between the rankings of gandμ(all pairs are concordant), andτ= 0indicates perfect disagreement (all pairs are discordant). Figure 3 shows the mutual information (solid) and Kendall rank correlation coefficient (shaded) between the generalization gap and various metrics. In both evaluation methods, margin-based metrics demonstrate stronger correlations than parameter-based metrics, indicating that margins are more effective in evaluating the generalization performance of QML models. Consistent with previous analyses, we used inverse margin values to measure correlation, as larger margins correspond to smaller generalization gaps. Further details on this experiment can be found in Appendix A.2. 4MARGINS ANDQUANTUMEMBEDDING Up to this point, we have focused on scenarios involving the classification ofn-qubit quantum data using QML models. Another important application of QML is the classification of classical data. To work with classical data in a quantum framework, thed-dimensional data,x∈R d , must be mapped to ann-qubit quantum state, represented by a density matrixρ(x)∈C 2 n ×2 n . This process, known asquantum embedding, typically involves applying a quantum embedding circuitU emb to the ground state, resulting inρ(x) =U emb (x)(|0⟩⟨0|) ⊗n U † emb (x). 7 Figure3.3: Mutualinformation(solid)andKendallrankcorrelationcoefficient(shaded)between the generalization gap and various metrics. The first three columns represent margin-based met- rics (lower quartile, median, and mean of the margin distribution), while the last three represent parameter-based metrics (total parameters, effective parameters at thresholds10 −1 and10 −2 ). Margin-based metrics show stronger correlations with the generalization gap than parameter- based metrics in these experiments. 3.3.4Connection to Quantum State Discrimination We now establish a connection between the margin and quantum state discrimination, bridging our generalization analysis with the quantum embedding theory developed in Chapter 2. Marginmeanandtracedistance.For binary classification, the margin mean can be expressed as: 휇 mean = 1 푚 ∑ 푖 2Tr(푈휌(푥 푖 )푈 † 퐸 푦 푖 ) − 1,(3.22) where푈is the optimized unitary and퐸 푦 푖 is the POVM element corresponding to class푦 푖 . Defining the effective POVM퐸 ∗ ±1 = 푈 † 퐸 ±1 푈and the class-averaged density matrices휌 ± = 1 푚 ± ∑ 푖 휌(푥 ± 푖 ), the margin mean becomes: 휇 mean = 2Tr(푝 + 휌 + 퐸 ∗ +1 ) + 2Tr(푝 − 휌 − 퐸 ∗ −1 ) − 1,(3.23) 70 where푝 ± are the class priors. Using퐸 ∗ +1 + 퐸 ∗ −1 = 퐼, the success-probability term symmetrizes as: Tr(푝 + 휌 + 퐸 ∗ +1 ) +Tr(푝 − 휌 − 퐸 ∗ −1 ) = 1 2 + 1 2 Tr [(푝 + 휌 + − 푝 − 휌 − )(퐸 ∗ +1 − 퐸 ∗ −1 )] .(3.24) Maximizing over binary POVMs퐸 ∗ ±1 recovers theHelstrom measurementfor discriminating 푝 + 휌 + from푝 − 휌 − [50], which yields: 휇 mean ≤ 2퐷 tr (푝 + 휌 + , 푝 − 휌 − ),(3.25) where퐷 tr is the trace distance, as in Chapter 2, and the inequality becomes an equality if and only if퐸 ∗ ±1 is the Helstrom measurement. Thus, the trace distance serves as an upper bound on the achievable margin mean. Quantum embeddingsthatproduce alargetracedistancebetweenclass ensembles—suchas thoseobtained through NQE (Chapter 2)—raise the ceiling on the margin, which can lead to tighter bounds in Theorem 3.1. This provides a theoretical explanation for the empirically observed relationship between large trace distances and improved generalization [48, 89, 90]: these embeddings make larger classification margins attainable. Experimental validation.We empirically examine this connection by comparing QCNNs with three different quantum embedding schemes on classical datasets (MNIST [91], Fashion- MNIST [92], and Kuzushiji-MNIST [93]): (1) a fixed Z feature map, (2) trainable quantum embedding (TQE) [89], and (3) Neural Quantum Embedding (NQE) [48]. These three schemes produce progressively increasing initial trace distances. Figure3.4supportsthetheoreticalprediction: thetracedistance(reddots)increasesprogres- sively from the fixed embedding to TQE to NQE, accompanied by a corresponding increase in 71 Understanding Generalization in Quantum Machine Learning with Margins Figure 4.A Tukey box-and-whisker plot illustrating the margin distributions of optimized 8-qubit Quantum Convolutional Neural Networks (QCNNs). The plot shows results for QCNNs using fixed quantum embedding (left), trainable quantum embedding (middle), and neural quantum embedding (right). The QCNNs were trained on a binary classification task using the MNIST (bottom), Fashion-MNIST (middle), and Kuzushiji-MNIST (top) datasets. In addition to the margin distributions, the mean of the margins is indicated by a black cross, and the trace distance between ensemble quantum states is marked by a red circle. tum Embedding (NQE), which leverages classical neural networks to optimize quantum embeddings. Their study highlighted the inherent limitations of TQE in maximizing trace distance and empirically demonstrated that NQE can effectively identify quantum embeddings with a large initial trace distance, outperforming the capabilities of TQE. While both TQE and NQE are theoretically designed to im- prove the lower bound of sample training loss, they have also been shown to significantly enhance test accuracy, thereby improving generalization performance. Although a posi- tive relationship between a large initial trace distance and improved generalization has been observed empirically, a theoretical explanation for this relationship was previously missing. Margin generalization offers this missing theoreti- cal basis, as the trace distance serves as an upper bound on the margin mean. In binary classification, the margin mean simplifies to ̄μ= 2 N ! i Tr(Uω(x i )U † E y i )→1, whereUdenotes the optimized unitary of QNN after the training. By defining E → ±1 =U † E ±1 U, the margin mean can be reformulated as, ̄μ= 2 N " # N + $ i=1 Tr(ω(x + i )E → +1 )+ N → $ i=1 Tr(ω(x ↑ i )E → ↑1 ) % & →1 = 2Tr(p + ω + E → +1 ) + 2Tr(p ↑ ω ↑ E → ↑1 )→1 ↑D tr (p + ω + ,p ↑ ω ↑ ),(2) where the final inequality becomes equality if and only if E → ±1 corresponds to the Helstrom measurement (see Ap- pendixA.4). In other words, TQE and NQE enhance the model’s generalization capability by identifying quantum embeddings with a large trace distance, thereby increasing the upper bound of the margin mean. This margin-based perspective on generalization enables a systematic optimiza- tion of the generalization performance of QML models by selecting quantum embeddings with large trace distances. Figure4presents the classification of MNIST (LeCun et al.,2010), Fashion-MNIST (Xiao et al.,2017), and Kuzushiji-MNIST (Clanuwat et al.,2018) datasets using 8-qubit QCNN with various quantum embedding schemes. The comparison highlights how the initial trace distance af- fects the margin distribution and the model’s generalization performance. Three quantum embedding schemes are em- ployed to vary the initial trace distance: the fixed quantum embedding (Z Feature Map introduced byHavl ́ ıcek et al. (2019)), TQE, and NQE. As anticipated from previous research, we observed a pro- gressively increasing trace distance (indicated by a red dot) in the order of fixed quantum embedding, TQE, and NQE. This increase in trace distance corresponds to higher margin mean values (indicated by a black cross). Larger margin means, coupled with right-skewed margin distributions, are associated with improved test accuracies, aligning with the principles of margin-based generalization. In summary, using an optimal quantum embedding with a large initial trace distance not only reduces sample training loss but also results in larger margins and enhanced generalization. Experimental details are provided in AppendixA.2. 6. Discussion In this work, we proposed a margin-based framework to bet- ter understand generalization in QML models. We began by 8 Figure 3.4: Tukey box-and-whisker plots of margin distributions for optimized 8-qubit QCNNs on binary classification tasks using three quantum embedding schemes: fixed Z feature map (left), trainable quantum embedding (middle), and Neural Quantum Embedding (right). Results are shown for MNIST (bottom), Fashion-MNIST (middle), and Kuzushiji-MNIST (top). Each legend entry reports the test accuracy with the corresponding generalization gap in parentheses. The black cross indicates the margin mean, and the red circle indicates the unweighted trace distancebetweenclassensembles. Highertracedistancescorrespondtolargerattainablemargins and higher test accuracies—while the generalization gap remains small throughout—in these experiments, consistent with the connection established in Equation (3.23). the margin mean (black crosses) and rightward shift of the margin distribution. Larger margin means, coupled with right-skewed margin distributions, are associated with higher test accu- racies while the generalization gap stays small across all three embeddings, aligning with the margin-based generalization framework. This result connects the training-loss and generalization perspectives: NQE (Chapter 2) re- duces the achievable training loss by increasing the trace distance, and the same increased trace distance can support larger classification margins. The margin-based perspective thus links state distinguishability and generalization through the quantum embedding. 72 3.4Summary In this chapter, we developed a margin-based framework for understanding generalization in quantum machine learning. Our main contributions are: 1.Margin generalization bound for QNNs(Theorem 3.1): We established a generalization bound for multiclass classification with QNNs that depends on the margin distribution. Unlike parameter-count bounds that are often too vacuous for the empirical memoriza- tion comparisons studied here, the margin bound yields tighter estimates when the model classifies training data with large margins. 2.Empirical comparison of margin-based metrics: Through experiments on the quantum phase recognition task, we found that margin-based metrics (lower quartile, median, and mean of the margin distribution) are stronger predictors of generalization performance than parameter-based metrics in the tested settings, as measured by both mutual informa- tion and the Kendall rank correlation coefficient. 3.Connection to quantum state discrimination: We showed that the margin mean is up- per bounded by the trace distance between class ensembles (Equation (3.25)). This es- tablishes a theoretical link between the generalization framework of this chapter and the trace-distanceframeworkofChapter 2: embeddings withlargetracedistancesmakelarger margins attainable. In Part I, we reverse the direction and explore how artificial intelligence techniques can ad- dress fundamental challenges in quantum computing, beginning with neural decoders for quan- tum error correction in Chapter 4. 73 Part I AI for Quantum 74 CHAPTER 4 NEURAL DECODERS FOR QUANTUM ERROR CORRECTION The results in this chapter are based onScalable Neural Decoders for Practical Real-Time Quantum Error Correction[94]. The numerical data underlying the real-time decoding results for the Transformer and Mamba decoders are publicly available in the GitHub repos- itory athttps://github.com/qDNA-yonsei/NeuralDecoder_v1. InPartIofthisthesis, weexploredhowquantumcomputingcanenhancemachinelearning— from neural quantum embeddings (Chapter 2) to generalization theory for quantum neural net- works (Chapter 3). In Part I, we reverse this direction and ask:how can artificial intelligence help solve quantum problems?The two Part I chapters examine this question at complemen- tary levels: Chapter 4 studies real-time quantum error correction, while Chapter 5 studies neural representations and optimizers for quantum many-body ground states. A central bottleneck on the path to fault-tolerant quantum computation isquantum error correction decoding—the classical inference task of identifying and correcting er- rors from noisy syndrome measurements. As quantum processors scale to hundreds and eventually thousands of physical qubits, decoders must provide throughput comparable to the syndrome-generation rate while maintaining high accuracy, so that the classical backlog remains bounded [95]. This chapter presents a neural decoder based on the Mamba architecture [96], a 75 state-space model that replaces the attention mechanism of Transformer-based decoders with a selective scan of풪(푑 2 )complexity, where푑is the code distance. The experiments show that this architectural choice preserves comparable decoding accuracy relative to the reproduced Transformer baseline in the tested settings while improving the asymptotic scaling relative to the풪(푑 4 )attention mechanism used by Transformer decoders such as AlphaQubit [97]. Under the simulated latency model introduced in this chapter, this improved scaling yields a higher finite-size effective threshold. The chapter is organized as follows. Section 4.1 introduces the basics of quantum error cor- rection, focusing on surface codes and the computational challenges of decoding. Section 4.2 surveys neural network approaches to decoding, with particular attention to AlphaQubit and its limitations. Section 4.3 presents our Mamba-based decoder—its architecture, training proce- dure, and experimental results on both hardware data and simulated real-time scenarios. Sec- tion 4.4 summarizes the chapter. 4.1Introduction to Quantum Error Correction Quantum information is inherently fragile: interactions with the environment, imperfect gate operations, and faulty measurements introduce errors that accumulate rapidly during a computa- tion. Unlikeclassicalbits, whichcanbecopiedandchecked, quantumstatescannotbecloned[1], making direct error detection impossible without carefully designed encoding schemes. Quantum error correction (QEC) overcomes this obstacle by encoding alogicalqubit re- dundantly across manyphysicalqubits, so that errors can be detected and corrected without de- stroying the encoded quantum information [98, 99, 100, 101]. The field has matured from early theoretical proposals to experimental demonstrations of error suppression in regimes relevant to fault tolerance [102], yet a critical classical bottleneck remains: thedecoderthat interprets syn- 76 drome measurements and prescribes corrections must keep pace with syndrome production—a challenge that grows with code size. 4.1.1Surface Codes Among the many families of quantum error-correcting codes, thesurface code[103, 104, 105] stands out as the leading candidate for near-term fault-tolerant quantum computing. Its appeal rests on three properties: (i) it requires only nearest-neighbor interactions on a two-dimensional lattice; (i) under standard circuit-noise assumptions it has one of the highest known fault- tolerance thresholds, approximately1%per physical gate [104, 105]; and (i) syndrome ex- traction requires only local measurements. Stabilizer formalism.The surface code is defined within thestabilizer formalism[101]. An [[푛, 푘, 푑]]stabilizer code encodes푘logical qubits into푛physical qubits and can detect any error acting on fewer than푑qubits, where푑is thecode distance. The code is specified by an Abelian stabilizer group풮 = ⟨푔 1 , 푔 2 , ... , 푔 푛−푘 ⟩ ⊂ 풫 푛 , where풫 푛 is the푛-qubit Pauli group. The codespaceis the simultaneous+1eigenspace of all stabilizer generators: 풞 = ∣휓⟩ ∈ (ℂ 2 ) ⊗푛 ∶ 푔 푖 ∣휓⟩ = ∣휓⟩ ∀ 푖.(4.1) An error퐸 ∈ 풫 푛 is detectable if it anticommutes with at least one stabilizergenerator, producing a−1measurement outcome that flags the error without revealing the encoded information. Rotated surface code.Throughout this chapter we focus on therotated surface code[106], the variant used in the experimental benchmarks below and in current superconducting-qubit demonstrations. A distance-푑rotated patch encodes a single logical qubit using2푑 2 − 1physical qubits:푑 2 data qubits and푑 2 − 1measurement (ancilla) qubits that mediate stabilizer readout. 77 Its stabilizers come in two types,푋-type and푍-type.Logical operatorsare Pauli strings that commute with all stabilizers but are not themselves in풮. On the surface code, logical ̄ 푋( ̄ 푍) is a chain of푋(푍) operators spanning the lattice from one boundary to the opposite boundary. Syndrome measurement.In a QEC cycle, every stabilizer generator is measured once, yield- ing a binary outcome푠 푖 ∈ 0, 1for each generator. The collections= (푠 1 , 푠 2 , ... , 푠 푛−푘 )is called thesyndrome. In an ideal setting,푠 푖 = 0for all푖indicates no detectable error. In practice, syndrome measurements are themselves noisy—ancilla qubits and measurement gates are imperfect. To reliably extract the syndrome, the measurement is repeated푑times (matching the code distance), creating a three-dimensionalspacetimevolume of syndrome data. Adetection eventis defined as a change in a stabilizer outcome between consecutive rounds: 훿 푖,푡 = 푠 푖,푡 ⊕ 푠 푖,푡−1 ,(4.2) where푠 푖,푡 is the outcome of stabilizer푖at round푡and⊕denotes addition modulo 2. Detection events, rather than raw syndromes, form the input to modern decoders because they are invariant to the initial syndrome configuration and directly encode the locations of faults in spacetime. 4.1.2The Decoding Problem Given a syndrome historys= 푠 푖,푡 , the task of thedecoderis to infer a correction operatorℛ that returns the system to the codespace. Because many different physical errors can produce the same syndrome (they differ by elements of풮), the decoder only needs to identify the correct equivalence classof the error, not the exact physical error. 78 EmbeddingEmbeddingEmbedding RNN coreRNN coreRNN core EmbeddingEmbedding RNN coreRNN coreReadout 푃 퐿 푑 0 푆 0 푆 1 푆 2 푆 푁 푆 푁−1 Syndrome Mixer Gated dense block Scatter To 2D Gatter To 2D Dilated Convs Mixer Τ 푑 푙−1 +푆 푙−1 2 푑 푙 ⋯ ⋯ M X 푡=0 푡=1 푡=2푡=푁−1 푡=푁 h 0 Figure 1: The overall recurrent architecture of the neural decoder. At each cyclen, stabilizer mea- surement embeddingsS n are processed by an RNN core, updating the decoder’s hidden state from h n toh n+1 . The final state is passed to a readout network to predictP L , the probability of a logical error. Further details on the architecture and notations are provided in the Methods section. ing [4], and correlated matching [5]—can correct errors quickly, but do not fully capture complex correlations in the syndrome data. Meanwhile, more exhaustive approaches like tensor-network (TN) decoders [6] can provide highly accurate solutions but scale poorly with code distance, un- less strong bond-dimension truncations are applied, which in turn degrade accuracy. To overcome these trade-offs, machine learning techniques have recently emerged as a promising alternative for decoding [7–28]. Among these,AlphaQubit[29] represents a major milestone, demonstrating state-of-the-art decod- ing performance in memory experiments. Leveraging a recurrent Transformer-based architecture, AlphaQubit learns to predict logical errors from syndrome data, surpassing existing decoders in both accuracy and adaptability. Through a two-stage training process—pretraining on synthetic noise models followed by fine-tuning on real experimental data—the model adapts to complex, hardware- specific noise sources, such as cross-talk, leakage, and soft readouts. This ability to directly process soft information and account for non-standard error channels like leakage provides a distinct advan- tage over traditional decoders, which are often restricted to binary syndrome outcomes [30]. In memory experiments using real hardware data from the Sycamore processor for distance-3 and distance-5 surface codes, AlphaQubit outperformed all other decoders, including the highly accurate TN decoder. For larger codes up to distance 11, which are inaccessible on current devices, evalua- tions were performed on simulated data. In these simulations, AlphaQubit continued to outperform correlated matching decoders. This success underscores the power of data-driven adaptation and demonstrates how machine learning can push QEC decoding beyond the limits of human-designed algorithms. However, AlphaQubit also exposes a fundamental limitation: the computational burden of its Transformer-based architecture [31]. The self-attention mechanism at the core of the Transformer scales asO(n 2 )with input length, leading to an overallO(d 4 )for surface codes, as the syndrom bitstring length scales withd 2 →1. While AlphaQubit performs well in memory experiments, where decoding is deferred until the end, itsO(d 4 )complexity makes it impractically slow for real-time error correction. Superconducting quantum processors require immediate feedback corrections on a microsecond-scale threshold. For distance-5 codes, AlphaQubit’s latency is already an order of magnitude slower than this requirement, and this speed penalty only worsens as the code distance in- creases. This bottleneck highlights the need for an alternative architecture that achieves comparable accuracy while drastically improving inference speed. In this work, we introduce and analyze theMamba decoder, a novel neural decoder architecture de- signed to address the computational bottlenecks of Transformer-based models like AlphaQubit. Our decoder is built upon Mamba [32], a state-space model that replaces the explicit attention mechanism with selective state updates. By maintaining a continuous representation of historical context in the decoder’s hidden states, Mamba avoids full pairwise interactions and reduces computational com- 2 Figure 4.1: Spacetime view of surface-code syndrome extraction for recurrent neural decoding. Each quantum error-correction (QEC) cycle produces stabilizer measurements and detection events, which are embedded as spatial features, processed recurrently over time, and decoded into a logical-error probability. Maximum likelihood decoding.The optimal decoding strategy ismaximum likelihood(ML) decoding, which selects the equivalence class with the highest total probability: ̂푐 =argmax 푐∈0,1 ∑ 퐸∈푐 Pr(퐸 ∣s),(4.3) where푐labels the equivalence class (trivial or logical error). ML decoding is #P-hard in gen- eral [107], motivating the development of efficient approximate decoders. Decoder taxonomy.A variety of decoding algorithms have been developed, broadly catego- rized as follows: •Matching-based decoders.Minimum-weight perfect matching (MWPM) [108, 109] maps the syndrome to a graph problem and finds the minimum-weight set of edges connecting detection events. MWPM runs in풪(푛 3 )worst-case time and has been the workhorse decoder for surface codes. Standard matching formulations are most direct 79 for independent graphlike error models. Extensions such as correlated MWPM [110] and belief matching [109] incorporate correlations between푋and푍errors at the cost of increased complexity. •Tensor network decoders.By contracting a tensor network representation of the error model, these decoders can approximate ML decoding with tunable accuracy [111, 107]. However, the bond dimension required for accuracy grows with code distance, limiting scalability. •Neural decoders.Machine learning models trained on syndrome–error pairs can learn complex noise correlations directly from data [112, 113, 97]. Their accuracy can match or exceed traditional decoders, but inference latency is a concern for real-time operation (discussed in Section 4.2). Real-time constraints.In a fault-tolerant quantum computer, error correction operates con- tinuously as each QEC cycle produces new syndrome information. For memory experiments and Clifford operations, Pauli-frame updates can often be deferred, so the decoder does not al- ways need to finish strictly before the next cycle. Nevertheless, a scalable processor still requires decoding throughput comparable to the syndrome-generation rate so that the backlog remains bounded [95]. Latency becomes especially consequential when classical feedforward is needed, and delayed corrections can be modeled as an additional effective noise source. This latency constraint makes decoding speed an essential companion to accuracy for practical QEC. Data-drivendecoding.A recent paradigm for decoder design is thedata-drivenapproach [97, 114]: a neural decoder is pretrained on large synthetic datasets generated from a calibrated noise model, then fine-tuned on experimental hardware data. This two-stage pipeline allows the de- 80 coder to learn the specific noise characteristics of the target device—including cross-talk, leak- age, and measurement-induced errors—without requiring an explicit analytical noise model. Performance metrics.The quality of a decoder is measured by thelogical error rate(LER), denoted휀, which is the probability of a logical error per QEC cycle. The LER is estimated by fitting the measured logical fidelity퐹(푛)after푛cycles to the model log퐹(푛) =log퐹 0 + 푛log(1 − 2휀),(4.4) where퐹 0 accounts for state preparation and measurement errors [97]. A good code–decoder pair is usually characterized by anerror threshold푝 th : the maximum physical error rate below which increasing the code distance푑exponentially suppresses the LER in an asymptotic setting. Theerror suppression ratioΛ = 휀(푑)/휀(푑 + 2)quantifies how much benefit each increase in code distance provides. 4.2Neural Decoders The idea of using neural networks for QEC decoding dates back to 2017, when the earliest de- coders ranged from generative Boltzmann machines [112] to feedforward networks for small codes [115]. Since then, the field has grown rapidly, with architectures ranging from simple multilayer perceptrons to recurrent transformers. For a broader review of machine learning ap- proaches to decoding topological quantum codes, see Lee et al. [116]. This section surveys the neural decoder literature (Section 4.2.1) and then discusses AlphaQubit (Section 4.2.2), a high-performing Transformer-based decoder that motivates our Mamba-based approach. 81 4.2.1Overview of Neural Decoder Literature Early feedforward approaches.Torlai and Melko [112] introduced one of the first neural decoders, using a restricted Boltzmann machine to decode the toric code, demonstrating that unsupervised generative models could learn the error distribution from syndrome data. Con- currently, Krastanov and Jiang [113] proposed a deep neural network probabilistic decoder for stabilizer codes that could handle depolarizing noise. Varsamopoulos et al. [115] systematically studied feedforward networks for small surface codes, followed by comparisons across architec- tures [117] and distributed decoder designs [118]. Recurrent and specialized architectures.Baireuther et al. [119] applied recurrent neural net- works to decode correlated qubit errors in a topological code, showing that temporal correla- tions across QEC cycles could be exploited. This was extended to color codes with circuit-level noise [120]. Chamberland and Ronagh [121] designed deep neural decoders specifically for near-term fault-tolerant experiments, while Maskara et al. [122] demonstrated the versatility of neural-network decoding for various topological codes. Ni [123] scaled neural decoders to large- distance 2D toric codes, and Liu and Poulin [124] introduced neural belief-propagation decoders that combined neural networks with message-passing algorithms. Modern approaches.More recently, Gicev et al. [125] developed a scalable artificial neu- ral network syndrome decoder optimized for speed. Lange et al. [114] applied graph neural networks (GNNs) to QEC decoding, exploiting the natural graph structure of stabilizer codes. Sweke et al. [126] formulated decoding as a reinforcement learning problem, and Cao et al. [127] used generative pre-trained transformers for decoding. Egorov et al. [128] proposed equivariant neural decoders that respect the symmetries of the code, improving sample efficiency. Fur- ther work has explored hardware cost-performance tradeoffs [129], symmetry-aware architec- 82 tures [130], and deep reinforcement-learning decoders [131]. Despite this progress, a persistent tension remains betweenaccuracyandlatency. Highly accurate neural decoders tend to use large models with expensive inference, while lightweight decoders sacrifice accuracy for speed. The breakthrough of AlphaQubit showed that neural decoders can be highly competitive with strong traditional decoders on important benchmarks, but at a computational cost that challenges real-time operation. 4.2.2AlphaQubit AlphaQubit [97] is a recurrent Transformer-based neural decoder developed by Google Deep- Mind and Google Quantum AI. It represents a landmark in neural decoding: it demonstrated that a data-driven neural decoder can achieve highly competitive performance on real hardware data from Google’s Sycamore processor. Architecture.AlphaQubit follows a recurrent encoder architecture with three components (cf. Figure 4.2): 1.Stabilizer Embedder:At each QEC cycle푛, raw stabilizer measurements, detection events, and optionally analog I/Q readout values and leakage flags are embedded into 푑 model -dimensional feature vectorss 푛,푖 for each stabilizer푖, via linear projections com- bined with positional encodings and processed through a small ResNet. 2.RNNCore:Thesequenceofstabilizerembeddings푆 푛 = s 푛,푖 |풮| 푖=1 isprocessedrecurrently across QEC cycles. At each cycle, the core updates a hidden state: ℎ 푛+1 = 푓 core (ℎ 푛 , 푆 푛 ),(4.5) where푓 core consists of multipleSyndrome Mixerlayers. In AlphaQubit, each Syndrome 83 Mixer usesmulti-head attention(MHA) [16] as the core mixing operation, followed by gated dense layers and dilated 2D convolutions. 3.Readout Network:After the final cycle, the hidden stateℎ 푁 is scattered onto a 2D grid, converted to the data qubit lattice via convolutions, and passed through a deep ResNet to produce the logical error probability푃 퐿 . The multi-head attention in the Syndrome Mixer computes: Attention(푄, 퐾, 푉 ) =softmax ⎛ ⎜ ⎜ ⎝ 푄퐾 ⊤ √ 푑 푘 ⎞ ⎟ ⎟ ⎠ 푉 ,(4.6) where푄,퐾,푉 ∈ ℝ 푛 푠 ×푑 푘 are the query, key, and value matrices derived from the푛 푠 = |풮| stabilizer embeddings, and푑 푘 is the head dimension. This allows the model to learn arbitrary pairwise correlations between stabilizers. Training.AlphaQubit employs a two-stage data-driven training pipeline. Inpretraining, the model is trained on large synthetic syndrome datasets generated from a detector error model (DEM) calibrated using cross-entropy benchmarking (XEB) data from the Sycamore processor. Infine-tuning, the pretrained model is then adapted on real experimental data from Sycamore memory experiments, allowing it to learn hardware-specific noise patterns including cross-talk, leakage, and correlated errors. Results on Sycamore data.On distance-3 and distance-5 surface codes, AlphaQubit achieved logical error rates competitive with the strongest tested decoders:휀 = 2.901 × 10 −2 at distance 3 and휀 = 2.748 × 10 −2 at distance 5, corresponding to an error suppression ratio ofΛ = 1.056. The full AlphaQubit result outperformed both matching-based and tensor-network decoders on the Sycamore data. 84 The latency bottleneck.Despite its accuracy, AlphaQubit has a fundamental scalability lim- itation rooted in its attention mechanism. The self-attention operation in Equation (4.6) com- putes pairwise interactions between all푛 푠 stabilizers, with complexity풪(푛 2 푠 )per layer. For a distance-푑surface code, the number of stabilizers scales as푛 푠 ∝ 푑 2 , giving an overall per-cycle complexity of풪(푑 4 ). AlphaQubit’s latency is already approximately40 휇s at distance 9—an order of magnitude slower than the∼ 1 휇s QEC cycle time of superconducting qubits. This latency bottleneck moti- vates the central contribution of this chapter: replacing the attention mechanism with a Mamba- based selective state-space model that achieves풪(푑 2 )complexity—a quadratic improvement— while preserving comparable accuracy. 4.3Mamba Decoder We now present the Mamba decoder [94], a neural decoder that replaces the multi-head atten- tion in AlphaQubit’s Syndrome Mixer with a Mamba module—a selective state-space model with linear complexity. We first introduce state-space models and the Mamba architecture (Sec- tion4.3.1), thendescribethedecoderarchitectureandtraining(Section4.3.2), andfinallypresent experimental results (Section 4.3.3). 4.3.1Mamba and State Space Models State-space models (SSMs) are a class of sequence models inspired by continuous-time dynami- cal systems. They have recently emerged as a compelling alternative to Transformers for model- ing long sequences, offering linear-time complexity while maintaining competitive performance across language, audio, and genomics tasks [132, 96]. 85 Continuous-time SSM.A linear time-invariant (LTI) state-space model maps an input signal 푥(푡) ∈ ℝto an output푦(푡) ∈ ℝthrough a latent stateℎ(푡) ∈ ℝ 푁 : ℎ ′ (푡) = 퐴 ℎ(푡) + 퐵 푥(푡),(4.7) 푦(푡) = 퐶 ℎ(푡),(4.8) where퐴 ∈ ℝ 푁×푁 is the state transition matrix,퐵 ∈ ℝ 푁 ×1 is the input projection, and퐶 ∈ ℝ 1×푁 is the output projection. Discretization.To process discrete sequences, the continuous system is discretized using a step sizeΔ > 0. Under the zero-order hold (ZOH) assumption: ̄ 퐴 =exp(Δ퐴),(4.9) ̄ 퐵 = (Δ퐴) −1 (exp(Δ퐴) − 퐼) ⋅ Δ퐵,(4.10) yielding the discrete recurrence: ℎ 푘 = ̄ 퐴 ℎ 푘−1 + ̄ 퐵 푥 푘 ,(4.11) 푦 푘 = 퐶 ℎ 푘 .(4.12) This recurrence can be computed in풪(퐿)time for a sequence of length퐿. Alternatively, unrolling the recurrence yields a convolution푦 = ̄ 퐾 ∗ 푥with kernel ̄ 퐾 = (퐶 ̄ 퐵, 퐶 ̄ 퐴 ̄ 퐵, 퐶 ̄ 퐴 2 ̄ 퐵, ...), which can be computed in풪(퐿log퐿)via the Fast Fourier Transform. Thisdual view— recurrence for inference, convolution for training—is a key advantage of SSMs. 86 S4: Structured State Spaces.The S4 model [132] introduced structured initialization of the state matrix퐴using the HiPPO (High-order Polynomial Projection Operator) framework, en- abling SSMs to capture long-range dependencies that challenge RNNs and even Transformers. S4 demonstrated state-of-the-art performance on the Long Range Arena benchmark, but its LTI nature means that the same dynamics are applied regardless of input content—the model cannot selectively attend to or ignore parts of the sequence. Mamba: Selective State Spaces.Mamba [96] addresses this limitation by making the SSM parametersinput-dependent. Specifically, the matrices퐵,퐶, and the step sizeΔare computed as functions of the input: 퐵 푘 =Linear 퐵 (푥 푘 ),퐶 푘 =Linear 퐶 (푥 푘 ),Δ 푘 =softplus(Linear Δ (푥 푘 )).(4.13) Thisselectivemechanism allows the model to perform content-aware filtering: it can dynam- ically decide which information to propagate through the hidden state and which to discard, analogous to the gating mechanisms in LSTMs [14] but within the SSM framework. The input-dependent parameters break the LTI structure, precluding the use of the convo- lution mode during training. Mamba compensates with ahardware-aware parallel scan. The key observation is that the linear recurrenceℎ 푘 = ̄ 퐴 푘 ℎ 푘−1 + ̄ 퐵 푘 푥 푘 isassociative: each step can be written as a pair( ̄ 퐴 푘 , ̄ 퐵 푘 푥 푘 )composed under the rule(퐴 2 , 푏 2 ) ∘ (퐴 1 , 푏 1 ) = (퐴 2 퐴 1 , 퐴 2 푏 1 + 푏 2 ), so the entire sequence can be evaluated with a parallel prefix scan [133] in풪(log퐿)parallel depth rather than퐿sequential steps. This recovers parallelism across the time dimension even though the parameters are now time-varying. The implementation ishardware-aware: because the selective SSM expands each input channel to an푁-dimensional state, naïvely storing all intermediate states in the GPU’s high-bandwidth memory (HBM) would make the operation 87 memory-bandwidth bound. Mamba instead fuses discretization, the scan, and the output pro- jection into a single kernel that keeps the expanded states in fast on-chip SRAM and writes only the final outputs back to HBM—similar in spirit to FlashAttention [134]—yielding near-linear scaling in sequence length in practice. Mamba block.The Mamba block processes its input through two parallel paths: 1.SSM path:The input is linearly projected to an expanded dimension, passed through a 1D convolution for local feature extraction, activated by SiLU (Sigmoid Linear Unit), and then processed by the selective SSM. 2.Gating path:A parallel linear projection with SiLU activation provides a multiplicative gate. The two paths are combined via element-wise multiplication and projected back to the original dimension: MambaBlock(푥) =Linear out (SSM(Conv1D(Linear 1 (푥))) ⊙SiLU(Linear 2 (푥))).(4.14) 4.3.2Architecture and Training The Mamba decoder shares the same overall recurrent structure as AlphaQubit (Figure 4.2)— a Stabilizer Embedder, an RNN Core with Syndrome Mixer layers, and a Readout Network (described in Section 4.2.2). The key difference is the replacement of multi-head attention with a Mamba module in each Syndrome Mixer. Hyperparameters.Table 4.1 lists the hyperparameters for both the Transformer and Mamba decoder variants across the two experimental settings (Sycamore memory experiments and real- time decoding simulations). 88 RNN coreStabilizer Embedder Readout network Scatter To 2DConv to data Line mean pool ResNet Syndrome Mixer Syndrome Mixer Gated dense block Scatter to 2D Gather from 2D Dilated Convs Mixer block Τ 푑 푛 +푆 푛 2 푑 푛+1 SSM Conv Projection 휎 휎 Mamba Concat Linear Linear Linear Scaled Dot-Product Attention Multi-Head Attention or Syndrome Mixer Syndrome Mixer Linear Linear ResNet Positional Encoding Measurement Detection event ×0.707 푑 푛+1 푑 푛 Projection 푃 퐿 푑 푁 Linear × (a) (b) (d) (c) h N h n h n+1 h n h n+1 h n Figure 5: (a) Stabilizer Embedder: Raw stabilizer measurements and detection events are passed through linear layers and combined with positional encodings before being processed by a ResNet to create an embedding S n for each cycle. (b) RNN Core: The core processes the sequence of embeddings recurrently. At each step, the previous decoder’s hidden state h n is combined with the current syndrome embedding S n and passed through three Syndrome Mixer layers to produce the updated state h n+1 . A scaled skip connection is used. (c) Syndrome Mixer: This is the central processing unit. The input is first processed by a Mixer Block, which can be either a Multi-Head Attention module (for the Transformer architecture) or a Mamba module. The output is then passed through a gated dense block and reshaped into a 2D grid to be processed by dilated convolutions, capturing spatial correlations. (d) Readout Network: The final decoder’s hidden state h N is pro- cessed by a series of convolutional and dense layers to produce the final prediction for the logical error probability, P L . ever, minimizes this effect. This result underscores that for practical QEC, decoding speed is as fundamental to corrective power as accuracy—a balance the Mamba architecture achieves. This interplay between speed and performance was further quantified in our error threshold analysis. The Mamba decoder achieved a higher threshold of 0.0104 compared to 0.0097 for the Transformer, indicating that it maintains fault-tolerant scaling under higher physical error rates. A higher thresh- old directly translates to more lenient hardware noise requirements for achieving fault tolerance. In this sense, the Mamba architecture not only improves decoding speed but also enhances the overall practicality of QEC. Overall, our results suggest that attention-free architectures represent a promising direction for scalable quantum error correction. Future work could focus on further optimizing the Mamba ar- chitecture—exploring model scaling, advanced training regimes, and hyperparameter tuning—to enhance accuracy without sacrificing speed. In addition, integrating Mixture-of-Experts (MoE) models [38–40] presents an intriguing avenue for dynamically allocating computational resources, thereby balancing accuracy and efficiency adaptively. Beyond the surface code, the same principles can be extended to other stabilizer codes, including color codes [11, 19] and Quantum Low-Density Parity-Check (QLDPC) codes [41, 42] such as Bivariate Bicycle (B) codes, which are actively in- vestigated as high-rate alternatives but present distinct decoding challenges. Ultimately, applying these efficient and scalable decoders to complex subroutines like magic state distillation [43] will be a critical step toward realizing universal fault-tolerant quantum computation. METHODS MODEL ARCHITECTURE Our decoder follows the recurrent-transformer design of AlphaQubit (see Figure 1). At each error- correction cycle n, for each stabilizer i, the raw stabilizer measurements and detection events (and optionally analogue I/Q readouts or leakage flags) are embedded into a d model -dimensional feature s n,i by a stabilizer embedding network. The set of embeddings S n = s n,i l i=1 , together with the 7 Figure 4.2: Architecture of the Mamba decoder. (a) The Stabilizer Embedder converts raw measurements and detection events into푑 model -dimensional embeddings via linear projections, positional encodings, and a ResNet. (b) The RNN Core processes each QEC cycle through 퐿 = 3Syndrome Mixer layers with scaled skip connections. (c) Each Syndrome Mixer contains a Mamba-based mixer block, gated dense block, and dilated 2D convolutions. (d) The Readout Network maps the final hidden state to the logical-error probability푃 퐿 . Training procedure.The training procedure differs between the two experimental settings: Sycamore memory experiments.Following AlphaQubit, we employ a two-stage training pipeline: 1.Pretraining:We generate up to 100 million synthetic syndrome samples from a detector error model (DEM) whose parameters are derived from a Pauli noise model calibrated us- ing cross-entropy benchmarking (XEB) data from the Sycamore processor. The pretrain- ing dataset includes QEC sequences of varying cycle counts푟 ∈ 1, 3, ... , 25, covering both푋- and푍-basis memory experiments. We employ acurriculum learningstrategy: the model initially trains on shorter sequences푟 ∈ 1, 3, 5, 7, 9, and every 150,000 it- erations the training set is expanded to include four additional cycle counts, eventually covering the full range up to푟 = 25. The Lion optimizer [135] is used with an initial learning rate of5 × 10 −6 and weight decay 89 Table 4.1: Decoder hyperparameters. Top: model-specific parameters for Transformer and Mamba variants in Sycamore memory experiments (Syc.) and real-time decoding simulations (RT). Bottom: shared training and architecture parameters. Transformer-specificMamba-specific Param.Syc. RTParam.Syc. RT 푑 model 320 256푑 model 320 256 퐻4 4푑 state 16 16 푑 푏 48 48푑 conv 4 4 푑 attn 32 32푤 mamba 1 1 푑 mid 32 32 Shared Hyperparameters Param.SycamoreParam.Real-time 퐿 stab 2퐿 stab 2 퐿 res 16 퐿 res 16 푑 read 64푑 read 48 푤 gate 5푤 gate 5 퐷 conv (푑 = 3)[1, 1, 1] 퐷 conv (푑 = 3)[1, 1, 1] 퐷 conv (푑 = 5)[1, 1, 2]퐷 conv (푑 = 5)[1, 1, 2] 퐷 conv (푑 = 7)[1, 2, 4] of1 × 10 −5 . The learning rate follows a cosine annealing schedule. We apply gradient clipping with a maximum norm of 1 and maintain an exponential moving average (EMA) of model weights with a decay rate of 0.9999. 2.Fine-tuning:The pretrained decoder is adapted on the actual Sycamore experimental dataset (50% train / 50% evaluation split) for 10 epochs with a reduced learning rate of 2 × 10 −6 and increased weight decay of7 × 10 −5 . Real-time decoding experiments.For the simulated real-time setting, we train decoders from scratch on synthetic data generated on-the-fly using the Superconducting-Inspired Circuit Depolarizing Noise (SI1000) model [136] via the Stim circuit simulator [137]. Separate models 90 are trained for code distances푑 ∈ 3, 5, 7at a base physical error rate of푝 = 0.002. Train- ing runs for 500,000 iterations with a batch size of 256, using the Lion optimizer with cosine annealing. For finite-size effective threshold analysis under the specified latency model, we fine-tune the baseline models (trained at푝 = 0.002) at higher physical error rates푝 ∈ 0.006, 0.008, 0.010, 0.012for 250,000 iterations each, using a transfer learning approach that significantly reduces the computational cost of training models at multiple noise levels. 4.3.3Experimental Results We evaluate the Mamba decoder across two experimental settings: memory experiments on real hardware data (Section 4.3.3) and simulated real-time decoding with decoder-induced noise (Section 4.3.3). Memory Experiments on Sycamore Data We benchmark our Mamba decoder against Transformer-based, matching-based, and tensor net- work decoders using the Sycamore memory experiment dataset from AlphaQubit [97, 102]. This publicly available dataset was obtained on the 72-qubit Sycamore superconducting pro- cessor comprising four distance-3 surface code blocks and a single distance-5 block. Both푋- and푍-basis memory experiments were conducted for up to 25 QEC cycles, with 50,000 experi- ments performed for each odd-numbered cycle count푛 ∈ 1, 3, ... , 25. The logical error rate is estimated by fitting the measured fidelity to Equation (4.4). Figure 4.3a summarizes the results. Our Mamba decoder achieves LERs of approximately2.98 × 10 −2 at distance 3 and3.03 × 10 −2 at distance 5, closely tracking the reproduced Transformer baseline. At distance 3, the reproduced neural baselines are competitive with the strongest listed decoders. At distance 5, 91 35 Code Distance 0.025 0.030 0.035 0.040 0.045 Mean logical error per round MWPM (PyMatching) MWPM-Corr MWPM-BP TN Transformer Mamba (a) 371115192327313539 Code Distance 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 Time (s) MHAttention block Mamba block (b) Figure 2: (a) Logical error per round on the Sycamore dataset for various decoders at code distances 3 and 5. (b) A comparison of inference time for a Mamba block versus a Multi-Head Attention (MHA) block as code distance increases, measured on a local RTX 4090 GPU. plexity from quadratic to linear, while still capturing essential correlations within the syndrome data. This architecture is particularly well-suited for QEC decoding, which requires efficient processing of long syndrome sequences. This mechanism makes our Mamba-based decoder an effective solution that balances the need to model complex error chains with the stringent low-latency requirements of real-time decoding. The schematic of the overall architecture is provided in Figure 1. Our investigation proceeds in two parts. First, we validate our Mamba decoder by applying it to the Sycamore memory experiments, a key benchmark for quantum error correction. Using the publicly available hardware dataset for distance-3 and distance-5 codes, we test our decoder’s performance on real-world experimental data. In this setting, our Mamba decoder’s performance matches that of the Transformer-based architecture, demonstrating that its superior computational efficiency does not come at the cost of performance. Second, we shift our focus to the more practical real-time decoding scenario. We simulate this by introducing decoder-induced noise at each correction round, with the noise strength scaling according to the decoder’s computational complexity—O(d 2 ) for Mamba andO(d 4 ) for the Transformer, a choice justified by wall-clock time comparisons. In these simulations, using the SI1000 noise model, the Mamba decoder outperforms the Transformer for distances 3 and 5. Furthermore, by analyzing the error threshold under these real-time conditions, we find that the Mamba decoder exhibits a significantly higher threshold (0.0104) compared to the Transformer decoder (0.0097). These results strongly suggest that Mamba-based architectures are a compelling choice for scalable, real-time decoding, a crucial component for the realization of a fault-tolerant quantum computer. RESULTS MEMORY EXPERIMENTS To evaluate the performance of our Mamba-based decoder, we benchmark it against Transformer- , matching-, and TN–based decoders using the Sycamore memory experiment dataset from Al- phaQubit [29]. This publicly available dataset, provided by Google Quantum AI [33], was obtained on a 72-qubit superconducting processor comprising four distance-3 surface-code blocks and a sin- gle distance-5 block. Both X - and Z-basis memory experiments were conducted for up to 25 cy- cles of error correction, with 50,000 experiments performed for each odd-numbered cycle count n ∈ 1, 3,..., 25. This setup is designed to trace the logical qubit’s fidelity as it decays over an increasing number of correction cycles. Decoder performance is quantified by the logical error per round (LER), which indicates the proba- bility that the decoder will fail per error correction cycle. To estimate the LER, we perform a linear regression on the logarithm of the measured fidelities as a function of the number of error correction cycles n, using the model logF(n) = logF 0 + n log (1− 2ε). Here, ε corresponds to the logical error per round. 3 (a) 35 Code Distance 0.025 0.030 0.035 0.040 0.045 Mean logical error per round MWPM (PyMatching) MWPM-Corr MWPM-BP TN Transformer Mamba (a) 371115192327313539 Code Distance 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 Time (s) MHAttention block Mamba block (b) Figure 2: (a) Logical error per round on the Sycamore dataset for various decoders at code distances 3 and 5. (b) A comparison of inference time for a Mamba block versus a Multi-Head Attention (MHA) block as code distance increases, measured on a local RTX 4090 GPU. plexity from quadratic to linear, while still capturing essential correlations within the syndrome data. This architecture is particularly well-suited for QEC decoding, which requires efficient processing of long syndrome sequences. This mechanism makes our Mamba-based decoder an effective solution that balances the need to model complex error chains with the stringent low-latency requirements of real-time decoding. The schematic of the overall architecture is provided in Figure 1. Our investigation proceeds in two parts. First, we validate our Mamba decoder by applying it to the Sycamore memory experiments, a key benchmark for quantum error correction. Using the publicly available hardware dataset for distance-3 and distance-5 codes, we test our decoder’s performance on real-world experimental data. In this setting, our Mamba decoder’s performance matches that of the Transformer-based architecture, demonstrating that its superior computational efficiency does not come at the cost of performance. Second, we shift our focus to the more practical real-time decoding scenario. We simulate this by introducing decoder-induced noise at each correction round, with the noise strength scaling according to the decoder’s computational complexity—O(d 2 ) for Mamba andO(d 4 ) for the Transformer, a choice justified by wall-clock time comparisons. In these simulations, using the SI1000 noise model, the Mamba decoder outperforms the Transformer for distances 3 and 5. Furthermore, by analyzing the error threshold under these real-time conditions, we find that the Mamba decoder exhibits a significantly higher threshold (0.0104) compared to the Transformer decoder (0.0097). These results strongly suggest that Mamba-based architectures are a compelling choice for scalable, real-time decoding, a crucial component for the realization of a fault-tolerant quantum computer. RESULTS MEMORY EXPERIMENTS To evaluate the performance of our Mamba-based decoder, we benchmark it against Transformer- , matching-, and TN–based decoders using the Sycamore memory experiment dataset from Al- phaQubit [29]. This publicly available dataset, provided by Google Quantum AI [33], was obtained on a 72-qubit superconducting processor comprising four distance-3 surface-code blocks and a sin- gle distance-5 block. Both X - and Z-basis memory experiments were conducted for up to 25 cy- cles of error correction, with 50,000 experiments performed for each odd-numbered cycle count n ∈ 1, 3,..., 25. This setup is designed to trace the logical qubit’s fidelity as it decays over an increasing number of correction cycles. Decoder performance is quantified by the logical error per round (LER), which indicates the proba- bility that the decoder will fail per error correction cycle. To estimate the LER, we perform a linear regression on the logarithm of the measured fidelities as a function of the number of error correction cycles n, using the model logF(n) = logF 0 + n log (1− 2ε). Here, ε corresponds to the logical error per round. 3 (b) Figure 4.3: Accuracy and speed of the Mamba decoder.(a)Logical error per round on the Sycamore memory experiment dataset for various decoders at code distances푑 = 3and푑 = 5; Mamba tracks the reproduced Transformer baseline while replacing multi-head attention with a Mamba block.(b)Wall-clock inference time for a single Mamba block versus a Multi-Head Attention (MHA) block as a function of code distance, measured on a local RTX 4090 GPU. both reproduced neural baselines underperform the tensor network decoder (휀 = 2.915 × 10 −2 ), while outperforming the matching-based decoders. We attribute this gap in part to the size of the pretraining dataset: while AlphaQubit is described as being pretrained on up to 1 billion samples, our models were pretrained on 100 million samples due to computational constraints. With a larger pretraining dataset, the neural baselines might reduce this remaining gap. The keytakeawayfrom the memory experiments is thatreplacing multi-head attention with a Mamba module preserves comparable decoding accuracy. This is a necessary condition for the Mamba decoder to be useful: its speed advantage matters only if the accuracy remains close to the reproduced Transformer baseline. Inference Speed To quantify the computational advantage of the Mamba block over multi-head attention, we measure wall-clock inference time on an NVIDIA RTX 4090 GPU as a function of code distance (Figure 4.3b). 92 The measured inference times track the complexity of the two blocks. The curves deviate near푑 ≈ 21, beyond which the Mamba block becomes increasingly faster, reaching roughly3× the speed of the MHA block at푑 = 39and widening further as푑grows. This is highly relevant as fault-tolerant algorithms targeting practical applications are expected to require substantially larger code distances [105], precisely the regime in which the quadratic vs. quartic scaling gap translates into large latency reductions. The Mamba block is thus better positioned than attention for the long-sequence decoding demanded by future large-distance processors. Real-Time Decoding with Decoder-Induced Noise Experimental setup.We simulate real-time decoding for surface codes of distance푑 ∈ 3, 5, 7using the SI1000 noise model [136] with a base physical error rate of푝 = 0.002. Each model is trained on sequences of2푑 + 1QEC cycles and evaluated over8푑 + 4cycles (four repe- titions of the training-length block). To simulate the effect of decoder latency,decoder-induced noiseis injected after every2푑 + 1cycles. Decoder-induced noise model.The strength of the decoder-induced noise is calibrated to re- flect each architecture’s computational complexity. For large code distances (푑 > 20), the overall inference time is dominated by the most computationally intensive component: the MHA block (for the Transformer) or the Mamba block. We model the decoder-induced error probability as: 푝 dec = ⎧ ⎨ ⎩ 훼 ⋅ 푑 4 (Transformer), 훼 ⋅ 푑 2 (Mamba), (4.15) where훼 = 7.623 × 10 −6 is a scaling constant based on AlphaQubit’s reported∼ 40 휇s latency at 푑 = 9. At this distance,훼푑 4 ≈ 25푝for the base physical error rate푝 = 0.002. Since the SI1000 93 →4 →(2d+ 1) ... S |ω↑ L Data qubit Syndrome Generation Decoding noise R |A↑ Meas qubit Decoder 2 (a) (b) Figure 3: (a) Experimental scheme for real-time decoding simulation. The evaluation runs for8d+4 cycles, structured as four repetitions of a2d+1cycle block. After each block, decoding noise is injected to simulate the impact of decoder latency before a correctionRis applied. (b) Comparison of Mamba and Transformer decoders in terms of Logical Error per Round (LER) under real-time decoding scenarios, evaluated at code distances of 3, 5, and 7. Performance is shown with decoder- induced noise, whose strength is scaled according to each decoder’s computational complexity. For reference, LERs obtained without decoder-induced noise are shown in the inset. constant ofω=7.623→10 →6 , which was then applied consistently across all distances for both decoder models. Figure 3b shows the LERs for both decoders at distances 3, 5, and 7, both with and without the simulated decoder-induced noise. In the absence of this induced noise, the Mamba and Trans- former decoders exhibit nearly identical performance, underscoring their comparable accuracy on the underlying physical noise model. However, the introduction of decoder-induced noise reveals a stark divergence. The Transformer’s prohibitiveO(d 4 )latency leads to a severe accumulation of errors, causing a significant degradation in its LER as the code distance increases. In contrast, the Mamba decoder’s more favorableO(d 2 )scaling results in a much smaller penalty. This resilience to latency-induced errors allows it to substantially outperform the Transformer, confirming its superior scalability and robustness for practical, real-time quantum error correction. ANALYSIS OF THEERRORTHRESHOLD UNDERREALTIMEDECODING A critical metric for any quantum error correction scheme is the error threshold (p th ), defined as the maximum physical error rate which increasing the code distance (d) leads to an exponential sup- pression of the logical error rate. A decoder that exhibits a higher error threshold is fundamentally more robust, as it can achieve fault tolerance with noisier physical qubits, a significant advantage for near-term hardware. In a real-time setting, a decoder’s computational latency introduces an addi- tional noise channel—decoder-induced noise—which effectively lowers this threshold. Therefore, an ideal real-time decoder must not only be accurate but also fast enough to minimize this perfor- mance degradation. We investigate the effective error threshold—that is, the threshold accounting for additional decoder- induced noise due to finite latency—of our Mamba decoder compared to the Transformer-based architecture under real-time decoding. This experiment is designed to determine the maximum 5 Figure 4.4: Real-time decoding performance.Main:Logical error per round (LER) for Mamba and Transformer decoders under simulated real-time conditions with decoder-induced noise pro- portional to computational complexity.Inset:LER without decoder-induced noise, showing comparable baseline accuracy. model assigns measurement error probability5푝, this corresponds to a Transformer-induced error rate five times larger than the physical measurement-error rate. Results.As a baseline, we first compare the two decoderswithoutdecoder-induced noise (Fig- ure 4.4, inset). In this setting, the Mamba and Transformer decoders exhibit nearly identical logical error rates at all distances, confirming that the Mamba architecture does not sacrifice ac- curacy when latency is not a factor. When decoder-induced noise is included (Figure 4.4, main panel), the two architectures diverge. At푑 = 3, both decoders experience mild noise penalties, with comparable LERs. At푑 = 5, the Transformer’s LER begins to degrade more rapidly. At 푑 = 7, the Transformer’s LER rises to approximately10 −1 , while the Mamba decoder maintains an LER of approximately10 −2 in this simulation. Finite-size effective threshold analysis.Here, we estimate a finite-size effective threshold under the specified decoder-induced-noise/latency model, rather than the asymptotic surface- 94 0.0060.0070.0080.0090.010.0110.012 Physical Error Rate 10 2 2 × 10 2 3 × 10 2 4 × 10 2 6 × 10 2 10 1 Logical Error Per Round d = 3 d = 5 Threshold : 0.0097 (a) 0.0060.0070.0080.0090.010.0110.012 Physical Error Rate 10 2 2 × 10 2 3 × 10 2 4 × 10 2 6 × 10 2 10 1 Logical Error Per Round d = 3 d = 5 Threshold : 0.0104 (b) Figure 4: Analysis of the effective error threshold under real-time decoding with decoder-induced noise. The plots show the Logical Error Per Round (LER) as a function of the Physical Error Rate (p) for code distances d = 3 and d = 5. The effective error threshold is the crossover point where the LER for d = 5 (orange line) becomes higher than the LER for d = 3 (blue line). (a) The Transformer-based decoder, showing a threshold of 0.0097. (b) The Mamba-based decoder, showing a higher threshold of 0.0104. physical error rate at which each decoder architecture can still provide effective error suppres- sion. To ensure that each decoder achieves its optimal performance at each noise level, we em- ployed a fine-tuning strategy. Starting with the baseline models pre-trained at a physical er- ror rate of p=0.002, we fine-tuned separate model instances for each higher physical error rate: p∈0.006, 0.008, 0.010, 0.012. The results, shown in Figure 4 for code distances 3 and 5, demonstrate the substantial impact of computational efficiency on the effective error threshold. The defining characteristic of operating below the threshold is that a larger code distance yields a lower logical error rate (P L (d = 5) < P L (d = 3)). As shown in Figure 4a, the LER curves for the Transformer architecture cross at a physical error rate of approximately p th ≈ 0.0097. In contrast, the Mamba decoder’s curves (Figure 4b) cross at a significantly higher physical error rate of p th ≈ 0.0104. This crossover point, found by comparing d = 3 and d = 5, represents the effective threshold for this specific comparison. However, as the code distance continues to increase, the unfavorable computational scaling of the Transformer decoder will cause this threshold to degrade rapidly. The Mamba decoder, with its more efficient scaling, will experience a much slower degradation. This superior scalability and higher initial threshold explicitly demonstrate that the Mamba architecture is a more robust and viable solution for practical, real-time quantum error correction. DISCUSSION Transformer decoders, while highly accurate, suffer from a computational complexity that scales as O(d 4 ) with code distance d, creating a significant bottleneck for real-time quantum error correction. In this work, we introduced a neural decoder based on Mamba—a state-space model with O(d 2 ) complexity—and showed that it offers a substantially improved balance between speed and accuracy for practical QEC. In memory experiments using Sycamore hardware data, our Mamba-based neural decoder achieved a logical error rate that matched its Transformer-based counterpart. This finding is crucial, as it demonstrates that the computationally intensive self-attention mechanism can be replaced without compromising—and in fact while preserving—state-of-the-art decoding accuracy in memory exper- iments. Even before accounting for its speed advantage, the Mamba architecture provides a highly effective foundation for neural decoding. The Mamba decoder’s computational efficiency becomes a decisive advantage in simulated real-time scenarios. In this setting, latency introduces ”decoder-induced noise,” which severely degrades the Transformer’s performance. This penalty, caused by errors accumulating during the Transformer’s prolonged processing time, grows rapidly with code distance. The Mamba decoder’s speed, how- 6 Figure 4.5: Finite-size effective threshold under this latency model. Each panel shows logical error per round versus physical error rate for code distances푑 = 3and푑 = 5. The effective threshold is identified where the푑 = 5curve crosses above the푑 = 3curve. Under the assumed decoder-induced-noise model, the Transformer decoder yields푝 th ≈ 0.0097, while the Mamba decoder yields푝 th ≈ 0.0104. codethreshold. Decoder-inducednoisecanshiftthisfinite-sizecrossoverbyaddinganadditional noise channel proportional to the decoder’s modeled latency. To estimate the finite-size effective threshold under this latency model, we fine-tune both decoders at physical error rates푝 ∈ 0.006, 0.008, 0.010, 0.012and compare the LERs at distances푑 = 3and푑 = 5. The estimate is the crossover point where increasing code distance no longer helps in this finite-size comparison under the assumed latency-induced noise. As shown in (Figure 4.5), the Transformer reaches푝 th ≈ 0.0097, while the Mamba de- coder reaches푝 th ≈ 0.0104. A higher finite-size effective threshold means that, under the as- sumed decoder-induced noise scaling, the decoder can tolerate a larger physical error rate before increasing the code distance ceases to help. Moreover, as code distance increases, the Trans- former’s풪(푑 4 )decoder-induced noise would degrade the model-dependent effective threshold faster than the Mamba decoder’s풪(푑 2 )noise, making the scaling advantage increasingly im- portant at larger distances. 95 4.4Summary In this chapter, we introduced a neural decoder for quantum error correction based on the Mamba architecture—a selective state-space model that replaces the attention mechanism of Transformer-based decoders with a linear-complexity alternative. Our investigation established three key findings: 1.Comparable accuracy to the reproduced Transformer baseline.In memory experi- ments on real hardware data from Google’s Sycamore processor, the Mamba decoder achieved logical error rates of휀 ≈ 2.98 × 10 −2 (distance 3) and휀 ≈ 3.03 × 10 −2 (distance 5), closely tracking the reproduced Transformer-based decoder in this comparison. 2.Fasterinferencewithgrowingadvantage.Wall-clock benchmarks on an RTX 4090 GPU showed that inference time tracks the풪(푑 2 )vs.풪(푑 4 )complexity of the two blocks: the curves deviate near푑 ≈ 21, beyond which the Mamba block pulls increasingly ahead, reaching roughly3×faster at푑 = 39and widening further as푑grows. Constant-factor overhead masks this advantage at the small distances accessible today (푑 ≤ 5), so its significance is asymptotic, favoring the Mamba architecture at the larger code distances relevant for future processors. 3.Higher finite-size effective threshold under this latency model.Under simulated real- time decoding conditions with decoder-induced noise, the Mamba decoder achieved a finite-size effective threshold of푝 th ≈ 0.0104, compared to푝 th ≈ 0.0097for the Transformer—a 7% improvement under the assumed latency model. This higher thresh- old reflects the benefit of lower decoder-induced noise in that model and suggests that latency scaling could become more important as code distance increases. 96 In the next chapter, we turn from classical neural networks thatassistquantum hardware to neural networks thatrepresentquantum states directly. 97 CHAPTER 5 NEURAL QUANTUM STATES The results in this chapter are based on Tak Hur,Stochastic Reconfiguration as Statisti- cal Filtering for Overparameterized Neural Quantum States[138]. Code is available at https://github.com/takh04/sr_filter. In Chapter 4, neural networks were used to process syndrome data for quantum error cor- rection. This chapter studies how neural networks can serve as variational representations of quantum many-body wave functions. Neural quantum states (NQS) combine expressive neural- network ansatzes with variational Monte Carlo (VMC), making it possible to search for ground states in Hilbert spaces far beyond exact diagonalization. The chapter focuses on stochastic reconfiguration (SR), the standard optimizer for NQS. For modern NQS, the number of parameters can greatly exceed the number of Monte Carlo samples used in one optimization step. In this parameter-rich regime, we show that the diagonal shift in SR is more than a numerical safeguard: it plays a statistical role, controlling how well the optimizer generalizes from a finite number of Monte Carlo samples. This perspective motivates a multi-shift variant of SR (MS-SR), which combines several differently shifted updates to make better use of the available samples. Thechapterisorganizedasfollows. Section5.1introducestheNQS-VMCworkflowandSR. Section 5.2 develops the fixed-checkpoint regression view and the bias-variance role of the shift. Section 5.3 summarizes the exact and large-scale diagnostics supporting this view. Section 5.4 98 presents MS-SR and the main empirical evidence for the method. Section 5.5 summarizes the chapter. 5.1Neural Quantum States and Stochastic Reconfiguration The central obstacle in quantum many-body physics is the exponential size of the Hilbert space. A system of푁spin- 1 2 degrees of freedom has basis states |푥⟩ indexed by bit strings푥 ∈ ±1 푁 , so a generic state requires2 푁 complex amplitudes. Tensor-network methods such as density matrix renormalization group exploit low entanglement and are highly effective in many one- dimensional systems [139, 140]. However, frustrated and higher-dimensional systems often re- quire more flexible representations. NQS address this by parameterizing the wave-function am- plitude휓 휃 (푥)with a neural network. Since the RBM construction of Carleo and Troyer [141], NQS have expanded to recurrent, fermionic, transformer-based, and foundation-style wave func- tions [142, 143, 144, 145, 146, 147]. VMC approximates the ground state of a Hamiltonian ̂ 퐻by minimizing the Rayleigh quo- tient 퐸(휃) = ⟨휓 휃 | ̂ 퐻|휓 휃 ⟩ ⟨휓 휃 |휓 휃 ⟩ ,∣휓 휃 ⟩ = ∑ 푥 휓 휃 (푥) |푥⟩ .(5.1) The energy is estimated by sampling configurations from the Born distribution 휋 휃 (푥) = |휓 휃 (푥)| 2 ∑ 푥 ′ |휓 휃 (푥 ′ )| 2 (5.2) and averaging the local energy 퐻 loc (푥) = ⟨푥| ̂ 퐻 ∣휓 휃 ⟩ ⟨푥|휓 휃 ⟩ = ∑ 푥 ′ ⟨푥| ̂ 퐻|푥 ′ ⟩ 휓 휃 (푥 ′ ) 휓 휃 (푥) .(5.3) The resulting training loop is conceptually simple: sample from|휓 휃 | 2 , evaluate local energies 99 and wave-function derivatives, and update the parameters. Ordinary gradient descent treats the parameter vector as Euclidean, but the same Euclidean movement can correspond to very different changes in the represented quantum state. SR cor- rects this mismatch by using the geometry induced by the variational state manifold [148, 149, 150]. Let 푂 푐 (푥) = ∇ 휃 log휓 휃 (푥) − 피 휋 휃 [∇ 휃 log휓 휃 ](5.4) denote centered tangent features. For real wave functions and real parameters, the quantum geometric tensor (QGT) and SR force are 푆 = 피 휋 휃 [푂 푐 (푥)푂 푐 (푥) 푇 ] ,푔 = 피 휋 휃 [푂 푐 (푥)퐻 loc,푐 (푥)] ,(5.5) where퐻 loc,푐 (푥) = 퐻 loc (푥) − 퐸(휃). SR updates parameters by solving 푆훿 = 푔,휃 + = 휃 − 휂훿.(5.6) Geometrically, this is a natural-gradient step; physically, it is the projection of imaginary-time evolution onto the tangent space of∣휓 휃 ⟩[151, 152]. Inpractice, theexactexpectationsdefining푆and푔areunavailableandarereplacedbyMonte Carlo averages over a batch of푁 푠 samples푥 푗 . Stacking the centered features as rows of푂 ∈ ℝ 푁 푠 ×푃 , with row푗equal to푂 푐 (푥 푗 ) 푇 , and collecting the centered local energies inh∈ ℝ 푁 푠 with ℎ 푗 = 퐻 loc,푐 (푥 푗 ), the empirical QGT and force are ̂ 푆 = 1 푁 푠 푁 푠 ∑ 푗=1 푂 푐 (푥 푗 )푂 푐 (푥 푗 ) 푇 = 1 푁 푠 푂 푇 푂,̂푔 = 1 푁 푠 푁 푠 ∑ 푗=1 푂 푐 (푥 푗 )퐻 loc,푐 (푥 푗 ) = 1 푁 푠 푂 푇 h.(5.7) Because ̂ 푆is rank-deficient and ill-conditioned when estimated from a finite batch, SR regular- 100 izes the linear system with a diagonal shift휆, giving the ridge solution ̂ 훿 휆 = ( ̂ 푆 + 휆퐼) −1 ̂푔.(5.8) MinSR.The computational regime has changed with modern NQS. Classical VMC and early RBM calculations often had fewer parameters than samples. Transformer and foundation-style NQScaninsteadhave푃largerthanthesamplecount푁 푠 byonetothreeordersofmagnitude[153, 146, 147]. In this regime, forming and inverting the푃 × 푃matrix ̂ 푆 + 휆퐼of equation (5.8) becomes prohibitive. Kernel-form SR, often called minSR, instead solves in sample space using the empirical neural tangent kernel ̂ 푇 = 1 푁 푠 푂 푇 ∈ ℝ 푁 푠 ×푁 푠 .(5.9) By the push-through identity, the ridge solution of equation (5.8) is obtained equivalently as ̂ 훿 휆 = 1 푁 푠 푂 푇 ( ̂ 푇 + 휆퐼) −1 h,(5.10) which inverts an푁 푠 × 푁 푠 system with the same shift휆rather than the full푃 × 푃QGT. This reduces the linear-algebra dimension from푃 × 푃to푁 푠 × 푁 푠 , but forming the dense kernel still costs풪(푁 2 푠 푃)andsolvingitcosts풪(푁 3 푠 ). Increasingthebatchsizethereforeremainsexpensive, and each SR update is still estimated from a small random batch relative to a very expressive tangent space. 101 5.2SR as Statistical Spectral Filtering The key simplification is to analyze one SR step at a fixed checkpoint휃. At this checkpoint, the wave function, sampling distribution, tangent features, and local energies are fixed. For notational clarity, we present the case of real wave functions and Hamiltonians in the sampling basis. For complex wave functions, the regression uses modulus-squared residuals, with푄 = 피 휋 휃 [푂 푐 푂 푇 푐 ]and푓 = 피 휋 휃 [푂 푐 퐻 loc,푐 ]. Real parameter increments then satisfy Re(푄)훿 =Re(푓 ), equivalently a regression with stacked real and imaginary residuals. At a fixed checkpoint, SR fits tangent predictions푂 푐 (푥) 푇 훿to centered local energies. The ideal population SR direction is the least-squares solution 훿 ∗ = 푆 † 푔 ∈argmin 훿 피 휋 휃 [(푂 푐 (푥) 푇 훿 − 퐻 loc,푐 (푥)) 2 ] . (5.11) The centered local energy is not generally representable exactly by the current tangent fea- tures. The population residual of the best tangent-space fit is theexpressivity gap 휖(푥) = 퐻 loc,푐 (푥) − 푂 푐 (푥) 푇 훿 ∗ .(5.12) The normal equations imply 피 휋 휃 [푂 푐 (푥)휖(푥)] = 0,휎 2 gap = 피 휋 휃 [휖(푥) 2 ] =Var 휋 휃 (퐻 loc ) − 푔 푇 푆 † 푔.(5.13) Physically, this gap is the component of imaginary-time evolution that the current tangent space cannot express. Statistically, it behaves like residual noise: it is orthogonal to the tangent features in population, but finite Monte Carlo batches contain sampled values of휖(푥)that an overparam- eterized empirical solve can fit. 102 This regression view clarifies the role of the diagonal shift. The target is훿 ∗ , while the empirical ridge update ̂ 훿 휆 of equation (5.8) depends on a random batch. The relevant excess risk is ℰ 휆 = 피 퐷 [‖ ̂ 훿 휆 − 훿 ∗ ‖ 2 푆 ] .(5.14) This푆-seminorm excess risk is physically meaningful because it is directly related to theinfi- delity, a standard distance between quantum states. For normalized states, bounded updates, and a small step size휂, 퐼 휃 (훿, 훿 ′ ) = 1 − |⟨휓 휃−휂훿 |휓 휃−휂훿 ′ ⟩| 2 = 휂 2 ‖훿 − 훿 ′ ‖ 2 푆 + 푂(휂 3 ),(5.15) soℰ 휆 controlstheexpectedlocalinfidelitybetweentheidealpopulationupdateandtheempirical SR update, up to the factor휂 2 and higher-order terms. The excess risk has the exact decomposition ℰ 휆 = ∥피 퐷 [ ̂ 훿 휆 ] − 훿 ∗ ∥ 2 푆 + 피 퐷 [∥ ̂ 훿 휆 − 피 퐷 [ ̂ 훿 휆 ]∥ 2 푆 ] .(5.16) Writing푆 = 푉diag(푠 푖 )푉 푇 on the non-null QGT subspace and훽 ∗ 푖 = (푉 푇 훿 ∗ ) 푖 , a spectral approx- imation for random-design ridge regression [154] gives the bias–variance proxy ℰ 휆 ≈ ∑ 푖 푠 푖 ( 휆 푠 푖 + 휆 ) 2 (훽 ∗ 푖 ) 2 ⏟ shrinkage bias + 휎 2 gap 푁 푠 ∑ 푖 ( 푠 푖 푠 푖 + 휆 ) 2 ⏟ finite-sample variance .(5.17) This approximation replaces the empirical inverse by(푆 + 휆퐼) −1 and approximates the residual- noise covariance피 휋 휃 [푂 푐 푂 푇 푐 휖 2 ]by휎 2 gap 푆. It therefore isolates a noise mechanism rather than givinganexactfinite-sampleriskformula; theheld-outidentitiesbelowdonotrequirethisscalar- 103 noise approximation. The same scalar filter 푓 휆 (푠) = 푠 푠 + 휆 (5.18) controlsbothterms. Increasing휆dampsnoisysmall-eigenvaluedirectionsandreducesvariance, but it also shrinks useful components of훿 ∗ and increases bias. The diagonal shift is therefore a statistical regularizer, not only a numerical tolerance. For small Hilbert spaces,푆,훿 ∗ , and휎 2 gap can be computed by exact summation. For large NQS, the population quantities are unavailable, so we introduce held-out tangent predictions. For an independent validation set퐷 val , 푅 val (훿) = 1 |퐷 val | ∑ 푥∈퐷 val (푂 푐 (푥) 푇 훿 − 퐻 loc,푐 (푥)) 2 ,(5.19) with expectation 피 퐷 val [푅 val (훿)] = ‖훿 − 훿 ∗ ‖ 2 푆 + 휎 2 gap .(5.20) Thus validation residual tracks excess risk plus the irreducible expressivity-gap term. A comple- mentary diagnostic measures how much the update varies across independent training batches. Given푚SR updates ̂ 훿 휆,푗 obtained from independently resampled batches at the same check- point, with mean ̄ 훿 휆 = 푚 −1 ∑ 푗 ̂ 훿 휆,푗 , the multi-batch variance is 풱 mb (휆) = 1 푚 − 1 푚 ∑ 푗=1 1 |퐷 val | ∑ 푥∈퐷 val (푂 푐 (푥) 푇 ( ̂ 훿 휆,푗 − ̄ 훿 휆 )) 2 .(5.21) This is the sample variance of the tangent predictions푂 푐 (푥) 푇 ̂ 훿 휆,푗 across the independent up- dates, evaluated on held-out configurations. Because the average over푥 ∈ 퐷 val approximates the QGT seminorm,풱 mb isolates the batch-to-batch fluctuation of the update and, as푚and 104 |퐷 val |grow, converges to the variance component of the excess risk in Equation (5.14). Like the validation residual, it requires only held-out tangent-feature predictions, so it remains prac- tical at large scale. The filtering view predicts a U-shaped validation residual as휆varies and a monotone decrease in풱 mb as finite-batch noise is suppressed. 5.3Evidence Across System Scales Small-scale test.The first test uses an exactly tractable4 × 4spin- 1 2 Heisenberg graph, ̂ 퐻 = 퐽 1 ∑ (푖,푗)∈ℬ 1 ̂ 푆 푖 ⋅ ̂ 푆 푗 + 퐽 2 ∑ (푖,푗)∈ℬ 2 ̂ 푆 푖 ⋅ ̂ 푆 푗 ,퐽 1 = 1,퐽 2 = 0.5.(5.22) Here ̂ S 푖 = 휎 푖 /2. The graph has32periodic nearest-neighbor bonds, but only24diagonal bonds: the historical constructor omits the eight diagonals across one periodic seam. The reported data therefore concern this graph, rather than the fully periodic퐽 1 –퐽 2 lattice. All2 16 = 65,536basis statesareincludedwithoutamagnetizationrestriction, allowingexactBorn-weightedpopulation sums. The experiment compares an underparameterized RBM with푃/푁 푠 = 0.27against an overparameterized Vision Transformer with푃/푁 푠 = 3.28, using푁 푠 = 4096and a near-ridgeless shift휆 = 10 −9 . The networks represent complex wave functions, but these offline diagnostics score only the real-amplitude regression: tangent features are∇ 휃 Relog휓 휃 , and targets are Re퐻 loc , each Born-centered. The complex state is retained in the Born distribution and local energy. Phase derivatives and the imaginary local-energy target are excluded, so the following gap and risk values describe this restricted regression, rather than the complete complex-SR error or local infidelity. The excess-risk estimates are means with standard errors over100independent Born- resampled batches for matched noise and10for matched state. Two protocols separate the two effects of overparameterization. In the matched-noise com- 105 parison, RBM and ViT checkpoints are selected with comparable gap variance,휎 2 gap ≈ 0.8. The RBM remains close to the population-optimal amplitude direction, with excess risk approxi- mately0.53 ± 0.08. The ViT has comparable gap variance but much larger excess risk, approx- imately3.39 ± 0.73, because its larger tangent space gives the empirical solve more directions in which to fit finite-sample residuals. In the matched-state comparison, a ViT is fit to represent the same wave function as a partially converged RBM checkpoint, reaching fidelity퐹 ≥ 0.98. Now the represented state is nearly fixed while the tangent space changes. The ViT reduces the gap variance from1.95to0.816and lowers excess risk from approximately2.53 ± 0.19to 1.59 ± 0.08. Thus overparameterization helps when it reduces the expressivity gap, but it can hurt when a comparable gap remains and finite-sample SR overfits it. Large-scaletest.The large-scale test asks whether the same noisy-ridge mechanism appears in a modern NQS setting. The system is a periodic one-dimensional transverse-field Ising family, ̂ 퐻(ℎ) = −ℎ 퐿 ∑ 푖=1 ̂ 푋 푖 − 퐿 ∑ 푖=1 ̂ 푍 푖 ̂ 푍 푖+1 ,퐿 = 100,ℎ ∈ [0.8, 1.2].(5.23) A single foundation neural quantum state represents the conditional wave function log휓 휃 (푥; ℎ) across the Hamiltonian family [147]. The model has푃 = 198,144trainable parameters, while each SR step uses푁 푠 = 12,000samples, organized as6000training fields with two spin config- urations per field. The diagnostic protocol holds the wave function fixed and varies only the solve-time shift. Ten late-training checkpoints from a run trained with휆 train = 10 −4 are reused. For each check- point and each휆 ∈ 10 −8 , 10 −7 , ... , 10 −1 ,푚 = 100independent standard-SR updates are recomputed from fresh batches. Validation uses120,000held-out samples, arranged as6000 held-out fields disjoint from the training fields with20spin configurations per field. Predictions 106 RBM ( P/N S =0.27 ) ViT ( P/N S =3.28 ) Matched Noise σ 2 gap ≈0.8 Matched State σ 2 RBM =1.95 σ 2 ViT =0.816 F=0.9836 ( ' ' !" ' ' ! & ( # ! ' ' !" ' ' ! & ( # ' !" ' ' ' ! & ( # ' !" ' ' ' ! & ( # ( ' ' !" ' ' ! & ( # ! ' ' !" ' ' ! & ( # ' !" ' ' ' ! & ( # ' !" ' ' ' ! & ( # ( ' ' !" ' ' ! & ( # ! ' ' !" ' ' ! & ( # ' !" ' ' ' ! & ( # ' !" ' ' ' ! & ( # d ( ' !" ' ! & ( # ! ' !" ' ! & ( # ' !" ' ' ' ! & ( # ' !" ' ' ' ! & ( # ( ' !" ' ! & ( # ! ' !" ' ! & ( # ' !" ' ' ' ! & ( # ' !" ' ' ' ! & ( # a c b Figure 5.1: Exact amplitude-regression diagnostics for the4 × 4Heisenberg graph of Equa- tion (5.22), with32nearest-neighbor and24diagonal bonds. At푁 푠 = 4096and휆 = 10 −9 , matched noise exposes ViT overfitting at comparable amplitude-gap variance, while matched state shows lower gap variance and excess risk for the ViT. The plotted predictions and risks exclude the phase channel. and local energies are centered separately within each field. Figure 5.2 shows the predicted pat- tern: the validation residual is U-shaped in the solve-time shift, while the multi-batch variance decreases as휆increases. Because all shifts are evaluated at the same fixed checkpoints, this isolates the statistical effect of the ridge filter rather than comparing different training trajecto- ries. The large-scale implementation uses the energy-gradient target2퐻 loc,푐 and doubled update directions, so the raw residual and variance values in the large-scale figures are four times those under the normalization of Equations (5.19) and (5.21). This common factor leaves relative comparisons and shift selection unchanged. 107 1e-81e-71e-61e-51e-41e-31e-21e-1 Solve-time λ 0.9 1.0 1.1 1.2 1.3 1.4 ̂ val ×10 −2 1e-81e-71e-61e-51e-41e-31e-21e-1 Solve-time λ 2 4 6 mb ×10 −3 Figure5.2: Large-scalefixed-checkpointdiagnosticsforthe퐿 = 100transverse-fieldIsingmodel family using a foundation NQS. The validation residual exhibits a U-shaped dependence on the SR diagonal shift, while the multi-batch variance decreases with stronger regularization, as predicted by the noisy-ridge interpretation in Equation (5.17). 5.4Multi-Shift Stochastic Reconfiguration The spectral-filtering view suggests the importance of the diagonal shift in SR. Single-shift SR applies the one-parameter filter푓 휆 (푠) = 푠/(푠 + 휆)to every QGT eigendirection. MS-SR replaces this with a mixture of shifted solves computed on independent batches. For fixed shifts and weights, the population spectral approximation of MS-SR has the filter 푓 푤,휆 (푠) = 퐾 ∑ 푘=1 푤 푘 푠 푠 + 휆 푘 .(5.24) This filter family is richer than any single shifted SR solve. Independent-batch candidates have different empirical QGTs, so their average is not exactly a filter of one empirical matrix. Differ- ent shifts can suppress noisy low-eigenvalue directions while preserving useful high-eigenvalue directions more flexibly than a single휆. Independent batches provide a complementary vari- ance reduction: for fixed shifts and weights, independent candidate errors contribute∑ 푘 푤 2 푘 푉 푘 to the mixture variance, where푉 푘 is the variance of candidate푘in the QGT seminorm. Uni- 108 Algorithm 1Multi-Shift Stochastic Reconfiguration (MS-SR) Require:Parameters휃 푡 , batch size푁 푠 , number of shifts퐾, NTK quantiles푞 푘 퐾 푘=1 , learning rate휂 1:Draw independent batches퐷 1 , ... , 퐷 퐾 ∼ 휋 휃 푡 , each of size푁 푠 2:Use the positive NTK spectrum of퐷 1 to choose shifts휆 푘 from quantiles푞 푘 퐾 푘=1 3:for푘 = 1, ... , 퐾do 4:Solve the shifted minSR system on퐷 푘 to obtain ̂ 훿 푘 ←minSR(퐷 푘 , 휆 푘 ) 5:end for 6:Evaluate leave-one-batch-out tangent-space residuals for the candidate updates 7:Fit simplex-constrained stacking weights푤 ∈ Δ 퐾−1 from the held-out residuals 8:Apply휃 푡+1 ← 휃 푡 − 휂 ∑ 퐾 푘=1 푤 푘 ̂ 훿 푘 form weights give a factor-퐾reduction relative to the average candidate variance in this ideal independent-batch limit. Data-dependent shifts and weights, as well as sampling correlations, require empirical assessment. Computationally,퐾independent minSR solves of size푁 푠 cost 풪(퐾푁 3 푠 )andcanbeparallelized, whereasonesolveonabatchofsize퐾푁 푠 costs풪(퐾 3 푁 3 푠 ). MS- SR is most worthwhile when finite-batch variance is a visible bottleneck and multiple solve-time shifts cover distinct useful parts of the empirical spectrum. In the experiments,퐾 = 4and the shifts are chosen from positive empirical NTK-spectrum quantiles0.9, 0.7, 0.4, 0.1. The weights are fit by leave-one-batch-out stacking. Writing푂 푗 and h 푗 for the centered feature matrix and local-energy vector on batch퐷 푗 , the objective is min 푤∈Δ 퐾−1 , 푤 푗 <1 퐾 ∑ 푗=1 ∥ ∥ ∥ ∥ ∑ 푖≠푗 푤 푖 1 − 푤 푗 푂 푗 ̂ 훿 푖 −h 푗 ∥ ∥ ∥ ∥ 2 .(5.25) Each held-out prediction excludes the candidate fitted on that batch and renormalizes the remain- ing weights. The resulting mixture is convex, but this objective is not generally a convex function of the weights; numerical fitting from uniform weights does not guarantee a global minimum. 109 Checkpoint-local ablations.We evaluate the ingredients of MS-SR on the퐿 = 100TFIM foundation NQS of Section 5.3 and an8 × 8periodic퐽 1 –퐽 2 ViT at퐽 2 /퐽 1 = 0.5. The latter uses Pauli interactions휎 푖 ⋅ 휎 푗 ,128bonds of each type, and the zero-magnetization sector. Its raw energies are four times those of the spin-operator convention in Equation (5.22); squared diagnostics retain the corresponding Pauli units. The ViT has푃 = 147,216real parameters, and its source training progressively adds translations and퐶 4 rotations. Ten fixed source check- points per system and20resampling repeats per checkpoint are used for the ablations. The NTK quantile grid is estimated on a separately seeded reference batch once per checkpoint and reused across repeats. Standard SR uses one batch and the training shift휆 train = 10 −4 .Bagged SRuniformly averages four independent candidates at one common shift, isolating independent-batch averag- ing.Shared-batch multi-shift SRfits four shifts on shared data and uses uniform weights; its stackedcounterpart fits weights on held-out samples using the same candidates.Independent- batch multi-shift SRuses four independent fitting batches with uniform weights, and fullMS-SR learns the weights by Equation (5.25). The shared-batch TFIM comparison uses an independent weight-fit batch; the퐽 1 –퐽 2 shared-batch variants reserve one quarter of their sampled batch for weight fitting. All multi-shift methods use the same quantile grid. Bagging is evaluated both at휆 train and at a checkpoint-specific휆 tuned selected by an eight- point sweep over10 −8 , ... , 10 −1 . The latter minimizes the bagged residual on the reporting data and is therefore an oracle reference requiring additional solves and validation, rather than an independently tuned baseline. Relative to standard SR at the training shift, MS-SR reduces the validation residual and multi-batch variance by approximately39%and43%on TFIM, and by14%and66%on퐽 1 –퐽 2 . Oracle-tuned bagging reduces them by45%and49%, and by18% and80%, respectively, giving lower mean values than MS-SR in both systems. Shared-batch multi-shift SR reduces the TFIM residual by17%, but increases the퐽 1 –퐽 2 residual by2%. 110 SR λ train shared- batch uniform shared- batch stacked bagged SR λ train bagged SR λ tuned indep. batch uniform MS-SR 0.0 0.2 0.4 0.6 0.8 1.0 1.2 diagnostics value 1e−2 TFIM FNQS (L= 100) SR λ train shared- batch uniform shared- batch stacked bagged SR λ train bagged SR λ tuned indep. batch uniform MS-SR 0 1 2 3 4 5 6 J 1 -J 2 (8×8) ̂ val mb Figure 5.3: Checkpoint-local MS-SR ablations at퐾 = 4. Bars show raw validation residuals and multi-batch variances averaged over fixed source checkpoints, with one standard error across checkpoints. Standard SR uses휆 train = 10 −4 . Bagging is shown at both the training shift and an oracle shift selected on the reporting data. All multi-shift methods use the same NTK-quantile grid. Values retain the doubled-target convention and, for퐽 1 –퐽 2 , Pauli Hamiltonian units. On TFIM, MS-SR has27%lower residual than bagging at the training shift, with the same ordering on all ten checkpoints. Training-shift bagging has lower variance but higher total resid- ual, illustrating the cost of shrinkage bias. Bagging near10 −6 –10 −7 achieves a lower residual than MS-SR, showing that a well-chosen single shift can suffice. On퐽 1 –퐽 2 , the mean residuals are3.98for training-shift bagging,3.92for oracle-tuned bagging, and4.13for MS-SR. Thus the benefit of MS-SR is adaptation across regularization scales without a separate validation-based shift search; the results do not establish uniform superiority over well-tuned bagging. Robustness checks with퐾 ∈ 2, 3, 4, 5reduce the TFIM MS-SR residual from0.00854 to0.00638as퐾grows, with diminishing gains and one extra batch and solve per candidate. MS-SR beats training-shift bagging at each퐾, while oracle-tuned bagging remains better. Two alternative quantile grids change the residual by at most3.4%, and checkpoints trained at shifts 10 −3 and10 −5 give near-parity with the corresponding bagged baseline. 111 Online training comparisons.The main online comparison uses five paired continuations from one shared SR checkpoint at iteration2500of the8 × 8 퐽 1 –퐽 2 model. This is a sepa- rate ViT setup from the symmetry-ramp ablations: it has depth8, embedding dimension72, and 12attention heads, without explicit lattice-translation or rotation symmetrization. Both meth- ods use Cholesky solves,푁 푠 = 8192samples per batch, and constant learning rate휂 = 0.01. The Hamiltonian and energy values use the same Pauli convention and zero-magnetization sec- tor as the large-scale퐽 1 –퐽 2 ablations. The comparison concerns later refinement after initial standard-SR training. Standard SR uses휆 = 10 −4 . The historical MS-SR implementation instead uses four fixed shifts10 −2 , 10 −3 , 10 −4 , 10 −5 on independent candidate batches and fits stacking weights on an additional independent batch. It therefore differs from the NTK-quantile, leave-one-batch- out algorithm above. Figure 5.4a plots progress in nominal SR-equivalent updates: one SR step counts as one unit and one four-candidate MS-SR step as four, so500SR steps and125MS- SR steps both reach500units. This axis counts candidates and does not equate actual compute budgets. The historical callback also performs a base-driver SR solve on another batch, giving five solves and six sampled batches per MS-SR step:625solves and750batches for the MS-SR continuation, versus500of each for SR. Each final state is evaluated independently using two fresh-chain replicas, with8,388,608 retained samples in total. Figure 5.4b reports the endpoint energies and paired differences Δ = (퐸 MS-SR − 퐸 SR )/푁, with푁 = 64. MS-SR gives lower energy in three of five pairs, with mean Δ = −1.78×10 −5 and paired95%Student-푡interval[−7.05, 3.50]×10 −5 . The interval in- cludes zero and measures continuation variability conditional on this single source state. These measurements therefore support a qualified online comparison, without establishing a reliable final-energy advantage across independently trained source states or at equal wall time. 112 (a) Training history 0250500 SR-equivalent updates −1.9946 −1.9945 −1.9944 −1.9943 −1.9942 Energy / site Paired continuations (n=5) SR MS-SR (b) Endpoint energies per site PairSRMS-SRΔ (10 −5 ) 1−1.9944675−1.9944307 +3.69 ± 1.09 2−1.9944450−1.9945160−7.10 ± 1.07 3−1.9944733−1.9944722 +0.11 ± 1.11 4−1.9944593−1.9944668−0.75 ± 1.04 5−1.9944451−1.9944934−4.83 ± 1.02 Mean−1.9944580 −1.9944758−1.78 Figure5.4: OnlineSRandMS-SRtrainingonthe8×8 퐽 1 –퐽 2 model. Fivepairedcontinuationsstartfroma single shared SR checkpoint; energies per site are in Pauli units.(a)Training curves show means of traces binned in intervals of20nominal SR-equivalent updates, with±1across-continuation sample standard deviation. Diamonds mark independent endpoint means. The horizontal axis counts one candidate per SR step and four per MS-SR step; the historical implementation’s extra solve and sampled batches are excluded from this nominal count.(b)Independent endpoint energies for the same five pairs. Bold marks the lower energy. Differences areΔ = (퐸 MS-SR − 퐸 SR )/64. Differences and individual±values are in units of10 −5 , with the latter combining the two Monte Carlo standard errors in quadrature. The across- pair confidence interval is given in the text. Update-constructioncost.A separate benchmark at the TFIM step-2000checkpoint measures theNTK-quantile, leave-one-batch-outimplementationofMS-SRononeH10080GBGPU,with 푁 푠 = 12,000and퐾 = 4. All methods use the same eigendecomposition-based pseudoinverse solver. Timing includes sampling, local energies, candidate solves, and construction of the final direction, with parameters held fixed. The first NTK eigendecomposition supplies the quantile shifts and is reused for the first candidate solve; stacking reuses the four candidate batches. The medians over three paired device/seed settings are shown in Table 5.1. MS-SR takes about four times as long as one SR update and adds approximately1.03%over bagged SR in this implementation. 113 MethodBatches/solves Time (s) Relative to SR SR1/1 13.381.00× Bagged SR4/4 53.554.00× MS-SR4/4 54.094.04× Table 5.1: Steady-state update-construction cost on the TFIM foundation NQS using one H100 80GB GPU. Values are medians over three paired device/seed settings, each with two warm-ups and five synchronized timed repetitions. Bagged SR uses the training shift and excludes the cost of shift tuning. 5.5Summary This chapter analyzed SR in a regime increasingly common for modern NQS: expressive, over- parameterized networks trained from finite Monte Carlo samples. The main conclusion is that SR should be viewed as statistical spectral filtering. At a fixed wave function, SR is ridge re- gression from tangent features to centered local energy. The tangent-space expressivity gap is orthogonal to the tangent space in population, but finite batches make it behave as residual noise that empirical SR can fit. The diagonal shift therefore controls a bias-variance tradeoff: it sup- presses variance from sampled gap residuals while shrinking useful components of the ideal tangent-space update. The experiments support this mechanism across scales. Exact4 × 4amplitude-regression experiments separate the two effects of overparameterization: a larger tangent space helps when it reduces the expressivity gap, but it can hurt when it overfits a comparable gap. Large-scale foundation-NQS diagnostics show the predicted U-shaped validation residual and decreasing multi-batch variance as the solve-time shift increases. Finally, the filtering view leads to MS-SR, whose independent multi-shift solves reduce validation residuals and update variance relative to fixed-shift SR. 114 CHAPTER 6 CONCLUSION This thesis studied the interface between quantum computing and artificial intelligence from two directions. Part I asked how quantum models can be made useful for supervised learning; Part I asked how machine learning can help with quantum problems such as real-time decoding and quantum many-body physics. Each chapter addressed a different bottleneck at this interface: representation, generalization, decoding latency, and finite-sample optimization. In Part I, Chapter 2 showed that the data embedding sets an empirical-risk floor for any downstream quantum classifier, and that Neural Quantum Embedding can lower this floor by re- shaping the representation before the data enter the quantum circuit: on binary MNIST the trace distance between class ensembles increased from approximately0.27to0.84, translating into classification accuracy improving from52.7%to96.1%on IBM hardware. The DQC1/NMR extension reached98%accuracy on an NMR quantum device, compared to54%without NQE. Chapter 3 addressed the complementary finite-data question: margin-based metrics predicted generalization in quantum phase recognition more reliably than parameter-based metrics, and the connection between margin mean and trace distance links the two chapters—embeddings with larger state distinguishability can make larger margins attainable. In Part I, Chapter 4 used machine learning for quantum error correction, where the bottle- neck is real-time inference under physical timing constraints. On Sycamore memory-experiment data, the Mamba decoder matched the accuracy of the reproduced Transformer baseline while its inference cost scales as풪(푑 2 )rather than풪(푑 4 )in the code distance. Under simulated 115 real-time decoding with decoder-induced noise, this latency advantage became an accuracy ad- vantage, improving the finite-size effective threshold from푝 th ≈ 0.0097to푝 th ≈ 0.0104. Chapter 5 studied neural quantum states, where the bottleneck is finite-sample optimization. At a fixed wave function, stochastic reconfiguration is tangent-space ridge regression from sam- pled features to centered local energies, and the diagonal shift acts as a statistical spectral filter rather than only a numerical stabilizer. At fixed large-system checkpoints, multi-shift stochas- tic reconfiguration (MS-SR) reduced the validation residual and multi-batch variance relative to standard SR at the training shift by approximately39%/43%on TFIM and14%/66%on퐽 1 –퐽 2 . In five paired online continuations from one shared8×8 퐽 1 –퐽 2 checkpoint, independent endpoint evaluations favored MS-SR in three pairs, with mean energy difference per site−1.78 × 10 −5 and paired95%Student-푡confidence interval[−7.05, 3.50] × 10 −5 . The overall conclusion of this thesis is that quantum computing and AI benefit from a two- way exchange, in which each field supplies principled tools for the other’s hardest problems. Quantum computing challenges AI to operate with new mathematical objects and physical con- straints; AI challenges quantum computing to adopt data-driven representations, scalable archi- tectures, and statistical regularization for problems such as real-time decoding and many-body physics. 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Park, “Multi-channel convolutional neural quantum embed- ding”, Advanced Quantum Technologies9, e00575 (2026). 134 Appendices 135 APPENDIX A SUPPLEMENTARY TECHNICAL DETAILS FOR MAIN CONTRIBUTIONS This appendix collects selected technical details, emphasizing derivations and reproducibility notes that are useful for reading the thesis but too detailed for the main narrative. A.1Additional Details for Neural Quantum Embedding A.1.1Implicit Fidelity Loss and Trace Distance Section 2.2.2 explains that the NQE loss is a pairwise fidelity objective used as a tractable proxy for increasing the trace distance between class-averaged embedded states. For a class푦 ∈ +, −, the empirical ensemble is 휌 푦 = 1 푁 푦 푁 푦 ∑ 푖=1 ∣푥 푦 푖 ⟩ ⟨푥 푦 푖 ∣ .(A.1) The purity of this ensemble is determined by within-class fidelities: Tr[(휌 푦 ) 2 ] = 1 (푁 푦 ) 2 푁 푦 ∑ 푖,푗=1 ∣⟨푥 푦 푖 |푥 푦 푗 ⟩∣ 2 .(A.2) Thus, thesame-labelpartoftheNQElossincreasesclasspuritybypushingwithin-classfidelities toward one. This matters because the trace distance between mixed class ensembles is upper bounded by the trace distance between purifications, and the bound is tightest when the empirical 136 ensembles approach pure states. For different-label pairs, the loss pushes cross-class fidelities toward zero. In a balanced binary dataset with paired examples, the convexity of trace distance gives 퐷 tr ⎛ ⎜ ⎝ 1 푁 푁 ∑ 푖=1 ∣푥 − 푖 ⟩ ⟨푥 − 푖 ∣ , 1 푁 푁 ∑ 푖=1 ∣푥 + 푖 ⟩ ⟨푥 + 푖 ∣ ⎞ ⎟ ⎠ ≤ 1 푁 푁 ∑ 푖=1 √ 1 − ∣⟨푥 + 푖 |푥 − 푖 ⟩∣ 2 .(A.3) Therefore, reducing cross-class fidelities increases the right-hand side of Equation (A.3), while increasingsame-classfidelitiesmakesthemixedensemblesclosertopurerepresentatives. Taken together, these two effects explain why the implicit fidelity objective is aligned with the trace- distance objective used in the empirical-risk lower bound. We also relates the trace-distance-oriented linear-loss analysis to mean-squared error objec- tives often used in QML training. Let푌 = (푦 1 , ... , 푦 푁 )and푓 (푋) = (푓 (푥 1 ), ... , 푓 (푥 푁 )). If 퐿 1 = ‖푌 − 푓 (푋)‖ 1 ,퐿 MSE = ‖푌 − 푓 (푋)‖ 2 2 ,(A.4) then the standard vector-norm inequalities imply 1 푁 퐿 2 1 ≤ 퐿 MSE ≤ 퐿 2 1 .(A.5) Consequently, improving the embedding-induced lower bound for the linear misclassification loss also improves the range in which the empirical MSE loss can lie. Table A.1 summarizes the protocol details that are most relevant for reproducing the Chapter 2 NQE experiments. 137 Table A.1: Selected reproducibility details for the NQE experiments in Hur et al. [48]. ComponentConfigurationPurpose in the thesis argument PCA-NQE networkPCA reduces the input to푛features. A fully connected ReLU network with two hidden layers maps these to2푛 feature-map parameters. Shows that the NQE effect is not tied to a large classical frontend. CNN-NQE networkA two-dimensional CNN processes the original image with pooling stages reducing spatial scale from28 × 28to 14 × 14and7 × 7, followed by an output layer of size2푛. Provides the image-native variant used when the input is not first compressed by PCA. Real-device NQE training Four-qubit hardware runs used SGD for 50 iterations, learning rate0.1, batch size 10. The loss was evaluated on ibmq_toronto using qubits selected by high CNOT fidelity. The Z feature map used one repetition with nearest-neighbor two-qubit gates. Explains why the hardware experiment is a shallow-circuit test of embedding quality rather than a deep quantum-classifier benchmark. Noiseless QCNN classification QCNN classifiers used a general SU(4) convolutional ansatz, Nesterov momentum for 1000 iterations, learning rate0.01, batch size 128, and five random initializations. Separates embedding quality from hardware noise and optimization variability. Hardware QCNN classification Hardware classifiers used a basic convolutional ansatz with two푅 푦 rotations and a CNOT, omitted pooling gates to reduce circuit depth, trained for 50 iterations (lr0.1, batch size 10), and evaluated 500 test samples with 1024 shots. Supports the claim that increasing trace distance can matter more than moderate device noise in the reported setting. 138 Feature-map ansatz robustness.The standard Z feature map used in the NQE-DQC1 exper- iments is a Hamiltonian-inspired encoding. The single-repetition circuit block 푉 (휙) = ⎡ ⎢ ⎣ exp ⎛ ⎜ ⎝ 푖 ∑ 푘 휙 푘 푍 푘 + 푖 ∑ 푘<푙 휙 푘,푙 푍 푘 푍 푙 ⎞ ⎟ ⎠ 퐻 ⊗푛 ⎤ ⎥ ⎦ 푀 (A.6) is, via the conjugation identity퐻푍퐻 = 푋, equivalent to a first-order Trotter expansion of time evolution under a Hamiltonian containing both푋-type and푍-type interactions: 퐻 푋푍 = ∑ 푘 훼 푘 (푋 푘 + 푍 푘 ) + ∑ 푘,푙 훽 푘,푙 (푋 푘 푋 푙 + 푍 푘 푍 푙 ).(A.7) TotestwhetherNQEperformanceissensitivetothisparticularPaulistructure, wealsoevalutated three alternative Hamiltonian-inspired encoding circuits using푋푌,푌 푍, and푋푌 푍interactions. Each circuit preserves the same layer structure—alternating single-qubit rotations and nearest- neighbor two-qubit entangling terms—with the Pauli operators exchanged: 푉 푋푌 (휙) = ⎧ ⎨ ⎩ exp ⎡ ⎢ ⎣ 푖 ∑ 푘 휙 푘 푌 푘 + 휙 푛+푘 푌 푘 푌 푘+1 ⎤ ⎥ ⎦ exp ⎡ ⎢ ⎣ 푖 ∑ 푘 휙 푘 푋 푘 + 휙 푛+푘 푋 푘 푋 푘+1 ⎤ ⎥ ⎦ ⎫ ⎬ ⎭ 푀/2 ,(A.8) 푉 푌 푍 (휙) = ⎧ ⎨ ⎩ exp ⎡ ⎢ ⎣ 푖 ∑ 푘 휙 푘 푍 푘 + 휙 푛+푘 푍 푘 푍 푘+1 ⎤ ⎥ ⎦ exp ⎡ ⎢ ⎣ 푖 ∑ 푘 휙 푘 푌 푘 + 휙 푛+푘 푌 푘 푌 푘+1 ⎤ ⎥ ⎦ ⎫ ⎬ ⎭ 푀/2 ,(A.9) 푉 푋푌 푍 (휙) = exp ⎡ ⎢ ⎣ 푖 ∑ 푘 휙 푘 푍 푘 + 휙 푛+푘 푍 푘 푍 푘+1 ⎤ ⎥ ⎦ exp ⎡ ⎢ ⎣ 푖 ∑ 푘 휙 푘 푌 푘 + 휙 푛+푘 푌 푘 푌 푘+1 ⎤ ⎥ ⎦ ⋅exp ⎡ ⎢ ⎣ 푖 ∑ 푘 휙 푘 푋 푘 + 휙 푛+푘 푋 푘 푋 푘+1 ⎤ ⎥ ⎦ 푀/2 .(A.10) 139 The effective Hamiltonians underlying these three circuits are, respectively, 퐻 푋푌 = ∑ 푘 훼 푘 (푋 푘 + 푌 푘 ) + ∑ 푘 훽 푘 (푋 푘 푋 푘+1 + 푌 푘 푌 푘+1 ),(A.11) 퐻 푌 푍 = ∑ 푘 훼 푘 (푌 푘 + 푍 푘 ) + ∑ 푘 훽 푘 (푌 푘 푌 푘+1 + 푍 푘 푍 푘+1 ),(A.12) 퐻 푋푌 푍 = ∑ 푘 훼 푘 (푋 푘 + 푌 푘 + 푍 푘 ) + ∑ 푘 훽 푘 (푋 푘 푋 푘+1 + 푌 푘 푌 푘+1 + 푍 푘 푍 푘+1 ).(A.13) All three circuits use depth푀 = 4and are tested on MNIST and Fashion-MNIST in place of the standard퐻 푋푍 map. Figure A.1 shows the training curves for each ansatz. Across all six dataset–ansatz combina- tions, the same qualitative behavior holds:퐿 NQE decreases while the class trace distance (inset) increases, and the downstream PQC loss퐿 PQC converges to a lower value than the without-NQE baseline. The classification accuracies with NQE are0.95,0.92, and0.99on MNIST for퐻 푋푌 , 퐻 푌 푍 , and퐻 푋푌 푍 respectively, compared with without-NQE baselines of0.49,0.55, and0.81. On Fashion-MNIST, NQE achieves0.87,0.80, and0.90against baselines of0.49,0.54, and0.50. These results confirm that NQE robustly improves embedding quality regardless of the specific Pauli-interaction structure in the feature-map ansatz. Multi-class extension.For a label set with more than two classes, the NQE-DQC1 objective can be extended by standard one-vs-one or one-vs-rest reductions. A direct multi-class objective is also possible by replacing the binary target with a Kronecker delta: 퐿 NQE = ∑ 푖,푗 [ 1 2 푛 Tr[푉 (푔(푥 푖 ))푉 † (푔(푥 푗 ))] − 훿 푦 푖 푦 푗 ] 2 .(A.14) This remains DQC1-compatible because each term only requires estimating a Hilbert–Schmidt inner product between two data-dependent unitaries. 140 7 MNIST Fashion-MNIST 0.2 0 0.4 0.6 0.6 0.2 1.0 510150 510150 Training set 510150 Test set 0.2 0 0.4 0.6 0.6 0.2 1.0 510150 Training set 510150 Test set 0.6 0.2 1.0 510150 Training set 510150 Test set 0.2 0 0.4 0.6 0.2 0 0.4 0.6 0.2 0 0.4 0.6 0.2 0 0.4 0.6 0.2 0 0.4 0.6 0.2 0 0.4 0.6 0.2 0 0.4 0.6 0.2 0 0.4 0.6 0.2 0 0.4 0.6 0.2 0 0.4 0.6 0.6 0.2 1.0 510150 Training set 510150 Test set 0.6 0.2 1.0 510150 Training set 510150 Test set 0.6 0.2 1.0 510150 Training set 510150 Test set Without NQE (acc=0.49) With NQE (acc=0.95) Without NQE (acc=0.55) With NQE (acc=0.92) Without NQE (acc=0.54) With NQE (acc=0.80) Without NQE (acc=0.49) With NQE (acc=0.87) Without NQE (acc=0.50) With NQE (acc=0.90) Without NQE (acc=0.51) With NQE (acc=0.99) Itera�on 510150 Itera�on 510150 Itera�on 510150 Itera�on FIG. S5. NQE-DQC1 training results with three alternative feature maps (H XY ,H YZ , andH XYZ ) on MNIST and Fashion-MNIST datasets. The results show consistent convergence across different ansatz choices. As explained in the main text, the Z-feature map used in our implementation is inspired by Hamiltonian dynamics. More generally, the ansatz can be written in a compact form as: V(ω)= ! " exp # i $ k ω k Z k +i $ k,l ω k,l Z k Z l % H →n & ' M , which, by using the identityHZH=X, is equivalent to a first-order Trotter expansion of a time evolution under a Hamiltonian containing bothXandZterms: H XZ = $ k ε k (X k +Z k )+ $ k,l θ k,l (X k X l +Z k Z l ). Building on this Hamiltonian perspective, we explored other ansatzes constructed by varying the Pauli terms in the effective Hamiltonian. Specifically, besides the standardH XZ ansatz (Z-feature map), we tested three additional cases inspired by Hamiltonians with different interactions:H XY ,H YZ ,andH XYZ . These encoding circuits preserve the structure of Eq. (2) from the main text, comprising layers of single-qubit and two-qubit gates with the same connectivity. The explicit forms are given by: V XY (ω)= ( exp ) i $ k ω k Y k +ω n+k Y k Y k+1 * exp ) i $ k ω k X k +ω n+k X k X k+1 *+ M/2 ,(6) V YZ (ω)= ( exp ) i $ k ω k Z k +ω n+k Z k Z k+1 * exp ) i $ k ω k Y k +ω n+k Y k Y k+1 *+ M/2 ,(7) V XYZ (ω)= ( exp ) i $ k ω k Z k +ω n+k Z k Z k+1 * exp ) i $ k ω k Y k +ω n+k Y k Y k+1 * exp ) i $ k ω k X k +ω n+k X k X k+1 *+ M/2 . (8) For consistency, all ansatzes used a circuit depth ofM=4. We trained the NQE-DQC1 protocol with each ansatz on the two datasets: MNIST and Fashion-MNIST. The training curves, shown in Fig.S5, reproduce the key results of Figure 2(b) and 3(b) in the main text for each ansatz-dataset pair. As seen, both cases demonstrate stable training: the NQE loss decreases while the trace distance between classes increases steadily. Figure A.1: NQE-DQC1 training results for the three alternative Hamiltonian-inspired fea- ture maps (퐻 푋푌 ,퐻 푌 푍 ,퐻 푋푌 푍 ) on MNIST (left four columns) and Fashion-MNIST (right four columns). For each ansatz row, the left panel of each dataset shows the NQE loss퐿 NQE with the class trace distance during training in the inset; the right panel shows the downstream PQC loss 퐿 PQC with (solid) and without (dashed) NQE. All ansatzes use circuit depth푀 = 4. The consis- tent decrease in퐿 NQE and improvement in퐿 PQC confirm that the NQE benefit is not specific to the푍(퐻 푋푍 ) feature map. A.2Additional Details for Margin Generalization A.2.1Proof Structure and Mixed-State Extension Theorem 3.1 is derived by combining a ramp-loss margin argument with covering-number bounds for the QNN-induced function class. First, define the ramp-loss class ℱ 훾 = (휌, 푦) ↦ 푙 훾 (ℳ(ℎ(휌), 푦)) ∶ ℎ ∈ ℋ .(A.15) 141 The standard Rademacher-complexity bound gives, with probability at least1 − 훿, a uniform bound on the expected ramp loss in terms of the empirical ramp loss, the sample Rademacher complexity ofℱ 훾 , and the concentration term √ ln(2/훿)/(2푚). Since the ramp loss upper bounds the zero-one loss and is upper bounded by the empirical margin error at threshold훾, the remaining problem is to controlℜ(ℱ 훾 | 푆 ). For pure input states, write the classifier output as 푔(푈푥) = (푥 † 푈 † 퐸 1 푈푥, ... , 푥 † 푈 † 퐸 푘 푈푥) .(A.16) The composed ramp-margin map is Lipschitz with constant proportional to퐸/훾, where퐸 = √ ∑ 푘 푖=1 ‖퐸 푖 ‖ 2 휎 .Thus, the covering number of the restricted ramp-loss class can be controlled by the covering number of the set of transformed training states푈푋 ∶ 푈 ∈ 핌 QNN : ln풩(ℱ 훾 | 푆 , 휖, ‖ ⋅ ‖ 2 ) ≤ ⌈ 32푚푏 2 퐸 2 휖 2 훾 2 ⌉ ln(4푁 2 ),(A.17) where푁 = 2 푛 is the Hilbert-space dimension and푏bounds the distance of the QNN unitary from a reference unitary in the ‖ ⋅ ‖ 2,1 sense. Applying Dudley’s entropy integral to Equation (A.17) yields the Rademacher term ℜ(ℱ 훾 | 푆 ) = ̃ 푂( 푏퐸 훾 √ 푛 푚 ) , (A.18) which gives the theorem in Equation (3.18). The same argument extends to mixed input states by vectorization. For 휌 = ∑ 푖,푗 휌 푖푗 |푖⟩ ⟨푗∣ ,|휌⟩ = ∑ 푖,푗 휌 푖푗 |푖⟩ ⊗ ∣푗⟩ ,(A.19) 142 the unitary action satisfies |푈휌푈 † ⟩ = (푈 ⊗ 푈 ∗ )|휌⟩.(A.20) The proof then proceeds with the distance bound applied to푈 ⊗ 푈 ∗ rather than푈, and with the measurement-size term evaluated using the Frobenius norms of the POVM elements. A.2.2Experimental Protocol Notes TableA.2recordstheexperimentalchoicesfromHurandPark[65]thatareusefulforinterpreting the margin plots in Chapter 3. Table A.2: Selected reproducibility details for the margin-generalization experiments. ExperimentProtocol details Quantum phase recognition The QPR task uses an eight-qubit generalized cluster Hamiltonian with four phases. The training set has 20 data points evenly split across the four classes, and the test set has 1000 samples. Label corruption levels of 0%, 50%, and 100% probe the overfitting regime. QPR optimizationQCNNs are trained with Adam, learning rate0.001, and full-batch updates for up to 5000 iterations. Early stopping compares consecutive 500-iteration loss windows. Margin-distribution plots average over 15 repetitions with different training samples. Ansatz comparisonsThe main comparison uses QCNNs with distinct two-qubit PQC parameters. The supplement repeats the margin-distribution test with shared-parameter QCNNs and Strongly Entangling Layers, showing the same qualitative leftward margin shift as label corruption increases. Embedding comparison The classical-data experiment uses the first two classes of MNIST, Fashion-MNIST, and Kuzushiji-MNIST. PCA reduces inputs to match the qubit count. The compared embeddings are a three-layer Z feature map, a trainable quantum embedding with푌and푌 푌rotations, and NQE with a fully connected ReLU network of layer dimensions[8, 16, 32, 32, 16, 8]. 143 A.3Additional Details for Stochastic Reconfiguration as Spectral Filtering A.3.1Fixed-Checkpoint Regression Identities Chapter 5 presents the real-valued fixed-checkpoint regression view of SR. The appendix of the SR paper [138] provides several identities that clarify the assumptions behind that view. Complex convention.For complex wave functions, define 푂 푐 (푥) = ∇ 휃 log휓 휃 (푥) − 피 휋 휃 [∇ 휃 log휓 휃 ],퐻 loc,푐 (푥) = 퐻 loc (푥) − 피 휋 휃 [퐻 loc ].(A.21) The Hermitian QGT and force are 푄 = 피 휋 휃 [푂 푐 (푥) 푂 푐 (푥) 푇 ],푓 = 피 휋 휃 [푂 푐 (푥) 퐻 loc,푐 (푥)].(A.22) For complex parameter increments, SR solves푄훿 = 푓in the QGT seminorm. For real parameter increments in a complex wave function, the equivalent real least-squares system is Re[푄] 훿 =Re[푓 ],(A.23) with푆 =Re[푄]and푔 =Re[푓 ]. Theregressionlossis피 휋 휃 |푂 푐 (푥) 푇 훿−퐻 loc,푐 (푥)| 2 . Theidentities below use real-valued features and targets. For the complex extension, scalar squares become modulus squares and the seminorm uses the corresponding real or Hermitian QGT. Validation residual.Let훿 ∗ be the population least-squares SR direction and define the ex- pressivity gap 휖(푥) = 퐻 loc,푐 (푥) − 푂 푐 (푥) 푇 훿 ∗ .(A.24) 144 The normal equations give피 휋 휃 [푂 푐 (푥)휖(푥)] = 0. Therefore, for any fixed update훿, 피 휋 휃 [(푂 푐 (푥) 푇 훿 − 퐻 loc,푐 (푥)) 2 ] = ‖훿 − 훿 ∗ ‖ 2 푆 + 휎 2 gap ,(A.25) where휎 2 gap = 피 휋 휃 [휖(푥) 2 ]. This is the identity behind Equation (5.20): held-out tangent resid- uals estimate excess risk plus a constant irreducible gap. Bias–variance decomposition.At a fixed checkpoint, let푍 = ̂ 훿 휆 be the random shifted-SR update obtained by resampling the Monte Carlo batch, and let휇 = 피 퐷 [푍]. Then 피 퐷 ‖푍 − 훿 ∗ ‖ 2 푆 = ‖휇 − 훿 ∗ ‖ 2 푆 + 피 퐷 ‖푍 − 휇‖ 2 푆 .(A.26) The multi-batch diagnostic in Equation (5.21) estimates the second term by evaluating the prediction-space disagreement among independent updates on a validation batch. Spectral proxy and local geometry.On the non-null QGT subspace, use the eigendecompo- sition 푆 = 푉diag(푠 푖 )푉 푇 ,훽 ∗ 푖 = (푉 푇 훿 ∗ ) 푖 .(A.27) The population ridge direction has shrinkage bias ‖훿 휆 − 훿 ∗ ‖ 2 푆 = ∑ 푖 푠 푖 ( 휆 푠 푖 + 휆 ) 2 (훽 ∗ 푖 ) 2 .(A.28) Under a scalar residual-noise approximation, the finite-sample variance scales as 휎 2 gap 푁 푠 ∑ 푖 ( 푠 푖 푠 푖 + 휆 ) 2 ,(A.29) 145 giving the spectral-risk proxy in Equation (5.17). The same푆-seminorm also controls local Fubini–Study infidelity. For nearby displacements푎and푏, 1 − |⟨Ψ(휃 + 푎)|Ψ(휃 + 푏)⟩| 2 = (푎 − 푏) 푇 푆(푎 − 푏) + 푂((‖푎‖ + ‖푏‖) 3 ).(A.30) WiththefullQGT,regressionexcessriskthereforemeasuresthediscrepancyfromthepopulation tangent-space imaginary-time step in the local state geometry. The small-system diagnostics in Chapter5insteadretainonlytheamplitudefeatures∇ 휃 Relog휓 휃 andthereallocal-energytarget, while evaluating local energies and Born probabilities from the complex state. Their covariance omits the phase channel, so those amplitude regression risks do not measure full complex-SR error or complete local infidelity. Sample centering.Practical SR uses sample-centered log-derivatives and sample-centered lo- cal energies. This is equivalent to ridge regression with an unpenalized intercept: min 푎,훿 1 푛 푛 ∑ 푗=1 (푎 + 푂(푥 푗 ) 푇 훿 − 퐻 loc (푥 푗 )) 2 + 휆‖훿‖ 2 2 .(A.31) Optimizing over푎subtracts the sample means of푂and퐻 loc . This removes the constant en- ergy/normalization component but does not remove the푥-dependent expressivity gap that drives the finite-sample overfitting effect. Large-system normalization.The large-scale implementation uses target2퐻 loc,푐 and corre- spondingly doubled SR directions. Its raw validation residuals and multi-batch variances are therefore four times those under the퐻 loc,푐 convention used in the regression derivation; relative comparisons and method ordering are unchanged. The large퐽 1 –퐽 2 experiments also use Pauli operators. Converting that Hamiltonian and its SR directions to the spin conventionS= 휎/2 146 would divide the squared diagnostics by16, independently of the doubled-force convention. The reported figures retain the original normalization. Table A.3 summarizes the reproducibility details from the SR paper [138] that are most relevant to the claims in Chapter 5. Table A.3: Selected protocol details for the SR spectral-filtering experiments. SettingDetails Exact4 × 4 퐽 1 –퐽 2 Archived graph with 32 periodic nearest-neighbor bonds and 24 diagonal bonds, omitting eight diagonal bonds across one seam. Couplings are 퐽 1 = 1,퐽 2 = 0.5in the spin conventionS= 휎/2. All2 16 = 65,536states enter the exact Born sums for the amplitude covariance, amplitude-optimal update, and amplitude gap variance. The RBM has 1120 real parameters at 푃/푁 푠 = 0.273with푁 푠 = 4096. The ViT has 13,448 real parameters at 푃/푁 푠 = 3.283. Matched-noise and matched-state diagnostics Matched-noise pairs independently trained RBM and ViT checkpoints with comparable amplitude gap variance and draws 100 independent batches at 휆 = 10 −9 . Matched-state first fits a ViT to a partially converged RBM wave function to fidelity퐹 = 0.983626, then compares amplitude regression diagnostics from 10 resampled batches. 1D TFIM foundational NQS The large-scale TFIM diagnostic uses a periodic퐿 = 100transverse-field Ising family withℎ ∈ [0.8, 1.2]. The conditional ViT foundation NQS has six layers, model dimension 72, 12 attention heads, patch size 4, translational invariance, and 198,144 parameters. Each SR step uses 12,000 samples from 6000 field replicas with two spin configurations per field. Large-scale diagnostics The shared-checkpoint run uses training shift휆 train = 10 −4 . Diagnostic checkpoints are steps 1100 through 2000 for TFIM. Solve-time shifts are swept over10 −8 , 10 −7 , ... , 10 −1 . The TFIM U-curve uses 100 independently resampled updates per shift; ablations use 20 repeats per checkpoint. Validation uses 120,000 held-out TFIM samples or 100,000 held-out퐽 1 –퐽 2 samples. MS-SR ablationsOffline ablations use퐾 = 4NTK-spectrum quantiles0.9, 0.7, 0.4, 0.1, cached per checkpoint from an independent reference batch. Comparisons include standard SR, shared-batch uniform and stacked multi-shift, independent-batch uniform multi-shift, and full MS-SR with leave-one-batch-out stacking. Bagged SR uses either the training shift or a checkpoint-tuned shift selected on the reporting data by the eight-point sweep above; the tuned variant is an oracle reference. 147 APPENDIX B RELATED NON-FIRST-AUTHOR CONTRIBUTIONS This appendix records two closely related collaborative papers in which the NQE methods intro- duced in Chapter 2 are extended and validated in application domains outside the primary thesis narrative. The goal is not to reproduce those papers in detail, but to summarize the connection to the thesis and the main empirical findings. Table B.1 gives a compact overview. Both papers extend the Neural Quantum Embedding perspective developed in Chapter 2: one applies trainable quantum embeddings to ligand-based virtual screening, and the other generalizes NQE to multi-channel image data. Table B.1: Summary of non-first-author contributions related to the main thesis. PaperThesis connection Data domainMethodHeadline result Optimizing Quantum Data Embeddings for Ligand-Based Virtual Screening [155] NQE application (Chapter 2) Molecular descriptors from LIT-PCBA and COVID-19 ligand datasets NQE with Z/XYZ feature maps, QCNN classifiers, quantum kernels, and quantum-pretrained classical embeddings NQE increased class trace distances; quantum or hybrid variants outperformed classical RBF/linear baselines, especially in limited-data or imbalanced settings. Multi-Channel Convolutional Neural Quantum Embedding [156] NQE extension (Chapter 2) CIFAR-10 and Tiny ImageNet binary image tasks Convolutional NQE interfaces for multi-channel data, followed by QCNN classification Trace distance and classification accuracy were strongly correlated. Parameter-efficient multi-channel interfaces achieved competitive accuracy with fewer trainable parameters. 148 B.1Quantum Embeddings for Ligand-Based Virtual Screening Choi et al. [155] studied whether optimized quantum data embeddings can improve ligand- based virtual screening, a setting where labeled biological data are often limited and class- imbalanced. The study used molecular descriptors computed from SMILES strings and eval- uated two benchmark collections: LIT-PCBA, containing multiple biological targets with se- vere activator/inactivator imbalance, and a smaller COVID-19 ligand dataset. This application is naturally connected to Chapter 2 because the central question is whether trainable quantum embeddings can produce more useful molecular representations than fixed or purely classical embeddings. For the LIT-PCBA experiments, the paper compared NQE models using Z and XYZ quan- tum feature maps against a classical neural-network parameterized RBF embedding. The NQE- pretrained embeddings were then evaluated with QCNN classifiers, while the classical embed- dings were evaluated with comparable shallow classical classifiers. In the Z-feature-map set- ting, NQE increased the trace distance between class ensembles after training across the reported conditions. For example, the GBA target in the balanced setting increased from approximately 5 × 10 −4 to about0.55on the test set. This mirrors the main mechanism of Chapter 2: improving the embedding geometry can lower the effective discrimination barrier faced by the downstream classifier. Figure B.1 illustrates the most direct connection to the trace-distance argument in Chapter 2. Beforetraining, theZZfeaturemaptypicallyproducesclass-ensembletracedistancesnear10 −4 – 10 −3 for these molecular descriptors. After NQE training, the distances move into the10 −1 range or higher for several targets, including GBA, ESR1 antagonist, MAPK1, FEN1, PKM2, and VDR. The effect is visible in both the training and test panels, indicating that the learned embedding is not merely separating the sampled training pairs but also improving the geometry 149 10 FIG. 4. Trace distance changes before and after training of NQE with the Z feature map. “Before” and “After” indicate the trace distances of NQE prior to and following training, respectively. “1:1” and “1:6” denote the class ratios of the dataset. TABLE VIII. Classification performances of SVM with quantum kernels evaluated on the COVID-19 dataset. Condition Balanced accuracy (4-qubits / 8-qubits) PCA + SVM (Z feature map kernel)0.71±0.00 / 0.76±0.00 PCA + SVM (PQK a with the Z feature map) 0.83±0.00 / 0.80±0.00 PCA + SVM (XYZ feature map kernel)0.50±0.00 / 0.50±0.00 PCA + SVM (PQK with the XYZ feature map) 0.72±0.00 / 0.73±0.00 PCA + SVM (linear kernel)0.59±0.00 PCA + SVM (RBF kernel)0.60±0.00 SVM (linear kernel)0.62±0.00 SVM (RBF kernel)0.65±0.00 a PQK:theprojectedquantumkernel bedding method, which utilizes NQE, and its classical counterpart, which employs RBF kernel with the classical neural network. Since the same classical neural network architecture was used in both NQE and RBF kernel, the key di!erence between the two approaches lies solely in their embedding methods, that is, the Z feature map and XYZ feature map for the quantum approach versus RBF kernel for the classical approach. Based on this di!erence, the classification performance of the binary classifiers was compared as an evaluation of embedding performance (the 8-qubit QCNN for the quantum model and the single-layer network for the classical model). Among the 32 comparison conditions (1:1 and 1:6 class ratio conditions for each biological target and two types of quantum feature maps), the QCNN with NQE achieved higher classification performance than the single-layer network with RBF kernel in most conditions, except for the 1:1 ratio case with the XYZ feature map. While the XYZ feature map-based NQE under 1:1 ra- tio showed lower performance than its classical counter- part for all biological targets, the overall trend indicates that quantum embedding methods utilizing classically intractable quantum feature maps generally outperform classical methods across various feature maps and class ratios. In addition, for the COVID-19 dataset analyzed using SVM with the 4- and 8-qubits Z feature map ker- nel, the XYZ feature map kernel, and the PQK showed improved classification performances compared to SVMs Figure B.1: Trace-distance increase obtained by NQE training with the Z feature map on LIT- PCBA targets. The plot compares trace distances before and after NQE training for balanced (1:1) and imbalanced (1:6) activator/inactivator settings, separately for training and test sets. The log scale highlights that the untrained embeddings produce nearly indistinguishable class ensembles, while NQE raises the class separation by several orders of magnitude for many tar- gets. of held-out ligand examples. The paper also explored quantum-pretrained classical embeddings, where neural networks trained through NQE were reused in classical classifiers. These transfer-style variants often im- proved performance in limited-data or class-imbalanced cases, suggesting that the representation learnedduringquantumembeddingoptimizationcanbeusefulevenoutsideafullyquantumclas- sifier. On the COVID-19 dataset, projected quantum kernels with the Z feature map achieved balanced accuracies of0.83and0.80for 4- and 8-qubit configurations, respectively, compared 150 with classical SVM baselines in the range0.59–0.65. The main takeaway is that NQE-style em- bedding optimization can be adapted to molecular screening as a data-efficient representation- learning tool, although hardware execution and biological interpretability remain directions for future work. B.2Multi-Channel Convolutional Neural Quantum Embedding Kim et al. [156] extended the NQE framework to multi-channel image data through convolu- tional neural quantum embedding (CNQE). Whereas the original NQE experiments in Chapter 2 focused mainly on low-dimensional or single-channel inputs after preprocessing, CNQE studies how a classical convolutional interface should map multi-channel data, such as RGB images, into quantum feature-map parameters. The work introduced three interface structures, denoted 푔 푎 ,푔 푏 , and푔 푐 , that differ in whether channels are processed jointly or separately before quantum embedding. The main theoretical and empirical point is that the classical-to-quantum interface controls both parameter efficiency and the separability of the resulting quantum states. The paper com- pared fidelity-based and Hilbert–Schmidt-based NQE losses, multiple embedding circuits, and QCNN classifiers on binary tasks from CIFAR-10 and Tiny ImageNet. Across the tested config- urations, trace distance and downstream classification accuracy were strongly correlated, with Pearson correlation푟 = 0.7926and Spearman correlation휌 = 0.8184reported across model configurations. This supports the trace-distance view developed in Chapter 2: embeddings that better separate class ensembles tend to give the classifier a better achievable operating point. Figure B.2 is the key empirical diagnostic from the CNQE study. The figure does not show a single model comparison; instead, it aggregates many design choices and asks whether the NQE quantity optimized before classifier training remains predictive once a QCNN classifier 151 w.advancedsciencenews.comwww.advquantumtech.com Lorentz rotations are expressed as J x = ! " " " # !""" "!"" ""cos!!sin! ""sin!cos! $ % % % & ,J y = ! " " " # !""" "cos!"sin! ""!" "!sin!"cos! $ % % % & , J z = ! " " " # !""" "cos!!sin!" "sin!cos!" """! $ % % % & , (#") whereJ x ,J y ,andJ z represent rotations around thex,y,andz axes, respectively. These rotations can be implemented using the two-qubit gatesCRY ji ,CRY ij ,R X+Y ij defined as CRY ji (#!)= ! " " " # !""" "!"" ""cos!!sin! ""sin!cos! $ % % % & , CRY ij (#!)= ! " " " # !""" "cos!"sin! ""!" "!sin!"cos! $ % % % & , R X+Y ij (#!,""#)= ! " " " # !""" "cos!!sin!" "sin!cos!" """! $ % % % & . (#!) Using this framework, we construct circuits composed of single- qubit rotation layers around thex-andy-axes followed by the cor- responding Lorentz rotation layer around the same axis, which is defined as U $ (!)= ' i CRY i # i (## # i (!)) exp(i ( i # i (!)X i ), U % (!)= ' i CRY i # (## # i (!)) exp(i ( i # i (!)Y i ), (##) where# i (!)=! i ,# # i (!)=! i+n andi # =i+!(mod n). Each unit has #ninput parameters. The three-layer embedding circuit is then given by U NCL (! (!) ,! (#) ,! (&) )=U $ (! (&) )U % (! (#) )U $ (! (!) ).(#&) This structure is referred to as a NCL (noncommutative Lorentz) feature map. !. Numerical Experiments We evaluate CNQE under various configurations, di'erent in- terface models, loss functions, and embedding circuits. Experi- ments are conducted on binary classification tasks using CIFAR- !" [(&] and Tiny ImageNet, [(&] because of their size and class di- versity (see also Ref. [!)] for another study applying quantum models to CIFAR-!"). Three binary classification tasks are se- lected for evaluation: Airplane–Automobile and Frog–Ship from Figure !.Relationship between trace distance and classification accuracy across datasets. The solid black line represents the linear regression fit with !"% confidence interval shaded. CIFAR-!", and School Bus–Maypole from Tiny ImageNet. Fur- ther details regarding the datasets and training procedures are provided in AppendixC. The main simulation results are summarized inTable!and were obtained on a (-qubit system. To examine scalability, we ad- ditionally performed experiments on $- and *-qubit systems us- ing the CIFAR-!"frog–ship pair and the Tiny ImageNet class pair. As reported in AppendixD, these results show that perfor- mance remains stable as the number of qubits and trainable pa- rameters increase, indicating that CNQE-QCNN maintains e'ec- tiveness without degradation from the larger state space. Perfor- mance is assessed using two metrics. The trace distance is used to evaluate the embedding quality of CNQE, and the classifica- tion accuracy is measured after training a QCNN ansatz on the embedded states. Supplementary results for precision, recall, and F!-score across all binary classification experiments are provided in AppendixG.Theseadditionalevaluationsareconsistentwith the accuracy-based conclusions, further reinforcing the robust- ness of our findings. In this analysis, first, we examine the overall relationship be- tween trace distance and classification accuracy. Next, we inves- tigate how the trace distance varies depending on the interface model, quantum loss function, and embedding circuit. We also analyze the impact of di'erent interface models on classification accuracy. In addition, we compare our models with purely clas- sical neural networks. We further consider models that employ classical dimensionality reduction methods instead of CNQE. Fi- nally, we evaluate the performance and noise robustness of our CNQE-based model relative to these baselines. !.". Trace Distance-Accuracy Correlation Figure"shows the correlation between trace distance and classi- fication accuracy across all configurations. Each data point corre- sponds to a configuration, with the median trace distance and mean classification accuracy computed over five independent runs. Tables#presents the corresponding numerical values. The figure demonstrates a strong positive correlation sup- ported by Pearson’sr=".%+#$(p<".""!) and Spearman’s$= ".*!*((p<".""!). These results indicate tendency of larger trace Adv.QuantumTechnol.#$#%,!,e##"$"e$$&'&('of "()© %#%"The Author(s). Advanced Quantum Technologies published by Wiley-VCH GmbH 25119044, 2026, 1, Downloaded from https://advanced.onlinelibrary.wiley.com/doi/10.1002/qute.202500575 by Western Institute Of Technolog, Wiley Online Library on [04/06/2026]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Figure B.2: Relationship between class trace distance and QCNN classification accuracy across CNQE configurations on CIFAR-10 and Tiny ImageNet binary tasks. Each point corresponds to a choice of dataset, classical-to-quantum interface, loss function, and embedding circuit. The positive trend supports the use of trace distance as an embedding-quality diagnostic for multi- channel NQE. is trained afterward. The observed monotone trend is important because it links architectural engineeringchoices, suchaschannel-wiseversusjoint-channelinterfaces, backtothesamestate- discrimination quantity used throughout Chapter 2. Empirically, CNQE-QCNN models achieved high accuracy on the selected multi-channel tasks, including results near95%on CIFAR-10 frog–ship and Tiny ImageNet school-bus– maypole binary classification tasks in favorable configurations. The channel-separated interface 푔 푐 was particularly interesting: although it produced smaller trace distances than푔 푎 and푔 푏 in some statistical comparisons, it used the fewest trainable parameters and still achieved com- petitive classification accuracy. The paper therefore broadens NQE from a basic embedding 152 optimizer into a practical design space for image-like, multi-channel inputs. 153