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One Qubit Can Beat One Bit: Quantum Advantage for Post-Training Quantization

Yuma Ichikawa, Moeto Mishima

Year: 2026Venue: arXiv preprintArea: quant-phType: PreprintEmbeddings: 170

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 89%

Last extracted: 8/8/2026, 2:41:46 AM

Summary

The paper introduces Quantum Random Access Quantization (QRAQ), a framework for post-training quantization (PTQ) that leverages quantum random access codes (QRACs) to overcome the 'shared-sign constraint' of classical one-bit quantization. In classical one-bit PTQ, a single binary sign matrix is shared across all deployment contexts, which is suboptimal when different contexts (e.g., attention heads, tasks) favor different sign patterns. QRAQ encodes context-dependent signs into a quantum state (logical qubit) and retrieves them via context-matched Pauli measurements. The authors prove that QRAQ achieves a strictly lower ideal reconstruction risk than classical one-bit PTQ when optimal context-wise signs are incompatible. They also derive conditions for this advantage under finite-shot noise and depolarizing noise, identifying measurement incompatibility as the key resource rather than quantization alone.

Entities (8)

Quantum Random Access Quantization · method · 98%Post-Training Quantization · concept · 95%Quantum Random Access Code · concept · 92%Pauli Measurement · operation · 90%Shared-Sign Constraint · problem · 88%Measurement Incompatibility · resource · 87%Reconstruction Risk · metric · 85%depolarizing noise · noise-model · 82%

Relation Signals (6)

Quantum Random Access Quantization → solves → Shared-Sign Constraint

confidence 95% · We study this shared-sign constraint and introduce Quantum Random Access Quantization (QRAQ).

Quantum Random Access Quantization → uses → Quantum Random Access Code

confidence 93% · This framework encodes context-dependent signs in a quantum random-access code...

Quantum Random Access Quantization → outperforms → Post-Training Quantization

confidence 90% · QRAQ achieves a strictly lower ideal reconstruction risk... separation from shared-sign one-bit PTQ

Pauli Measurement → enables → Context-Dependent Sign Retrieval

confidence 88% · retrieves them via context-matched Pauli measurements.

Measurement Incompatibility → provides → Quantum Advantage

confidence 85% · the relevant resource in this model is measurement incompatibility rather than quantization alone.

Quantum Random Access Quantization → isaffectedby → depolarizing noise

confidence 80% · derive finite-shot and calibrated-noise conditions under which this separation is retained.

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Abstract

Abstract:One-bit post-training quantization represents each weight using only its sign, requiring all deployment contexts to share the same binary weight matrix even when their activation statistics favor different sign patterns. We study this shared-sign constraint and introduce Quantum Random Access Quantization (QRAQ). This framework encodes context-dependent signs in a quantum random-access code and retrieves them via context-matched Pauli measurements. Under an explicit fresh-copy logical readout model, QRAQ produces an unbiased, context-specific binary surrogate with a tractable shot-noise penalty. We prove a row-wise separation from shared-sign one-bit PTQ with signed per-row scales. When the optimal context-wise signs are incompatible, QRAQ achieves a strictly lower ideal reconstruction risk. We also derive finite-shot and calibrated-noise conditions under which this separation is retained. Fixed-readout quantum schemes are classically simulable, so the relevant resource in this model is measurement incompatibility rather than quantization alone. Finally, we characterize the role of scale granularity, provide finite-sample certificates, and evaluate the predicted ideal, finite-shot, noisy, and multi-context regimes in simulator experiments.

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One Qubit Can Beat One Bit: Quantum Advantage for Post-Training Quantization Yuma Ichikawa Fujitsu Limited, RIKEN for AIP &Moeto Mishima Fujitsu Limited Abstract One-bit post-training quantization represents each weight using only its sign, requiring all deployment contexts to share the same binary weight matrix even when their activation statistics favor different sign patterns. We study this shared-sign constraint and introduce Quantum Random Access Quantization (QRAQ). This framework encodes context-dependent signs in a quantum random-access code and retrieves them via context-matched Pauli measurements. Under an explicit fresh-copy logical readout model, QRAQ produces an unbiased, context-specific binary surrogate with a tractable shot-noise penalty. We prove a row-wise separation from shared-sign one-bit PTQ with signed per-row scales. When the optimal context-wise signs are incompatible, QRAQ achieves a strictly lower ideal reconstruction risk. We also derive finite-shot and calibrated-noise conditions under which this separation is retained. Fixed-readout quantum schemes are classically simulable, so the relevant resource in this model is measurement incompatibility rather than quantization alone. Finally, we characterize the role of scale granularity, provide finite-sample certificates, and evaluate the predicted ideal, finite-shot, noisy, and multi-context regimes in simulator experiments. 1 Introduction Post-training quantization (PTQ) compresses foundation models by replacing each linear layer with a low-bit surrogate that is calibrated to match the full-precision outputs of layers on representative activations (Touvron et al., 2023; Grattafiori et al., 2024; Gong et al., 2024; Frantar et al., 2023; Lin et al., 2024; Chee et al., 2023; Tseng et al., 2024; Ashkboos et al., 2024). From the perspective of machine learning, this is the standard layer-wise reconstruction problem: selecting a compact codebook such that W^X WX remains close to WXWX. From the perspective of quantum information, it is a finite-dimensional quadratic projection problem under a stringent memory constraint. In the extreme one-bit-per-weight setting, each memory cell can store only a sign; the same sign matrix must be reused across all deployment contexts. This constraint is particularly limiting when a single layer is invoked under different attention heads, routing patterns, task heads, prompt clusters, or token regimes, each of which may induce activation covariances that favor different signs for the same row. The research question is intentionally forward-looking: if future deployment hardware could allocate one logical qubit per weight and support calibrated repeated state preparation, could that qubit serve as a more effective context-dependent sign memory than a classical bit? Quantum random access codes (QRACs) provide a natural framework for studying this question. A QRAC encodes multiple classical bits into a single quantum state and retrieves a requested bit, with noise, by selecting the corresponding measurement (Ambainis et al., 2002, 2008; Nayak, 1999; Farkas et al., 2025). Its role is not to reveal multiple bits simultaneously. Rather, incompatible measurements can reveal different classical functions of the same stored state (Carmeli et al., 2020; Heinosaari et al., 2016; Designolle et al., 2019). Quantum Random Access Quantization (QRAQ) brings this mechanism to one-bit PTQ. Each weight slot stores a single logical QRAC register, and a context label selects the corresponding Pauli measurement. By averaging fresh readouts, QRAQ obtains an unbiased estimate of the context-specific binary surrogate at an explicit shot-noise cost. In this way, QRAQ preserves the algebraic structure of one-bit signed-scale quantization while replacing the globally shared classical sign with a measurement-selected sign. The construction should therefore be understood as a logical readout model, not as a claim of immediate hardware acceleration: it asks what a qubit could offer in terms of quantization if the required state-preparation and measurement primitives were available. Main takeaway. Bottleneck Resource Certificate A shared-sign bit must serve all contexts. Incompatible measurements read different signs from one logical QRAC state. A row-wise gap appears exactly when the context-wise optimal sign sets have no common representative, and it survives when this gap exceeds shot noise. For signed per-row scales, ideal QRAQ attains the average of the context-wise binary optima, whereas a classical one-bit PTQ baseline must select a single sign vector shared across all contexts. The resulting row-wise gap is therefore nonnegative, and it is strictly positive exactly when the context-wise optimal sign sets admit no common representative. Finite-shot and noise-robust variants follow by comparing this ideal gap against the corresponding variance-inflation term. Thus, the theory isolates the obstruction, identifies the role of measurement incompatibility, and gives a computable certificate for when QRAQ can improve over a shared classical sign. Our contributions are as follows: • We formulate context-aware one-bit PTQ and identify the shared-sign constraint addressed by QRAQ. • We prove a strict row-wise separation from shared-sign one-bit PTQ under signed per-row scales and extend the result to finite-shot estimation, depolarizing noise, and admissible Pauli-noise thresholds. • We characterize the scale granularities under which the separation is preserved, derive finite-sample certificates from calibration data, and empirically validate the predicted ideal, finite-shot, noisy, and multi-context regimes in simulation. The main text focuses on the core mechanism and key formulas; Appendix C provides the formal assumptions and complete proofs. Figure 1: QRAQ workflow for two contexts. Each weight entry stores one qubit whose X and Z readouts encode two target sign matrices, and the context selects the matched Pauli measurement before applying a per-row scale. 2 Related Work Layer-wise and one-bit PTQ. Modern PTQ methods compress trained models without retraining by solving local reconstruction problems for individual linear layers (Frantar et al., 2023; Lin et al., 2024; Chee et al., 2023; Tseng et al., 2024; Ashkboos et al., 2024). Recent approaches further refine this principle using error propagation, activation-aware weighting, incoherence, rotations, or later-layer matching (Arai and Ichikawa, 2025; Li et al., 2025; Zhang et al., 2026; Lin and Wan, 2025). The extreme one-bit setting builds on the lineage of binary and XNOR-style networks (Hubara et al., 2016; Rastegari et al., 2016). It has recently been extended to large language models through BitNet-style architectures and post-training binarization methods (Wang et al., 2023; Ma et al., 2024; Huang et al., 2024; Xu et al., 2024; Li et al., 2024; Yuan et al., 2024). These methods store a single classical sign per weight, together with shared or grouped scales. QRAQ preserves the same signed-scale reconstruction objective but changes the underlying memory primitive: the stored sign is no longer a single context-independent classical bit. Context awareness in quantization. Activation-aware PTQ reflects the fact that the optimal low-bit approximation depends on the input distribution. In standard practice, however, the calibration set is typically compressed into a single covariance estimate per layer. We instead partition the calibration data into deployment contexts, such as attention heads, routing patterns, tasks, or prompt clusters. This is a targeted extension of PTQ rather than a new training objective: all baselines and risks remain layer-wise quadratic reconstruction problems. The key distinction is that a classical one-bit quantizer must use the same sign matrix across all contexts, whereas QRAQ can retrieve context-dependent signs from the same logical quantum state. QRACs and quantum learning. QRACs were introduced in communication complexity (Ambainis et al., 2002), bounded by Nayak-type information-theoretic limits (Nayak, 1999; Farkas et al., 2025), and linked to measurement incompatibility (Carmeli et al., 2020; Heinosaari et al., 2016; Designolle et al., 2019; Lin and de Wolf, 2025). Related work in quantum learning establishes quantum-over-classical sample-complexity separations for learning properties of physical processes (Huang et al., 2021, 2022). Our results differ in both the object and the metric: they provide finite-dimensional reconstruction-risk separation for model quantization. The paper, therefore, lies at the interface of PTQ and QRACs. We use measurement incompatibility as a quantization resource and certify its value using the same calibration objective that underlies classical PTQ. 3 Preliminaries Appendix A provides a self-contained quantum-information primer for ML readers, while Appendix B provides a self-contained PTQ primer for quantum-information readers. Notation. We write ∥A∥F A _F, Tr⁡(A)Tr(A), A⊤A , Ai,:A_i,:, and A:,jA_:,j for the Frobenius norm, trace, transpose, i-th row, and j-th column of a matrix A, respectively. Let IdI_d denote the d×d× d identity matrix, J the all-ones matrix, sgn(⋅)sgn(·) the entry-wise sign function, and diag(v)diag(v) the diagonal matrix with diagonal entries v. We use A⪰0A 0 to denote positive semidefiniteness. Unless stated otherwise, expectations are taken over both the context prior to π and the measurement randomness. Layer-wise PTQ and the one-bit class. PTQ compresses a full-precision weight matrix W∈ℝN×MW∈R^N× M by replacing it with a surrogate W W whose entries lie in a codebook (Gong et al., 2024; Frantar et al., 2023; Lin et al., 2024; Chee et al., 2023; Tseng et al., 2024; Ashkboos et al., 2024; Dettmers et al., 2022; Hubara et al., 2016; Jacob et al., 2018; Esser et al., 2020; Wang et al., 2023; Ma et al., 2024). For a calibration activation matrix X, the standard layer-wise reconstruction error is ℛ(W,W^;X)=∥WX−W^X∥F2=Tr⁡((W−W^)XX⊤(W−W^)⊤).R(W, W;X)= WX- WX _F^2=Tr ((W- W)X (W- W) ). (1) This quadratic objective is the canonical local criterion used in PTQ (Frantar et al., 2023; Lin et al., 2024; Arai and Ichikawa, 2025; Lin and Wan, 2025; Zhang et al., 2026); Appendix B recalls the standard Lipschitz propagation argument that relates this layer-wise error to end-to-end model error. A one-bit quantizer represents each weight by a sign in ±1\± 1\, optionally multiplied by a scale from a scale class ⊆ℝS R: W^τ,ijC=ατ,g(i,j)Cij,C∈±1N×M,ατ,g∈. W^C_τ,ij= _τ,g(i,j)C_ij, C∈\± 1\^N× M, _τ,g∈S. (2) Here, g:[N]×[M]→[G]g:[N]×[M]→[G] specifies the scale granularity: for example, G=1G=1 corresponds to per-tensor scaling, G=NG=N corresponds to per-row scaling, G=MG=M corresponds to per-column scaling, G=NM/bG=NM/b corresponds to per-group scaling, and G=NMG=NM corresponds to per-entry scaling. Modern one-bit baselines (Wang et al., 2023; Ma et al., 2024; Huang et al., 2024; Xu et al., 2024; Li et al., 2024; Yuan et al., 2024) typically use signed per-row scaling, denoted row±S_row±, or a coarser scale class. The crucial constraint is that the sign matrix C is shared across all contexts. Context-aware PTQ. A context τ∈1,…,Kτ∈\1,…,K\ indexes a deployment regime with a distinct activation distribution, for example an attention head, MoE route, task head, or prompt cluster. It has prior π=(π1,…,πK)π=( _1,…, _K) and per-context calibration covariance Στ≔[XτXτ⊤]⪰0 _τ E[X_τX_τ ] 0. This context-aware extension is not a standard PTQ baseline, but it is the minimal way to expose the failure mode that motivates the paper: the activation covariance entering the reconstruction metric may change with the deployment regime, so the best binary signs may also change. A context-aware quantizer returns a surrogate W^τ W_τ and is evaluated by ℛ(W,W^τ;Στ,πτ)=∑τ=1KπτTr⁡((W−W^τ)Στ(W−W^τ)⊤),R (W,\ W_τ\;\ _τ, _τ\ )= _τ=1^K _τTr ((W- W_τ) _τ(W- W_τ) ), (3) which is our objective. Classical one-bit context awareness can adjust the scales ατ∈ _τ∈S, but still shares C across τ. QRAQ, introduced in Section 4, uses one logical QRAC register per weight and a context-matched measurement to produce a context-dependent sign. Assumption 4.1 formalizes the finite-shot readout model. Qubits and Pauli measurements. A qubit state is a density matrix ρ∈ℂ2×2ρ∈C^2× 2 with ρ=ρ†ρ=ρ , ρ⪰0ρ 0, and Tr⁡(ρ)=1Tr(ρ)=1 (Nielsen and Chuang, 2010); equivalently, it admits the Bloch form ρ=12(I+rxX+ryY+rzZ),∥r∥2≤1,X=(0110),Y=(0−i0),Z=(100−1),ρ= 12(I+r_xX+r_yY+r_zZ),~ r _2≤ 1,~X= pmatrix0&1\\ 1&0 pmatrix,Y= pmatrix0&-i\\ i&0 pmatrix,Z= pmatrix1&0\\ 0&-1 pmatrix, (4) with Pauli matrices X,Y,ZX,Y,Z being Hermitian, traceless, pairwise anticommuting, and squaring to I. For a self-inverse observable A and state ρ, a binary projective measurement returns m∈±1m∈\± 1\ with [m∣ρ,A]=Tr(ρA),Var[m∣ρ,A]=1−Tr(ρA)2,E[m ρ,A]=Tr(ρ A), [m ρ,A]=1-Tr(ρ A)^2, (5) the most general measurement is a POVM Ey\E_y\ with Pr⁡(y∣ρ)=Tr⁡(ρEy) (y ρ)=Tr(ρ E_y). The qubit depolarizing channel η(ρ)=ηρ+(1−η)I/2N_η(ρ)=ηρ+(1-η)I/2 shrinks each Pauli expectation by η∈[0,1]η∈[0,1]; on a d-dimensional logical register we use η(ρ)=ηρ+(1−η)Id/dN_η(ρ)=ηρ+(1-η)I_d/d. This is our default noise model. Appendix A expands tensor products, Jordan–Wigner strings (Jordan and Wigner, 1928), and asymmetric Pauli channels. 4 Method: Quantum Random Access Quantization Our quantizer, Quantum Random Access Quantization (QRAQ), has three separable stages. First, calibration chooses per-context target sign matrices B(τ)τ=1K⊂±1N×M\B^(τ)\_τ=1^K⊂\± 1\^N× M. Second, these signs are compiled into one logical QRAC register per weight entry. Third, inference applies a context-specific Pauli measurement and multiplies the readout by a classical scale, as illustrated in Figure 1. 4.1 QRAC encoding and context-aware measurement For the two-context case K=2K=2, pick B(1),B(2)∈±1N×MB^(1),B^(2)∈\± 1\^N× M and store, per weight coordinate (i,j)(i,j), the single-qubit state ρij=12(I+Bij(1)2X+Bij(2)2Z). _ij= 12 (I+ B^(1)_ij 2X+ B^(2)_ij 2Z ). (6) Because X and Z anticommute, ρij⪰0 _ij 0 for every sign pair, and the Bloch radius 1/21/ 2 is the largest consistent with positivity (Lemma C.1). Thus, the qubit encodes both Bij(1)B^(1)_ij and Bij(2)B^(2)_ij inside a single logical memory cell, while each context only queries one component. At inference time, context τ∈1,2τ∈\1,2\ selects P1=XP_1=X or P2=ZP_2=Z. The matched measurement produces mij(τ)∈±1m^(τ)_ij∈\± 1\ with [mij(τ)]=Bij(τ)/2E[m^(τ)_ij]=B^(τ)_ij/ 2 by Lemma C.3. Scaling the outcome by a raw scale γτ,g(i,j) _τ,g(i,j) defines W^τ,ijQ=γτ,g(i,j)mij(τ),[W^τ,ijQ]=μτ,g(i,j)Bij(τ),μτ,g≔cγτ,g,c=12. W^Q_τ,ij= _τ,g(i,j)m^(τ)_ij, E [ W^Q_τ,ij ]= _τ,g(i,j)B^(τ)_ij, _τ,g c _τ,g, c= 1 2. (7) QRAQ therefore has the same sign-scale algebraic form as a classical one-bit quantizer, except that B(τ)B^(τ) is context-dependent rather than shared. General K-context and multi-qubit construction. For any integer K≥2K≥ 2 and any pairwise anticommuting self-inverse observables Aτ=1K\A_τ\_τ=1^K on n qubits, store ρb(K)=12n(I+c∑τ=1KbτAτ),b∈±1K,c=1K, _b^(K)= 12^n (I+c _τ=1^Kb_τA_τ ), b∈\± 1\^K, c= 1 K, (8) which is positive semidefinite if and only if c≤1/Kc≤ 1/ K (Lemma C.1). The minimum number of qubits that host K pairwise anticommuting self-inverse observables is n=⌈(K−1)/2⌉n= (K-1)/2 (Theorem C.15). The Jordan–Wigner family described in Appendix A.4 realizes any such K (Jordan and Wigner, 1928). Single-qubit constructions with X,Z\X,Z\ (K=2K=2) and X,Y,Z\X,Y,Z\ (K=3K=3) give c=1/2c=1/ 2 and c=1/3c=1/ 3, respectively. 4.2 Finite-shot noise and QRAQ risk Assumption 4.1 (Fresh-copy readout model). All S≥1S≥ 1 measurement shots used to estimate a single weight entry are performed on independent identically prepared copies of the corresponding QRAC state, or equivalently on a device that can re-prepare the same QRAC state before each shot. Thus the finite-shot risk in Eq. (10) is a per-query risk under a fresh-copy or re-preparable-memory model. If physical storage rather than logical QRAC state size is counted, then S simultaneous shots require S physical preparations per weight entry unless state re-preparation is available. Each readout is averaged over S independent shots in the sense of Assumption 4.1, and the logical QRAC register undergoes a depolarizing channel ηN_η (Section 3), which shrinks the matched Pauli expectation by η∈(0,1]η∈(0,1]. From this point on, the calibrated scale is reparameterized as μτ,g≔ηcKγτ,g _τ,g η c_K _τ,g; in the ideal case η=1η=1, this agrees with Eq. (7). Lemma C.3, proved in Appendix C.3, shows that [W^τ,ijQ]=μτ,g(i,j)Bij(τ),Var⁡(W^τ,ijQ)=μτ,g(i,j)2SνK(η),νK(η)≔Kη2−1.E [ W^Q_τ,ij ]= _τ,g(i,j)B^(τ)_ij, ( W^Q_τ,ij )= _τ,g(i,j)^2S _K(η), _K(η) Kη^2-1. (9) Under this independent depolarizing model, the single coefficient νK(η) _K(η) is the only channel through which hardware non-ideality enters the finite-shot thresholds. In particular, ν2(1)=1 _2(1)=1 and ν3(1)=2 _3(1)=2. Different weight slots carry independent qubits, so Cov⁡(W^τQ)Cov( W^Q_τ) is diagonal (Lemma C.2); correlated cross-qubit noise is treated in Theorem C.16. Substituting Eq. (7)–Eq. (9) into Eq. (3) and applying the bias–variance decomposition from Lemma C.4 yields the QRAQ risk ℰQ(W,B(τ),μτ)=∑τ=1Kπτ[Rτ(W,Qτ)+νK(η)S∑i,jμτ,g(i,j)2(Στ)jj],E_Q (W,\B^(τ), _τ\ )= _τ=1^K _τ [R_τ(W,Q_τ)+ _K(η)S _i,j _τ,g(i,j)^2( _τ)_j ], (10) with Qτ,ij=μτ,g(i,j)Bij(τ)Q_τ,ij= _τ,g(i,j)B^(τ)_ij and Rτ(W,Qτ)=Tr⁡((W−Qτ)Στ(W−Qτ)⊤)R_τ(W,Q_τ)=Tr ((W-Q_τ) _τ(W-Q_τ) ). The classical one-bit risk is ℰC(W,C,ατ)=∑τπτRτ(W,ατ⊙C)E_C^S(W,C,\ _τ\)= _τ _τR_τ(W, _τ C) with (ατ⊙C)ij=ατ,g(i,j)Cij( _τ C)_ij= _τ,g(i,j)C_ij. The key structural difference is that C is shared across τ, while B(τ)B^(τ) is not. Resource-fair comparison. Under Assumption 4.1, S state preparations physically consume S quantum resources per weight entry. For a fair resource comparison, the classical quantizer should be permitted S bits of memory per weight entry. If S≥KS≥ K, the classical quantizer can store a separate sign for each context, so the shared-sign constraint disappears. Quantum advantage thus requires S<KS<K; for K=2K=2, the only non-trivial regime S=1S=1. The corresponding QRAQ risk is obtained by substituting S=1S=1 into Eq. (10). Scale-granularity lattice. Our analysis uses the partial order on scale classes induced by feasible-set inclusion: tensor±⊆row±⊆row×col±⊆entry±, S_tensor± S_row± S_row× col± S_entry±, (11) tensor±⊆col±⊆row×col±⊆entry±, S_tensor± S_col± S_row× col± S_entry±, (12) tensor±⊆group(g)±⊆entry±. S_tensor± S_group(g)± S_entry±. (13) Here group(g)±S_group(g)± denotes a fixed partition g; such a group class is comparable to row or column scaling only when the partition refines or coarsens the corresponding rows or columns. Thus row±S_row± and col±S_col± are generally incomparable. The subscript ± marks signed vs. non-negative scales; Appendix B.4 records the precise definitions. Modern one-bit quantizers (Frantar et al., 2023; Lin et al., 2024; Wang et al., 2023; Huang et al., 2024) exist in row±S_row± or coarser. 5 Theoretical Guarantees This section states the guarantees and explains their meanings. Appendix C contains the formal hypotheses, exact constants, and proofs. Fixed-readout no-go. A quantum memory is not useful for QRAQ by itself. If every context reads the stored state with the same measurement, that measurement produces one classical outcome per weight slot, and the outcome can be sampled by a classical stochastic decoder. Theorem 5.1 (Informal: fixed-readout classical simulability). Any quantum quantizer that uses a fixed POVM in every context is exactly simulable by a classical stochastic quantizer with the same outcome alphabet and decoder family. With a sign-symmetric binary decoder, this simulator is a stochastic signed one-bit quantizer; with a non-symmetric decoder, it is an affine one-bit quantizer with a zero-point; with more outcomes, it is a larger classical codebook. Thus any QRAQ advantage must come from context-dependent, incompatible measurements. The formal statement is Theorem C.5. Advantage criterion. The comparison is between two finite-dimensional calibration objectives. QRAQ may choose one sign matrix per context but pays a variance penalty. The classical baseline chooses one shared sign matrix and noiseless scales. Theorem 5.2 (Informal: quantitative sufficient condition). QRAQ strictly improves on a classical shared-sign scale class whenever the best context-dependent signed surrogate, along with its finite-shot variance term, has smaller calibration risk than the best shared-sign classical surrogate. This condition is directly computable from calibrated weights, covariances, signs, scales, shot budget, and noise coefficient. The formal statement is Theorem C.6. Main separation: signed per-row scales. Signed per-row scales are the central one-bit PTQ regime: each output channel has its own scale, but all contexts still share the same binary signs. QRAQ only removes that shared-sign constraint. Theorem 5.3 (Informal: row-wise separation). For signed per-row scales, the ideal QRAQ row risk is never worse than the best shared-sign classical row risk. It is strictly better exactly when the context-wise optimal sign sets for that row have no common representative, modulo the global sign symmetry. At finite shots, the same row maintains a strict advantage whenever its ideal gap exceeds the explicit shot-noise inflation. If the sign-disagreement condition fails, no finite-shot advantage is possible for that row. The formal statement is Theorem C.7. We refer to the empty-intersection sign condition in Theorem 5.3 as (D1) and the finite-shot margin inequality as (D2). Condition (D1) has a concrete interpretation: the classical row must commit to one sign vector for all contexts, whereas QRAQ can encode the context-wise optima into one logical random-access state and query the relevant sign by choosing the measurement axis. Condition (D2) is the shot budget required for the variance penalty not to erase the ideal gap. What the main theorem certifies. For row w=Wi,:w=W_i,: and sign vector b, define the context-wise best row error Jτ⋆(b)=wΣτw⊤−(bΣτw⊤)2bΣτb⊤.J_τ (b)=w _τw - (b _τw )^2b _τb . (14) The ideal row advantage is the gap between a shared sign and context-specific signs: Δi∞=minc∑τπτJτ⋆(c)−∑τπτminb⁡Jτ⋆(b). _i^∞= _c _τ _τJ_τ (c)- _τ _τ _bJ_τ (b). (15) Hence QRAQ wins exactly when “min after summing” is larger than “sum of mins.” A finite-shot sufficient condition is S>νK(η)∑τπτTτ(μτ⋆)2Δi∞,Tτ=∑j(Στ)jj.S> _K(η) _τ _τT_τ( _τ )^2 _i^∞, T_τ= _j( _τ)_j. (16) Corollary 5.4 (Informal: closed-form two-dimensional gap). In the two-context, two-input symmetric covariance family, the row-wise gap is zero below an explicit anisotropy threshold and increases linearly above it. For example, the row w=(1,3)w=(1,3) at anisotropy r=0.8r=0.8 has ideal classical risk 22 and ideal QRAQ risk 11, resulting in a 50%50\% reduction. The formal statement and formula are Corollary C.9. Corollary 5.5 (Informal: resource-fair closed-form threshold). Even under the resource-fair comparison with S=1S=1, a closed-form necessary and sufficient condition for quantum advantage is obtained in the two-context, two-input symmetric covariance family, as in Corollary 5.4. The threshold value of the anisotropy is stricter than in the ideal S=∞S=∞ setting, but there still exists a region of quantum advantage. For example, for w=(1,0)w=(1,0), (5−1)/2≃0.618<r<1( 5-1)/2 0.618<r<1 is the necessary and sufficient condition for quantum advantage in the resource-fair comparison, whereas any 0<r<10<r<1 yields advantage in the ideal setting. The formal statement is Corollary C.10. Corollary 5.6 (Informal: row additivity and finite-shot total gap). For signed per-row scales, both the classical and QRAQ objectives decompose over rows. The ideal total gap is therefore the sum of nonnegative row gaps and is positive if at least one row satisfies the sign-disagreement condition. At finite shots, the total advantage is the row-summed finite shot gap. A positive ideal row can be offset by shot noise on other rows; thus, the correct finite-shot criterion is a global margin comparison. The formal statement is Corollary C.11. Corollary 5.7 (Informal: finite-shot threshold). For any row with a positive ideal gap, a sufficient shot budget is obtained by dividing the row’s variance inflation at the ideal QRAQ optimizer by that ideal gap. Larger noise, more contexts, or larger effective scales require more shots. The formal statement is Corollary C.8. Scale-granularity boundary. The separation is universal only against classical scale classes contained in signed per-row scaling. More expressive scale classes can sometimes absorb part of the context disagreement, so they require direct margin certification. Theorem 5.8 (Informal: scale-granularity boundary). Whenever the signed-row ideal gap is positive, the same ideal separation automatically holds against shared-sign one-bit PTQ classes whose feasible set is contained in signed per-row scaling, including per-tensor and nonnegative row-scale variants. For scale classes that are incomparable to or finer than signed per-row scaling, such as per-column, row-times-column, and group scaling, no universal ordering exists; QRAQ wins exactly on instances where the directly computed margin over that class is positive and exceeds the finite-shot inflation. Per-entry scaling can reproduce the full-precision weights and, therefore, admits no positive reconstruction-risk advantage. The formal statement is Theorem C.12. Extensions and finite-sample certificate. The same mechanism extends beyond the single-qubit two-context case. Multi-qubit QRACs host up to 2n+12n+1 contexts on n qubits; arbitrary priors reduce to the active contexts, Gram-bounded non-anticommuting observables replace the noise coefficient with a spectral analog, and correlated or Pauli-diagonal noise changes only the variance threshold (Theorems C.15, C.13, C.14, C.16, and C.17). A complementary quantum-activation result shows that whenever the activation states are valid density matrices and are decoded through a fixed POVM, a classical receiver has an incompatibility floor, while context-matched quantum readout converges at rate O(1/S)O(1/S) (Theorem C.19). Theorem 5.9 (Informal: finite-sample certificate). With bounded calibration activations ∥xτ(s)∥2≤Bx x_τ^(s) _2≤ B_x and uniformly nondegenerate sign denominators, the empirical ideal gap uniformly approximates the population gap at a square-root sample rate. In particular, when the sample size is large enough that MεΣ≤λ0/2M _ ≤ _0/2, the confidence radius rad(δ)=2K(1+2MBx2λ0)2∥W∥F2MBx22log⁡(2KM2/δ)Nmin,rad(δ)=2K (1+ 2MB_x^2 _0 )^2 W _F^2\,MB_x^2 2 (2KM^2/δ)N_ , (17) Δ^∞>rad(δ) ^∞>rad(δ) certifies a positive population ideal gap with a probability of at least 1−δ1-δ. The finite-shot certificate subtracts the same variance inflation used in the main theorem. The formal statement is Theorem C.18. 6 Experiments Figure 2: Simulator validation of QRAQ. Panel a shows the ideal relative gap versus inter-context correlation for K=2K=2; panel b shows finite-shot reconstruction risk versus shots per entry at ρ=0ρ=0; panel c shows relative gap versus the number of contexts at several shot budgets; and panel d shows relative gap versus depolarizing fidelity across K. The experiments test whether the computable margins in Section 5 predict the behavior of QRAQ in the regimes used by the proofs: ideal reconstruction, finite-shot readout, multiple contexts, and calibrated noise. Figure 2 summarizes these four checks. The goal is not to benchmark hardware throughput but to verify the reconstruction-risk separation and its finite-shot degradation. Weights are i.i.d. Gaussian. Per-context covariances follow the shared-factor Wishart model given in Eq. (160) of Appendix E. We use the optimal signed per-row classical baseline, solved by exhaustive enumeration for the tested widths. The primary metric is the relative population gap (ℰCrow±−ℰQrow)/ℰCrow±(E^row±_C-E^row_Q)/E^row±_C. Appendix E provides the simulator details, unit tests, and additional scaling sweeps. In panel a of Figure 2, the ideal gap vanishes when the contexts collapse to the same covariance at |ρ|=1|ρ|=1 and peaks at 31.1%±6.6%31.1\%± 6.6\% near ρ=−0.4ρ=-0.4. The closed-form two-dimensional corollary is unit-tested to numerical agreement 10−1210^-12. In panel b, analytic and Monte-Carlo finite-shot curves cross the classical baseline between S=8S=8 and S=16S=16, consistent with the predicted threshold S0≈12S_0≈ 12, and approach the population limit by S≃256S 256. In panel c, the ideal gap increases from 28.2%28.2\% to 44.1%44.1\% for K∈2,3,4,5,7K∈\2,3,4,5,7\, with positive finite-shot gaps for S≥256S≥ 256. In panel d, the gap decreases monotonically as the depolarizing parameter η decreases, ordered by the coefficient νK(η) _K(η), and remains positive down to η≥0.40η≥ 0.40 in the tested setting. The empirical pattern is consistent with the theoretical predictions. QRAQ gains when contexts prefer incompatible signs, loses margin through the explicit shot-noise term, incurs a larger variance cost as K or noise increases, and approaches the ideal value when the shot budget is large enough. Additional experiments in Appendix E.1 examine the same qualitative behavior across weight distributions, asymmetric Pauli-noise coefficients, and larger K. Figure 3: Resource-fair comparison on the two-context symmetric covariance family Σ±=I±r(J−I) _±=I± r(J-I). Panel a shows row risk versus anisotropy for w=(1,0)w=(1,0), and panel b shows the regions in which QRAQ has lower risk in polar coordinates (r,θ)(r,θ) with w=(cos⁡θ,sin⁡θ)w=( θ, θ). Resource-fair comparison at S=1S=1. As discussed in Section 4, a fair resource comparison grants the classical quantizer S bits per weight entry. For K=2K=2 the only non-trivial regime is S=1S=1, where one qubit is compared against one classical bit. Figure 3 illustrates this regime on the two-context symmetric covariance family of Corollary 5.4. Panel a of Figure 3 plots the row risk of the classical baseline, ideal QRAQ (S=∞S=∞), and resource-fair QRAQ (S=1S=1) as a function of the anisotropy r for w=(1,0)w=(1,0). While the ideal QRAQ risk falls below the classical baseline for any r>0r>0, the S=1S=1 curve crosses the classical baseline at r=(5−1)/2≃0.618r=( 5-1)/2 0.618, matching the closed-form threshold of Corollary 5.5. Panel b shows, in polar coordinates (r,θ)(r,θ) with w=(cos⁡θ,sin⁡θ)w=( θ, θ), the region where QRAQ has lower risk than the classical baseline. The resource-fair region is a strict subset of the ideal region, as expected from the additional shot-noise cost, but it remains non-empty in this example. This illustrates that the resource-fair comparison can retain a reconstruction-risk advantage when the classical quantizer is granted the same number of memory cells per weight entry. 7 Deployment path, limitations, and applications How a device would realize the advantage. Calibration computes context-wise signs B(τ)B^(τ) and signed row scales μτ,i _τ,i from representative activations. A future device would then prepare one logical QRAC state ρij _ij per weight coordinate. At inference time, a classical router supplies the context label τ, the device measures each logical slot along the matched observable AτA_τ for S independently prepared shots, and the digital backend applies (ηcK)−1μτ,i(η c_K)^-1 _τ,i to the averaged outcomes. These shots are independent preparations of a known calibrated state, not clones of an unknown quantum state. The operational certificate is the same as the theoretical one: the ideal context-disagreement margin must exceed the finite-shot and noise inflation. Limitations. The result is a reconstruction-risk separation, not an immediate wall-clock acceleration claim for current GPU-only inference stacks. It assumes accurate context labels, representative calibration covariances, and logical QRAC states that can be freshly prepared or instantiated in parallel for shot averaging, in addition to calibrated readout noise. QRAQ has no certified advantage when the context-wise sign optima already agree, when per-entry classical scales are allowed, or when finite-shot and hardware-noise costs exceed the ideal margin. Correlated hardware noise and scale classes outside signed per-row scaling are covered only by direct margin tests in Theorems C.16 and C.12; the universal row-wise separation does not extend to them automatically. Applications. Potential targets are layers that already expose context labels and context-specific activation statistics: attention-head or route-specific blocks, task or adapter heads, and prompt-cluster caches. In these settings, the certificate can be evaluated from ordinary PTQ calibration data before any hardware deployment. Longer term, QRAQ suggests hybrid memory designs in which a classical accelerator performs dense arithmetic while a small quantum readout layer supplies context-dependent binary signs. It also motivates the co-design of sign codebooks, measurement axes, shot allocation, and hardware noise calibration. 8 Conclusion One-bit PTQ requires all contexts to share one binary weight matrix, even when their activation statistics prefer different signs. QRAQ replaces this shared-sign constraint with a logical QRAC readout: one stored quantum state is queried by context-matched incompatible measurements to produce a calibrated binary surrogate. Under the fresh-copy readout model, this provides an unbiased, context-specific estimator with an explicit shot-noise penalty. The resulting reconstruction-risk separation can be evaluated from the encoded state and the calibration data: signed per-row QRAQ strictly beats shared-sign one-bit PTQ when context-wise optimal signs disagree and the ideal gap exceeds finite-shot and hardware-noise costs. These results indicate that measurement incompatibility, rather than quantization alone, is the relevant quantum resource in this model. 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The rest of the appendix elaborates on these facts using the notation of the paper, following standard references [Nielsen and Chuang, 2010, Watrous, 2018, Heinosaari and Ziman, 2012]. Mapping to PTQ intuition. In classical one-bit PTQ, a memory cell stores a sign, and the decoder multiplies it by a scale. In QRAQ, the memory cell stores a density matrix, the decoder chooses a measurement axis from the context, and the measured sign is averaged over fresh copies before scaling. The formulas in Section 4 are, therefore, ordinary bias–variance calculations once the measurement mean and variance are known. A.1 Hilbert spaces, qubits, and density matrices A quantum system is mathematically described by a complex Hilbert space ℋH. The basic such space is ℂ2C^2, and a system described by ℂ2C^2 is called a qubit. Vectors in ℋH are written as |ψ⟩|ψ , and their Hermitian adjoints are written as ⟨ψ| ψ|, so ⟨ψ|ϕ⟩ ψ|φ represents the inner product, and |ψ⟩⟨ϕ||ψ φ| is a rank-one operator. A pure state is a unit vector in ℋH, or equivalently, a rank-one projector ρ=|ψ⟩⟨ψ|ρ=|ψ ψ|. A mixed state is a convex combination of pure states, which represents the most general state compatible with unit trace and positive semidefiniteness: ρ=∑kpk|ψk⟩⟨ψk|,pk≥0,∑kpk=1.ρ= _kp_k| _k _k|, p_k≥ 0, _kp_k=1. (18) Every mixed state is a density matrix, meaning it satisfies ρ⪰0ρ 0, ρ=ρ†ρ=ρ , and Tr⁡(ρ)=1Tr(ρ)=1. The set of all qubit density matrices is called the Bloch ball and is the image, under the Bloch representation Eq. (4), of the unit three-ball in ℝ3R^3. A.2 Pauli matrices and the qubit algebra The Pauli matrices X,Y,Z\X,Y,Z\ displayed in Eq. (4), together with the identity I, form a real orthogonal basis of the space of Hermitian 2×22× 2 matrices under the Hilbert–Schmidt inner product ⟨A,B⟩HS=Tr⁡(A†B) A,B _HS=Tr(A B), with common norm Tr⁡(I2)=Tr⁡(X2)=Tr⁡(Y2)=Tr⁡(Z2)=2Tr(I^2)=Tr(X^2)=Tr(Y^2)=Tr(Z^2)=2 (so I/2,X/2,Y/2,Z/2\I/ 2,X/ 2,Y/ 2,Z/ 2\ is the associated orthonormal basis). Every 2×22× 2 Hermitian matrix H can therefore be expanded as H=h0I+hxX+hyY+hzZH=h_0I+h_xX+h_yY+h_zZ with h0,hx,hy,hz∈ℝh_0,h_x,h_y,h_z∈R, and density matrices correspond to the special case h0=1/2h_0=1/2, hx2+hy2+hz2≤1/4h_x^2+h_y^2+h_z^2≤ 1/4. The Pauli matrices satisfy X2=Y2=Z2=I, X^2=Y^2=Z^2=I, XY=iZ,YZ=iX,ZX=iY, XY=iZ, YZ=iX, ZX=iY, (19) X,Y \X,Y\ =Y,Z=Z,X=0, =\Y,Z\=\Z,X\=0, where A,B=AB+BA\A,B\=AB+BA is the anticommutator. The pairwise-anticommuting property is the key identity through which our QRAC state Eq. (6) and Eq. (8) are positive semidefinite at the maximal Bloch radius c=1/Kc=1/ K. A.3 Projective measurements and POVMs A projective measurement of a self-adjoint observable A with spectrum ayy∈\a_y\_y∈Y and spectral projectors Πy\ _y\ returns outcome y with probability Tr⁡(ρΠy)Tr(ρ _y) when the system is in state ρ. For the Pauli observable P∈X,Y,ZP∈\X,Y,Z\, the spectrum is +1,−1\+1,-1\ and the spectral projectors are Π±=(I±P)/2 _±=(I± P)/2. The outcome m∈±1m∈\± 1\ has expectation [m]=Tr⁡(ρP)E[m]=Tr(ρ P) and variance Var⁡(m)=1−(Tr⁡(ρP))2Var(m)=1-(Tr(ρ P))^2. A positive operator-valued measure, or POVM, generalises projective measurement to non-projective operators. A POVM with classical outcomes indexed by a finite set Y is a collection Eyy∈\E_y\_y∈Y of positive semidefinite operators satisfying ∑yEy=I _yE_y=I, and the outcome y on state ρ occurs with probability Tr⁡(ρEy)Tr(ρ E_y). POVMs model every physically realisable measurement on a quantum system (Naimark’s theorem [Watrous, 2018]), so our no-go and lower-bound results are expressed against the most general POVM. A.4 Tensor products, many qubits, and the Jordan–Wigner family A system of n qubits has Hilbert space ℋ=(ℂ2)⊗n=ℂ2nH=(C^2) n=C^2^n. Operators on ℋH include tensor products of single-qubit operators, and in particular the Pauli strings of the form Pσ→=σ1⊗σ2⊗⋯⊗σnP_ σ= _1 _2 ·s _n for σi∈I,X,Y,Z _i∈\I,X,Y,Z\. Pauli strings are self-adjoint and square to I, and any two Pauli strings either commute or anticommute. The Jordan–Wigner construction of 2n+12n+1 pairwise anticommuting self-inverse observables on n qubits is A2k−1=Z⊗(k−1)⊗X⊗I⊗(n−k),A2k=Z⊗(k−1)⊗Y⊗I⊗(n−k),A2n+1=Z⊗n.k=1,…,n, aligned A_2k-1&=Z (k-1) X I (n-k),\\ A_2k&=Z (k-1) Y I (n-k),\\ A_2n+1&=Z n. aligned k=1,…,n, (20) which one verifies by direct computation. This construction realises every multi-qubit QRAC used in Theorem C.15. A.5 Noise channels: depolarizing and Pauli-diagonal A quantum channel is a completely positive trace-preserving linear map on density matrices. For a qubit, the depolarizing channel with parameter η∈[0,1]η∈[0,1] is η(ρ)=ηρ+(1−η)I/2,N_η(ρ)=ηρ+(1-η)I/2, (21) which shrinks the Bloch vector by η and leaves the trace invariant. For a logical d-dimensional register in the appendix proofs, the same notation denotes η(ρ)=ηρ+(1−η)Id/dN_η(ρ)=ηρ+(1-η)I_d/d. A Pauli-diagonal qubit channel is a map (ρ)=∑P∈I,X,Y,ZpPPρPN_ p(ρ)= _P∈\I,X,Y,Z\p_PPρ P with pP≥0p_P≥ 0 and ∑PpP=1 _Pp_P=1. It acts on the Bloch coordinates as (rx,ry,rz)↦(ηxrx,ηyry,ηzrz),ηx=pI+pX−pY−pZ,ηy=pI−pX+pY−pZ,ηz=pI−pX−pY+pZ.(r_x,r_y,r_z) ( _xr_x, _yr_y, _zr_z), aligned _x&=p_I+p_X-p_Y-p_Z,\\ _y&=p_I-p_X+p_Y-p_Z,\\ _z&=p_I-p_X-p_Y+p_Z. aligned (22) Conversely, a triple (ηx,ηy,ηz)∈[−1,1]3( _x, _y, _z)∈[-1,1]^3 defines a completely positive Pauli-diagonal channel if and only if the four numbers 1±ηx±ηy±ηz4(any choice of signs with even number of minus signs), 1± _x± _y± _z4 (any choice of signs with even number of minus signs), (23) namely 1+ηx+ηy+ηz4,1+ηx−ηy−ηz4,1−ηx+ηy−ηz4,1−ηx−ηy+ηz4, 1+ _x+ _y+ _z4,\ 1+ _x- _y- _z4,\ 1- _x+ _y- _z4,\ 1- _x- _y+ _z4, are all non-negative; these are the corresponding probabilities (pI,pX,pY,pZ)(p_I,p_X,p_Y,p_Z). In particular, the cubic prescription (ηx,ηy,ηz)∈[0,1]3( _x, _y, _z)∈[0,1]^3 is not sufficient for complete positivity: e.g. (ηx,ηy,ηz)=(0,1,1)( _x, _y, _z)=(0,1,1) yields pX=(1+0−1−1)/4=−1/4<0p_X=(1+0-1-1)/4=-1/4<0 and is not CP, whereas the depolarizing line ηx=ηy=ηz=η∈[−1/3,1] _x= _y= _z=η∈[-1/3,1] is CP. When the conditions of Eq. (23) are met we call (ηx,ηy,ηz)( _x, _y, _z) admissible; this is the model under which Theorem C.17 is stated. Figure 4: Bloch-sphere representation of the two-context QRAC state for a single qubit. Each state corresponds to a sign pair (Bij(1),Bij(2))∈±12(B^(1)_ij,B^(2)_ij)∈\± 1\^2 and is represented by a Bloch vector (Bij(1)/2, 0,Bij(2)/2)(B^(1)_ij/ 2,\,0,\,B^(2)_ij/ 2) in the X–Z plane. Thus, the four QRAC states lie on the great circle at polar angles corresponding to the sign pairs (+,+)(+,+), (+,−)(+,-), (−,+)(-,+), and (−,−)(-,-). A context-matched Pauli measurement recovers the requested sign in expectation: measuring X reads out the first sign, while measuring Z reads out the second, each with signal strength 1/21/ 2. Appendix B Post-training quantization primer for readers from quantum information This appendix provides a self-contained introduction to post-training quantization (PTQ) for readers with a background in quantum information. We use only linear algebra, elementary probability, and the same notation as the main body. B.1 What is a neural network layer and what does quantizing it mean? A feed-forward neural network of depth L is a composition f=fL∘fL−1∘⋯∘f1f=f_L f_L-1 … f_1, where each layer fℓf_ consists of an affine map x↦Wℓx+bℓx W_ x+b_ followed by a pointwise non-linearity ϕφ (for example the ReLU ϕ(z)=max⁡(0,z)φ(z)= (0,z) or a variant thereof). The matrices Wℓ∈ℝNℓ×MℓW_ ∈R^N_ × M_ and the biases bℓ∈ℝNℓb_ ∈R^N_ are the parameters of the network. A trained network has very many parameters (billions for modern foundation models), and storing them in full precision (e.g., 16 or 32 bits per entry) can be a major deployment cost. Quantization replaces each WℓW_ by a surrogate W^ℓ W_ whose entries are drawn from a small codebook, e.g., the two-point codebook ±1\± 1\ times a small set of real scales; the biases are typically kept in higher precision because they are few in number. The goal is to retain the network’s input–output behaviour while reducing the number of bits per weight entry. B.2 Layer-wise linear reconstruction and why it is the right objective Layer-wise PTQ uses the observation that quantization error can be analysed locally for each layer. Formally, let XℓX_ denote the activations entering layer ℓ , let the original layer apply fℓ(x)=ϕ(Wℓx+bℓ)f_ (x)=φ(W_ x+b_ ), and let its quantized counterpart apply f^ℓ(x)=ϕ(W^ℓx+bℓ) f_ (x)=φ( W_ x+b_ ). Under a Lipschitz assumption ∥ϕ(u)−ϕ(v)∥≤Lϕ∥u−v∥ φ(u)-φ(v) ≤ L_φ u-v (valid for ReLU with Lϕ=1L_φ=1 and for other bounded-slope activations with an appropriate constant), we have ∥fℓ(Xℓ)−f^ℓ(Xℓ)∥F≤Lϕ∥(Wℓ−W^ℓ)Xℓ∥F, f_ (X_ )- f_ (X_ ) _F≤ L_φ (W_ - W_ )X_ _F, (24) and composing along the L layers gives a telescoping bound ∥f(X1)−f^(X1)∥F≤∑ℓ=1L(∏ℓ′>ℓLϕ,ℓ′∥W^ℓ′∥)∥(Wℓ−W^ℓ)Xℓ∥F f(X_1)- f(X_1) _F≤ _ =1^L ( _ > L_φ, W_ ) (W_ - W_ )X_ _F. Thus, to control the final output error it suffices to control the layer-wise linear reconstruction error ∥(Wℓ−W^ℓ)Xℓ∥F2 (W_ - W_ )X_ _F^2, which equals Eq. (1) up to the label ℓ . Collecting activations into a second-moment covariance Σ=[XℓXℓ⊤] =E[X_ X_ ], the objective becomes ℒ(W,W^;Σ)≔Tr⁡((W−W^)Σ(W−W^)⊤),L(W, W; ) ((W- W) (W- W) ), (25) which is a simple generalised least-squares problem on the weight matrix: the classical solution is a projection of W onto the codebook under the Mahalanobis inner product induced by Σ . This is why making the linear reconstruction error small is the central quantitative goal of PTQ: it decouples the non-linear parts of the network and reduces quantization, for every linear layer, to a well-understood quadratic optimisation on the weight matrix. B.3 Context-aware PTQ and why classical sign sharing hurts A classical one-bit quantizer Eq. (2) stores a binary sign matrix C and per-context scales ατ\ _τ\. Its context-τ reconstruction risk is ℒ(W,ατC;Στ)=Tr⁡((W−ατC)Στ(W−ατC)⊤).L(W, _τC; _τ)=Tr ((W- _τC) _τ(W- _τC) ). (26) Differentiating in ατ _τ and setting to zero yields the per-context optimal scale ατ⋆(C)=Tr⁡(C⊤WΣτ)/Tr⁡(C⊤CΣτ) _τ (C)=Tr(C W _τ)/Tr(C C _τ) and the minimum ℒ⋆(W,C;Στ)=Tr⁡(W⊤WΣτ)−ατ⋆(C)2Tr⁡(C⊤CΣτ)L (W,C; _τ)=Tr(W W _τ)- _τ (C)^2Tr(C C _τ). Appendix C.7 gives the row-wise derivation. The sign matrix C is optimised by solving a combinatorial problem over ±1N×M\± 1\^N× M, usually approximately via ADMM [Boyd et al., 2011, Leng et al., 2018]. The key observation is that C has to be shared across τ: in the sum ∑τπτℒ⋆(W,C;Στ) _τ _τL (W,C; _τ), a single C must trade off against all contexts. A strict row-wise gap appears precisely when the per-context optimal sign sets have no common representative, modulo the global sign symmetry C∼−C -C. QRAQ removes this limitation because it stores a logical QRAC register whose incompatible readouts can carry one queried sign per context. B.4 The scale-granularity lattice The classical scale class S of Eq. (2) is parameterised by the grouping map g:[N]×[M]→[G]g:[N]×[M]→[G]. Writing 1⊆2S_1 S_2 whenever every matrix realisable in 1S_1 is realisable in 2S_2, the resulting set of classes is a partial order (not a total one): per-row and per-column groupings are incomparable because a per-row scale ατ,i _τ,i is only expressible as a per-column scale ατ,j _τ,j when it is constant in i (i.e., per-tensor). The classes tensor±,row±,col±,group(g)±,entry±S_tensor±,S_row±,S_col±,S_group(g)±,S_entry± are defined as in Eq. (2) with G groups corresponding respectively to 11, N, M, a fixed partition g, and NMNM; we additionally define the joint row-and-column class row×col±≔W^τ:W^τ,ij=ατ,iβτ,jCij,ατ,i,βτ,j∈ℝ,S_row× col± \ W_τ: W_τ,ij= _τ,i _τ,jC_ij,\ _τ,i, _τ,j∈R \, (27) i.e., a separable rank-(ατ,βτ)( _τ, _τ) scale acting multiplicatively on the sign matrix; this is the natural lattice element strictly above both per-row and per-column. The relevant unconditional inclusions are tensor±⊆row±⊆row×col±⊆entry±,tensor±⊆col±⊆row×col±⊆entry±,tensor±⊆group(g)±⊆entry±.S_tensor± S_row± S_row× col± S_entry±,\\ S_tensor± S_col± S_row× col± S_entry±,\\ S_tensor± S_group(g)± S_entry±. (28) A fixed group partition is comparable with row or column scaling only when that partition refines or coarsens the corresponding row or column partition. The classical optimum is monotone non-increasing along each valid inclusion chain of this partial order. Modern BitNet-style PTQ methods operate in row±S_row± (one signed scale per output channel) or coarser, because per-entry scales would be equivalent to storing the full-precision weight. Our main separation theorem is stated in row±S_row±; the granularity boundary of Theorem 5.8 characterises which cells of Eq. (28) preserve the advantage. B.5 ADMM codebook alignment (classical optimiser we reuse as a baseline) The classical baseline against which we measure our quantum advantage is obtained by solving minC∈±1N×M,ατ∈∑τ=1KπτTr⁡((W−ατC)Στ(W−ατC)⊤). _C∈\± 1\^N× M, _τ∈S _τ=1^K _τTr ((W- _τC) _τ(W- _τC) ). (29) The standard tool for Eq. (29) is ADMM [Boyd et al., 2011, Leng et al., 2018, Arai and Ichikawa, 2025]: introduce a continuous surrogate Z with Z=CZ=C, split the quadratic in Z from the constraint on C, and alternate between the closed-form quadratic minimisation of Z and a projection of Z onto ±1N×M\± 1\^N× M. We use the same solver for both the per-context classical baselines of Eq. (29) and, inside QRAQ, for finding the per-context sign matrices B(τ)B^(τ) row by row in Appendix D. The algorithmic structure is identical; only the number of sign matrices changes from one (classical) to K (QRAQ). Appendix C Full proofs This appendix collects the proofs of every statement in Sections 4 and 5. Throughout we write cK=1/Kc_K=1/ K, νK(η)=K/η2−1 _K(η)=K/η^2-1, and let d=2nd=2^n denote the ambient Hilbert-space dimension of the (multi-)qubit register that stores one QRAC state per weight entry. All proofs are presented as self-contained proof blocks; intermediate results that are reused later are isolated as numbered Lemmas with full proofs. C.1 Positivity and trace of the QRAC state Lemma C.1 (QRAC positivity and tightness). Let K≥2K≥ 2 and let Aτ=1K\A_τ\_τ=1^K be self-adjoint operators on ℂdC^d satisfying Aτ2=Ifor every τ∈[K],AτAτ′+Aτ′Aτ=0for every τ≠τ′.A_τ^2=I every τ∈[K], A_τA_τ +A_τ A_τ=0 every τ≠τ . (30) For every b∈±1Kb∈\± 1\^K and every c≥0c≥ 0 define Sb≔∑τ=1KbτAτ,ρb(c)≔1d(I+cSb).S_b _τ=1^Kb_τA_τ, _b^(c) 1d (I+cS_b ). (31) Then: (i) Tr⁡(Aτ)=0Tr(A_τ)=0 for every τ, and ρb(c) _b^(c) is Hermitian with Tr⁡(ρb(c))=1Tr( _b^(c))=1. (i) Sb2=KIS_b^2=KI; hence the spectrum of SbS_b equals −K,+K\- K,+ K\ with each eigenvalue of multiplicity d/2d/2. (i) ρb(c)⪰0 _b^(c) 0 for every b∈±1Kb∈\± 1\^K if and only if c≤1/Kc≤ 1/ K, and the bound is tight. Proof. (i) Tracelessness and unit trace. Fix τ∈[K]τ∈[K] and any τ′≠τ ≠τ (this exists because K≥2K≥ 2). Using Aτ′2=IA_τ ^2=I, cyclicity of the trace, and the anticommutation Aτ′Aτ=−AτAτ′A_τ A_τ=-A_τA_τ , Tr⁡(Aτ)=Tr⁡(Aτ′2Aτ)=Tr⁡(Aτ′AτAτ′)=−Tr⁡(AτAτ′2)=−Tr⁡(Aτ),Tr(A_τ)=Tr(A_τ ^2A_τ)=Tr(A_τ A_τA_τ )=-Tr(A_τA_τ ^2)=-Tr(A_τ), (32) hence 2Tr⁡(Aτ)=02Tr(A_τ)=0 and so Tr⁡(Aτ)=0Tr(A_τ)=0. Each AτA_τ is self-adjoint, so SbS_b and ρb(c) _b^(c) are Hermitian, and Tr⁡(ρb(c))=1dTr⁡(I)+cd∑τ=1KbτTr⁡(Aτ)=1.Tr ( _b^(c) )= 1dTr(I)+ cd _τ=1^Kb_τTr(A_τ)=1. (33) (i) Spectrum of SbS_b. Expand Sb2=∑τ=1Kbτ2Aτ2+∑τ≠τ′bτbτ′AτAτ′.S_b^2= _τ=1^Kb_τ^2A_τ^2+ _τ≠τ b_τb_τ A_τA_τ . (34) The diagonal sum equals ∑τ1⋅I=KI _τ1· I=KI since bτ2=1b_τ^2=1 and Aτ2=IA_τ^2=I. The off-diagonal sum vanishes term by term: pairing (τ,τ′)(τ,τ ) with (τ′,τ)(τ ,τ) gives bτbτ′AτAτ′+bτ′bτAτ′Aτ=bτbτ′(AτAτ′+Aτ′Aτ)=0b_τb_τ A_τA_τ +b_τ b_τA_τ A_τ=b_τb_τ (A_τA_τ +A_τ A_τ)=0 (35) by anticommutation. Hence Sb2=KIS_b^2=KI. Since SbS_b is self-adjoint and (Sb/K)2=I(S_b/ K)^2=I, every eigenvalue of SbS_b lies in −K,+K\- K,+ K\. Let m±m_± denote the multiplicities of ±K± K. Then m++m−=dm_++m_-=d and, by part (i), Tr⁡(Sb)=∑τbτTr⁡(Aτ)=0Tr(S_b)= _τb_τTr(A_τ)=0, so K(m+−m−)=0 K(m_+-m_-)=0 and m+=m−=d/2m_+=m_-=d/2. (i) Tight positivity bound. By (i), the smallest eigenvalue of I+cSbI+cS_b equals 1−cK1-c K, attained on the −K- K-eigenspace of SbS_b. Therefore ρb(c)⪰0 _b^(c) 0 if and only if 1−cK≥01-c K≥ 0, i.e., c≤1/Kc≤ 1/ K. Conversely, for c>1/Kc>1/ K any unit vector |ψ⟩|ψ in the −K- K-eigenspace of SbS_b satisfies ⟨ψ|ρb(c)|ψ⟩=(1−cK)/d<0 ψ| _b^(c)|ψ =(1-c K)/d<0, certifying both non-positivity and tightness. ∎ C.2 Independence of qubit-wise measurements Lemma C.2 (Factored product measurement). Let ρ=⨂(i,j)ρijρ= _(i,j) _ij be a product state on ⨂(i,j)ℋij _(i,j)H_ij, and let the global measurement be a product POVM ⨂(i,j)Eij,yijy\ _(i,j)E_ij,y_ij\_y with each Eij,yijyij\E_ij,y_ij\_y_ij a POVM on ℋijH_ij. Then: (i) the joint outcome (yij)i,j(y_ij)_i,j has a product distribution, i.e., the per-slot outcomes are mutually independent; (i) the per-slot signs mij(τ)m^(τ)_ij obtained by the matched Pauli measurement satisfy Cov⁡(mij(τ),mi′j′(τ))=0Cov (m^(τ)_ij,m^(τ)_i j )=0 for every (i,j)≠(i′,j′)(i,j)≠(i ,j ); (i) Cov⁡(W^τQ)Cov( W^Q_τ) (with W^τQ W^Q_τ vectorised) is diagonal. Proof. (i) By the Born rule and the tensor-product trace identity Tr⁡(A⊗B)=Tr⁡(A)Tr⁡(B)Tr(A B)=Tr(A)Tr(B), Pr⁡(y∣ρ)=Tr⁡[(⨂(i,j)ρij)(⨂(i,j)Eij,yij)]=∏(i,j)Tr⁡(ρijEij,yij), (y ρ)=Tr [ ( _(i,j) _ij ) ( _(i,j)E_ij,y_ij ) ]= _(i,j)Tr ( _ijE_ij,y_ij ), (36) which is a product over (i,j)(i,j) of per-slot marginals. (i)–(i) Independence is preserved under deterministic post-processing of each slot, in particular under yij↦mij(τ)y_ij m^(τ)_ij and under yij↦W^τ,ijQy_ij W^Q_τ,ij. Independence of two scalar random variables implies vanishing covariance, so all off-diagonal entries of Cov⁡(W^τQ)Cov( W^Q_τ) vanish. ∎ C.3 Noise-statistic derivation Lemma C.3 (Per-slot noise statistics). Fix a slot (i,j)(i,j) and let ρij _ij be the multi-qubit QRAC state of Eq. (8) with Bloch radius cK=1/Kc_K=1/ K and pairwise anticommuting self-inverse observables Aτ\A_τ\. Suppose the matched observable AτA_τ is measured after applying a depolarizing channel ηN_η with η∈(0,1]η∈(0,1], on each of S≥1S≥ 1 independent identically prepared copies (Assumption 4.1). Let m¯ij(τ) m^(τ)_ij denote the empirical average of the S outcomes, and define the quantized weight W^τ,ijQ≔γτ,g(i,j)m¯ij(τ) W^Q_τ,ij _τ,g(i,j) m^(τ)_ij together with the calibrated scale μτ,g≔ηcKγτ,g _τ,g η c_K _τ,g. Then [W^τ,ijQ] E [ W^Q_τ,ij ] =μτ,g(i,j)Bij(τ), = _τ,g(i,j)B^(τ)_ij, (37) Var⁡(W^τ,ijQ) ( W^Q_τ,ij ) =μτ,g(i,j)2SνK(η),νK(η)=Kη2−1. = _τ,g(i,j)^2S _K(η), _K(η)= Kη^2-1. (38) Proof. Step 1: noiseless single-shot mean. By Lemma C.1(i) and the observable expectation formula Eq. (5), [mij(τ)]=Tr⁡(ρijAτ)=1dTr⁡(Aτ)+cKd∑τ′=1KBij(τ′)Tr⁡(Aτ′Aτ).E [m^(τ)_ij ]=Tr( _ijA_τ)= 1dTr(A_τ)+ c_Kd _τ =1^KB^(τ )_ijTr(A_τ A_τ). (39) The first term vanishes by Lemma C.1(i). For the second sum: τ=τ′τ=τ contributes Bij(τ)Tr⁡(Aτ2)=Bij(τ)Tr⁡(I)=Bij(τ)dB^(τ)_ijTr(A_τ^2)=B^(τ)_ijTr(I)=B^(τ)_ijd; for τ≠τ′τ≠τ , cyclicity of the trace gives Tr⁡(Aτ′Aτ)=Tr⁡(AτAτ′)Tr(A_τ A_τ)=Tr(A_τA_τ ), while anticommutation gives Tr⁡(Aτ′Aτ)=−Tr⁡(AτAτ′)Tr(A_τ A_τ)=-Tr(A_τA_τ ), so Tr⁡(Aτ′Aτ)=0Tr(A_τ A_τ)=0. Substituting back into Eq. (39), [mij(τ)]=cKBij(τ).E [m^(τ)_ij ]=c_KB^(τ)_ij. (40) Step 2: noiseless single-shot variance. Since mij(τ)∈±1m^(τ)_ij∈\± 1\, [(mij(τ))2]=1E [(m^(τ)_ij)^2 ]=1, and using (Bij(τ))2=1(B^(τ)_ij)^2=1, Var⁡(mij(τ))=1−(cKBij(τ))2=1−cK2.Var (m^(τ)_ij )=1- (c_KB^(τ)_ij )^2=1-c_K^2. (41) Step 3: depolarizing channel. For the depolarizing channel η(ρ)=ηρ+(1−η)I/dN_η(ρ)=ηρ+(1-η)I/d applied before measurement, linearity of the trace and Tr⁡(Aτ)=0Tr(A_τ)=0 yield Tr⁡(η(ρ)Aτ)=ηTr⁡(ρAτ)+1−ηdTr⁡(Aτ)=ηTr⁡(ρAτ).Tr (N_η(ρ)A_τ )= (ρ A_τ)+ 1-ηdTr(A_τ)= (ρ A_τ). (42) Hence the per-shot mean becomes ηcKBij(τ)η c_KB^(τ)_ij, and since the outcome remains in ±1\± 1\ the per-shot variance becomes 1−(ηcKBij(τ))2=1−η2cK21-(η c_KB^(τ)_ij)^2=1-η^2c_K^2. Step 4: S-shot averaging. Under Assumption 4.1 the S shots are independent, so the empirical mean m¯ij(τ) m^(τ)_ij satisfies [m¯ij(τ)]=ηcKBij(τ),Var⁡(m¯ij(τ))=1−η2cK2S.E [ m^(τ)_ij ]=η c_KB^(τ)_ij, ( m^(τ)_ij )= 1-η^2c_K^2S. (43) Step 5: rescaling to μτ,g _τ,g. Substituting γτ,g=μτ,g/(ηcK) _τ,g= _τ,g/(η c_K) into W^τ,ijQ=γτ,g(i,j)m¯ij(τ) W^Q_τ,ij= _τ,g(i,j) m^(τ)_ij, by linearity of expectation and homogeneity of variance, [W^τ,ijQ] E [ W^Q_τ,ij ] =γτ,g⋅ηcKBij(τ)=μτ,gBij(τ), = _τ,g·η c_KB^(τ)_ij= _τ,gB^(τ)_ij, (44) Var⁡(W^τ,ijQ) ( W^Q_τ,ij ) =γτ,g2⋅1−η2cK2S=μτ,g2η2cK2⋅1−η2cK2S=μτ,g2S(1η2cK2−1). = _τ,g^2· 1-η^2c_K^2S= _τ,g^2η^2c_K^2· 1-η^2c_K^2S= _τ,g^2S ( 1η^2c_K^2-1 ). (45) Substituting cK2=1/Kc_K^2=1/K gives 1/(η2cK2)−1=K/η2−1=νK(η)1/(η^2c_K^2)-1=K/η^2-1= _K(η), which proves Eq. (38). ∎ C.4 Bias-variance decomposition Lemma C.4 (Bias-variance decomposition for the QRAQ risk). Let W^τ W_τ be a random estimator of W with finite second moment, write W¯τ≔[W^τ] W_τ E[ W_τ] and Uτ≔W^τ−W¯τU_τ W_τ- W_τ. Then for every Στ⪰0 _τ 0, Tr⁡[(W−W^τ)Στ(W−W^τ)⊤]=Tr⁡[(W−W¯τ)Στ(W−W¯τ)⊤]+∑i=1NTr⁡(Cov⁡(Uτ,i,:)Στ).ETr [(W- W_τ) _τ(W- W_τ) ]=Tr [(W- W_τ) _τ(W- W_τ) ]+ _i=1^NTr (Cov(U_τ,i,:) _τ ). (46) In particular, when W^τ W_τ is the QRAQ estimator with diagonal within-row covariance (Lemma C.2) and per-slot variance given by Lemma C.3, Eq. (46) reduces to the noise-augmented QRAQ risk Eq. (10). Proof. Write W−W^τ=(W−W¯τ)−UτW- W_τ=(W- W_τ)-U_τ with [Uτ]=0E[U_τ]=0, and expand Tr⁡[(W−W^τ)Στ(W−W^τ)⊤] ETr [(W- W_τ) _τ(W- W_τ) ] =Tr⁡[(W−W¯τ)Στ(W−W¯τ)⊤]−2Tr⁡[(W−W¯τ)ΣτUτ⊤]+Tr⁡[UτΣτUτ⊤], =Tr [(W- W_τ) _τ(W- W_τ) ]-2ETr [(W- W_τ) _τU_τ ]+ETr [U_τ _τU_τ ], (47) by linearity of the trace and of expectation. The middle term vanishes: (W−W¯τ)Στ(W- W_τ) _τ is deterministic and [Uτ⊤]=0E[U_τ ]=0, so Tr⁡[(W−W¯τ)ΣτUτ⊤]=Tr⁡[(W−W¯τ)Στ[Uτ⊤]]=0.ETr [(W- W_τ) _τU_τ ]=Tr [(W- W_τ) _τE[U_τ ] ]=0. (48) For the last term, decompose row-by-row: Tr⁡[UτΣτUτ⊤]=∑i=1N[Uτ,i,:ΣτUτ,i,:⊤]=∑i=1NTr⁡(Cov⁡(Uτ,i,:)Στ),ETr [U_τ _τU_τ ]= _i=1^NE [U_τ,i,: _τU_τ,i,: ]= _i=1^NTr (Cov(U_τ,i,:) _τ ), (49) where the second equality uses, for any zero-mean row vector u∈ℝMu∈R^M, [uΣu⊤]=∑j,kΣjk[ujuk]=∑j,kΣjkCov(u)jk=Tr(Cov(u)Σ).E[u u ]= _j,k _jkE[u_ju_k]= _j,k _jkCov(u)_jk=Tr (Cov(u) ). (50) Substituting Eq. (49) into Eq. (47) yields Eq. (46). QRAQ specialisation. Lemma C.2 makes Cov⁡(Uτ,i,:)Cov(U_τ,i,:) diagonal with diagonal entries Var⁡(W^τ,ijQ)=μτ,g(i,j)2νK(η)/SVar( W^Q_τ,ij)= _τ,g(i,j)^2 _K(η)/S from Lemma C.3. Hence Tr⁡(Cov⁡(Uτ,i,:)Στ)=∑jVar⁡(W^τ,ijQ)(Στ)jjTr(Cov(U_τ,i,:) _τ)= _jVar( W^Q_τ,ij)( _τ)_j, and summing over i gives the noise term in Eq. (10). ∎ C.5 Formal fixed-readout no-go Theorem C.5 (Fixed-readout classical simulability). Let a fixed-readout quantum quantizer store a product state ρ=⨂(i,j)ρijρ= _(i,j) _ij over weight slots and apply the same POVM Eyy∈\E_y\_y∈Y at every slot in every context, with the context τ entering only through a deterministic decoder fτ,g:→ℝf_τ,g:Y→R producing W^τ,ijQ=fτ,g(i,j)(Yij) W^Q_τ,ij=f_τ,g(i,j)(Y_ij). Then there exists a classical stochastic simulator that draws, per weight slot (i,j)(i,j), one shared classical random variable ξij∈ _ij∈Y with Pr⁡(ξij=y)=Tr⁡(ρijEy) ( _ij=y)=Tr( _ijE_y) and outputs W^τ,ijC=fτ,g(i,j)(ξij) W^C_τ,ij=f_τ,g(i,j)( _ij). The joint distributions of W^τ,ijCi,j\ W^C_τ,ij\_i,j and W^τ,ijQi,j\ W^Q_τ,ij\_i,j are identical for every context τ, and the two schemes have the same expected layer-wise reconstruction risk. If ||=2|Y|=2 and fτ,g(+1)=−fτ,g(−1)f_τ,g(+1)=-f_τ,g(-1), the simulator is a stochastic signed one-bit quantizer in the corresponding scale class; if ||=2|Y|=2 without sign symmetry, it is an affine one-bit quantizer with a context-dependent zero-point; if ||>2|Y|>2, it is a classical |||Y|-ary stochastic quantizer. Proof of Theorem C.5. Per the fixed-readout setup, the quantum memory stores one state per weight slot, so the global state factorises as ρ=⨂(i,j)ρijρ= _(i,j) _ij. Let ℰ=Eyy∈E=\E_y\_y∈Y be the fixed POVM applied at every slot in every context, and suppose context τ enters only through a deterministic decoder fτ,g(i,j):→ℝf_τ,g(i,j):Y→R that maps the measurement outcome to a quantized weight component, so W^τ,ijQ=fτ,g(i,j)(Yij) W^Q_τ,ij=f_τ,g(i,j)(Y_ij) where YijY_ij is the outcome at slot (i,j)(i,j). For each slot, the outcome distribution Pr⁡(Yij=y)=Tr⁡(ρijEy) (Y_ij=y)=Tr( _ijE_y) is independent of τ. Step 1: joint-distribution simulability. Define one classical random variable ξij∈ _ij∈Y per slot with law Pr⁡(ξij=y)=Tr⁡(ρijEy) ( _ij=y)=Tr( _ijE_y), mutually independent across (i,j)(i,j). By Lemma C.2(i) applied to the product state ⨂(i,j)ρij _(i,j) _ij and the product POVM ⨂(i,j)Ey _(i,j)\E_y\, the joint law of Yiji,j\Y_ij\_i,j also factorises into independent marginals identical to those of ξiji,j\ _ij\_i,j. Hence the random matrices W^τQ=(fτ,g(i,j)(Yij))i,j,W^τC≔(fτ,g(i,j)(ξij))i,j, W^Q_τ= (f_τ,g(i,j)(Y_ij) )_i,j, W^C_τ (f_τ,g(i,j)( _ij) )_i,j, (51) have identical joint distributions for every context τ. Since the reconstruction risk Eq. (3) is a deterministic quadratic functional of the random matrix W^τ W_τ, equality in distribution implies equality of the expected risks for every τ and hence of their π-weighted sum. Step 2: decoder-class identification. For ||=2|Y|=2, identify Y with +1,−1\+1,-1\ and parameterise the binary decoder by fτ,g(C)=βτ,g+ατ,gC,βτ,g=fτ,g(+1)+fτ,g(−1)2,ατ,g=fτ,g(+1)−fτ,g(−1)2.f_τ,g(C)= _τ,g+ _τ,gC, _τ,g= f_τ,g(+1)+f_τ,g(-1)2, _τ,g= f_τ,g(+1)-f_τ,g(-1)2. (52) The simulator of Step 1 then takes the form W^τ,ijC=βτ,g(i,j)+ατ,g(i,j)Cij W^C_τ,ij= _τ,g(i,j)+ _τ,g(i,j)C_ij with classical sign Cij≔2[ξij=+1]−1∈±1C_ij 21[ _ij=+1]-1∈\± 1\ and per-context-and-group scale ατ,g _τ,g, plus zero-point βτ,g _τ,g. The zero-point vanishes if and only if fτ,gf_τ,g is sign-symmetric (fτ,g(+1)=−fτ,g(−1)f_τ,g(+1)=-f_τ,g(-1)); in this case the simulator lives in row±S_row± (or in a coarser class, depending on g). Otherwise the simulator is an affine one-bit quantizer with a context-dependent zero-point. For general ||>2|Y|>2, the same identification produces a classical stochastic quantizer with |||Y| codebook entries per slot, encodable in ⌈log2⁡||⌉ _2|Y| classical bits per slot. Step 3: conclusion. In every case, the classical simulator matches the quantum scheme in distribution and risk, so no fixed-readout quantum scheme can strictly beat the corresponding classical stochastic decoder class with the same outcome alphabet and decoder family. ∎ C.6 Formal sufficient advantage condition Theorem C.6 (Sufficient advantage condition). Fix a scale class S and a context distribution Στ,πτ\ _τ, _τ\. Let ℰQ⋆E_Q and ℰC,⋆E_C^S, be the optima of the QRAQ and classical shared-sign risks over their admissible variables. A sufficient condition for ℰQ⋆<ℰC,⋆E_Q <E_C^S, is that there exist per-context sign matrices B(τ)∈(±1N×M)K\B^(τ)\∈(\± 1\^N× M)^K and scales μτ∈K\ _τ\∈S^K such that ∑τ=1KπτRτ(W,μτ⊙B(τ))+νK(η)S∑τ=1K∑i=1N∑j=1Mπτμτ,g(i,j)2(Στ)jj<minC,α∑τ=1KπτRτ(W,ατ⊙C). _τ=1^K _τR_τ (W, _τ B^(τ) )+ _K(η)S _τ=1^K _i=1^N _j=1^M _τ _τ,g(i,j)^2( _τ)_j< _C,α _τ=1^K _τR_τ(W, _τ C). (53) Proof of Theorem C.6. Any feasible pair B(τ),μτ∈(±1N×M)K×K\B^(τ), _τ\∈(\± 1\^N× M)^K×S^K produces, by Lemma C.4, the QRAQ risk upper bound ℰQ⋆≤∑τ=1KπτRτ(W,μτ⊙B(τ))+νK(η)S∑τ=1K∑i=1N∑j=1Mπτμτ,g(i,j)2(Στ)jj,E_Q ≤ _τ=1^K _τR_τ (W, _τ B^(τ) )+ _K(η)S _τ=1^K _i=1^N _j=1^M _τ _τ,g(i,j)^2( _τ)_j, (54) where the first sum is the bias and the second is the variance contribution from Lemma C.4. The classical one-bit minimum is ℰC,⋆=minC∈±1N×M,ατ∈K∑τ=1KπτRτ(W,ατ⊙C),E_C^S, = _C∈\± 1\^N× M, _τ∈S^K _τ=1^K _τR_τ(W, _τ C), (55) which equals the right-hand side of Eq. (53) by definition of the classical quantizer (the sign matrix C does not pick up a shot-noise regulariser because classical bits are noiseless). If Eq. (53) holds, the right-hand side of Eq. (54) is strictly less than Eq. (55), so ℰQ⋆<ℰC,⋆E_Q <E_C^S, . ∎ C.7 Formal proof of the main row-wise separation Theorem C.7 (Main separation: per-row signed scale). Assume the scale class row±S_row±, a pairwise anticommuting K-mode QRAC with cK=1/Kc_K=1/ K, depolarizing parameter η∈(0,1]η∈(0,1], context prior πτ>0 _τ>0 for every τ∈[K]τ∈[K], and Στ⪰0 _τ 0 with bΣτb⊤>0b _τb >0 for every b∈±1Mb∈\± 1\^M. For each row i, define ℰC,irow±≔minc∈±1M,ατ∈ℝ∑τ=1Kπτ(Wi,:−ατc)Στ(Wi,:−ατc)⊤,E^row±_C,i _c∈\± 1\^M, _τ∈R _τ=1^K _τ(W_i,:- _τc) _τ(W_i,:- _τc) , (56) define ℰQ,irow(S)E^row_Q,i(S) from Eq. (10) restricted to row i, and let Δi∞ _i^∞ and Δi(S) _i(S) be the corresponding classical-minus-QRAQ gaps at S=∞S=∞ and finite S. Then: (a) Δi∞≥0 _i^∞≥ 0, with Δi∞>0 _i^∞>0 if and only if ⋂τ=1Kℳτ=∅ _τ=1^KM_τ= modulo the global sign symmetry b∼−b -b, where ℳτ=argminb∈±1M⁡Jτ⋆(b)M_τ= *arg\,min_b∈\± 1\^MJ_τ (b) and Jτ⋆J_τ is defined in Eq. (61). (b) If the intersection condition in part (a) holds, then Δi(S)>0 _i(S)>0 whenever S>S0(i)≔νK(η)∑τ=1KπτTτ(μτ⋆)2Δi∞,Tτ≔∑j=1M(Στ)jj,S>S_0^(i) _K(η) _τ=1^K _τT_τ( _τ )^2 _i^∞, T_τ _j=1^M( _τ)_j, (57) for any ideal row-wise QRAQ optimizer Bτ⋆,μτ⋆τ\B_τ , _τ \_τ. If the intersection condition fails, then Δi(S)≤0 _i(S)≤ 0 for every finite S. Corollary C.8 (Finite-shot threshold). Under the sign-disagreement condition of Theorem C.7, the per-row gap is positive whenever S>S0(i)≔νK(η)∑τ=1KπτTτ(μτ⋆)2Δi∞,Tτ≔∑j=1M(Στ)jj,S>S_0^(i) _K(η) _τ=1^K _τT_τ( _τ )^2 _i^∞, T_τ _j=1^M( _τ)_j, (58) where Bτ⋆∈ℳτB_τ ∈M_τ and μτ⋆=ατ⋆(Bτ⋆) _τ = _τ (B_τ ) are any ideal QRAQ row optimizers. This is a sufficient threshold; it need not be necessary. Proof of Theorem C.7. We prove the theorem row by row. The Frobenius-norm risk decomposes additively as Tr⁡[(W−W^τ)Στ(W−W^τ)⊤]=∑i=1N(W−W^τ)i,:Στ(W−W^τ)i,:⊤,Tr [(W- W_τ) _τ(W- W_τ) ]= _i=1^N(W- W_τ)_i,: _τ(W- W_τ)_i,: , (59) and, by Lemma C.2(i), Cov⁡(W^τQ)Cov( W^Q_τ) is diagonal so the variance term in Lemma C.4 is also additive across rows. The total statement therefore follows from the per-row statement by summation (see Corollary C.11). Fix row i, write w≔Wi,:∈ℝMw W_i,:∈R^M, and for a sign vector b∈±1Mb∈\± 1\^M and a scalar α∈ℝα∈R define Jτ(b,α)≔(w−αb)Στ(w−αb)⊤,τ∈[K].J_τ(b,α) (w-α b) _τ(w-α b) , τ∈[K]. (60) Step 1: per-context optimal scale. The map α↦Jτ(b,α)α J_τ(b,α) is a quadratic with second derivative 2bΣτb⊤≥02b _τb ≥ 0. Under the hypothesis bΣτb⊤>0b _τb >0, the first-order condition ∂αJτ(b,α)=−2bΣτ(w−αb)⊤=0 _αJ_τ(b,α)=-2b _τ(w-α b) =0 yields the unique minimiser ατ⋆(b)=bΣτw⊤bΣτb⊤,Jτ⋆(b)≔Jτ(b,ατ⋆(b))=wΣτw⊤−(bΣτw⊤)2bΣτb⊤. _τ (b)= b _τw b _τb , J_τ (b) J_τ (b, _τ (b) )=w _τw - (b _τw )^2b _τb . (61) Note Jτ⋆(b)=Jτ⋆(−b)J_τ (b)=J_τ (-b), since flipping b flips the sign of ατ⋆(b) _τ (b) but leaves ατ⋆(b)b _τ (b)b invariant. Step 2: classical per-row minimum. In row±S_row±, the classical quantizer Eq. (2) for row i picks a single sign vector c∈±1Mc∈\± 1\^M shared across contexts and per-context signed scales ατ∈ℝ _τ∈R. Minimising Eq. (56) first over ατ\ _τ\ (independently per context, closed-form by Eq. (61)) yields ℰC,irow±=minc∈±1M⁡F(c),F(c)≔∑τ=1KπτJτ⋆(c).E^row±_C,i= _c∈\± 1\^MF(c), F(c) _τ=1^K _τJ_τ (c). (62) Step 3: ideal QRAQ per-row minimum. In the ideal regime S→∞S→∞, the noise term in the QRAQ per-row risk Eq. (10) vanishes and each B(τ)B^(τ) appears only in the τ-th summand: ℰQ,irow,∞=minB(τ),μτ∑τ=1KπτJτ(B(τ),μτ).E^row,∞_Q,i= _\B^(τ)\,\ _τ\ _τ=1^K _τJ_τ(B^(τ), _τ). (63) Since the summands are uncoupled across τ, we may minimise term-by-term: first in μτ _τ pointwise in B(τ)B^(τ) (closed-form by Eq. (61)), then in B(τ)B^(τ). This gives ℰQ,irow,∞=∑τ=1KπτminB(τ)∈±1M⁡Jτ⋆(B(τ)).E^row,∞_Q,i= _τ=1^K _τ _B^(τ)∈\± 1\^MJ_τ (B^(τ)). (64) Step 4: ideal gap is non-negative. Subtracting Eq. (64) from Eq. (62) gives Δi∞=minc∈±1M⁡F(c)−∑τ=1Kπτminb∈±1M⁡Jτ⋆(b). _i^∞= _c∈\± 1\^MF(c)- _τ=1^K _τ _b∈\± 1\^MJ_τ (b). (65) For every c∈±1Mc∈\± 1\^M and every τ, Jτ⋆(c)≥minb⁡Jτ⋆(b)J_τ (c)≥ _bJ_τ (b), so F(c)≥∑τπτminb⁡Jτ⋆(b)F(c)≥ _τ _τ _bJ_τ (b). Taking the minimum in c preserves the inequality and gives Δi∞≥0 _i^∞≥ 0. Step 5: Δi∞>0⇔(D1) _i^∞>0 (D1). For each τ define ℳτ≔argminb∈±1M⁡Jτ⋆(b)⊆±1M,M_τ *arg\,min_b∈\± 1\^MJ_τ (b) \± 1\^M, (66) regarded as a subset of ±1M/±\± 1\^M/\ 1\ after quotienting by the global sign (because Jτ⋆(b)=Jτ⋆(−b)J_τ (b)=J_τ (-b)). Condition (D1) is ⋂τ=1Kℳτ=∅ _τ=1^KM_τ= in this quotient. (⇐ ) If ⋂τℳτ≠∅ _τM_τ≠ , pick any c⋆∈⋂τℳτc ∈ _τM_τ. Then Jτ⋆(c⋆)=minb⁡Jτ⋆(b)J_τ (c )= _bJ_τ (b) for every τ, so F(c⋆)=∑τπτminb⁡Jτ⋆(b)F(c )= _τ _τ _bJ_τ (b) and Δi∞=0 _i^∞=0. (⇒ ) Assume ⋂τℳτ=∅ _τM_τ= . Then for every c∈±1Mc∈\± 1\^M there exists τc∈[K] _c∈[K] with c∉ℳτc∉M_ _c, i.e., Jτc⋆(c)>minb⁡Jτc⋆(b)J_ _c (c)> _bJ_ _c (b). The hypothesis πτc>0 _ _c>0 upgrades this to F(c)=πτcJτc⋆(c)+∑τ≠τcπτJτ⋆(c)>πτcminb⁡Jτc⋆(b)+∑τ≠τcπτminb⁡Jτ⋆(b)=∑τπτminb⁡Jτ⋆(b).F(c)= _ _cJ_ _c (c)+ _τ≠ _c _τJ_τ (c)> _ _c _bJ_ _c (b)+ _τ≠ _c _τ _bJ_τ (b)= _τ _τ _bJ_τ (b). (67) Since ±1M\± 1\^M is finite, this strict inequality is preserved when taking the minimum in c, so Δi∞>0 _i^∞>0. Step 6: necessity of (D1) at finite S. Suppose c⋆∈⋂τℳτc ∈ _τM_τ (i.e., (D1) fails). Then ℰC,irow±=∑τπτJτ⋆(c⋆)=ℰQ,irow,∞E^row±_C,i= _τ _τJ_τ (c )=E^row,∞_Q,i. For finite S the QRAQ objective at every feasible (B(τ),μτ)(B^(τ), _τ) exceeds the ideal objective pointwise because the shot-noise regulariser (νK(η)/S)μτ2Tτ≥0( _K(η)/S) _τ^2T_τ≥ 0. Hence minimisation preserves the inequality: ℰQ,irow(S)≥ℰQ,irow,∞=ℰC,irow±,E^row_Q,i(S) ^row,∞_Q,i=E^row±_C,i, (68) i.e., Δi(S)≤0 _i(S)≤ 0, with equality only in the limit S→∞S→∞. Thus (D1) is necessary for Δi(S)>0 _i(S)>0 at any finite S. Step 7: finite-shot QRAQ per-row risk. Including the depolarizing factor absorbed into μτ _τ (Lemma C.3), the finite-shot per-row QRAQ risk equals ℰQ,irow(S)=minB(τ),μτ∑τ=1Kπτ[Jτ(B(τ),μτ)+νK(η)Sμτ2Tτ],Tτ≔∑j=1M(Στ)jj.E^row_Q,i(S)= _\B^(τ)\,\ _τ\ _τ=1^K _τ [J_τ(B^(τ), _τ)+ _K(η)S _τ^2T_τ ], T_τ _j=1^M( _τ)_j. (69) Step 8: finite-shot threshold. Let Bτ⋆,μτ⋆τ\B_τ , _τ \_τ be a tuple of ideal row-wise QRAQ optimisers, i.e., Bτ⋆∈ℳτB_τ ∈M_τ and μτ⋆=ατ⋆(Bτ⋆) _τ = _τ (B_τ ) from Eq. (61). Plugging this feasible tuple into the right-hand side of Eq. (69) yields the upper bound ℰQ,irow(S)≤∑τ=1Kπτ[Jτ(Bτ⋆,μτ⋆)+νK(η)S(μτ⋆)2Tτ]=ℰQ,irow,∞+νK(η)S∑τ=1KπτTτ(μτ⋆)2,E^row_Q,i(S)≤ _τ=1^K _τ [J_τ(B_τ , _τ )+ _K(η)S( _τ )^2T_τ ]=E^row,∞_Q,i+ _K(η)S _τ=1^K _τT_τ( _τ )^2, (70) since ∑τπτJτ(Bτ⋆,μτ⋆)=∑τπτJτ⋆(Bτ⋆)=ℰQ,irow,∞ _τ _τJ_τ(B_τ , _τ )= _τ _τJ_τ (B_τ )=E^row,∞_Q,i by definition of the ideal optimum. Subtracting from ℰC,irow±E^row±_C,i and using Δi∞=ℰC,irow±−ℰQ,irow,∞ _i^∞=E^row±_C,i-E^row,∞_Q,i, Δi(S)≥Δi∞−νK(η)S∑τ=1KπτTτ(μτ⋆)2, _i(S)≥ _i^∞- _K(η)S _τ=1^K _τT_τ( _τ )^2, (71) which is strictly positive whenever S>S0(i)≔νK(η)∑τ=1KπτTτ(μτ⋆)2Δi∞,S>S_0^(i) _K(η) _τ=1^K _τT_τ( _τ )^2 _i^∞, (72) matching the form in Corollary C.8. Under (D1), Δi∞>0 _i^∞>0 by Step 5, so S0(i)<∞S_0^(i)<∞. Combined with Step 6, this proves parts (a) and (b) of Theorem C.7; the row-additive total gap of Corollary C.11 follows by summation over i. ∎ C.8 Formal closed-form two-context corollary Corollary C.9 (Closed-form gap, M=2M=2, K=2K=2, symmetric covariance). Let M=K=2M=K=2, Σ±=I±r(J−I) _±=I± r(J-I) with r∈[0,1)r∈[0,1), and π±=1/2 _±=1/2. Write w=(w1,w2)=Wi,:w=(w_1,w_2)=W_i,:. If w=0w=0, both classical and QRAQ row risks vanish, so Δi=0 _i=0. Otherwise, in the ideal regime S=∞S=∞, η=1η=1, the per-row gap satisfies Δi=r(w12+w22)−2|w1w2|2⋅[r>r0],r0≔2|w1w2|w12+w22∈[0,1]. _i= r(w_1^2+w_2^2)-2|w_1w_2|2·1 [r>r_0 ], r_0 2|w_1w_2|w_1^2+w_2^2∈[0,1]. (73) The gap is strictly positive whenever r>r0r>r_0 and is monotone increasing in r above that threshold. Proof of Corollary C.9. Fix M=K=2M=K=2, uniform prior π±=1/2 _±=1/2, and context covariances Σ±=I±r(J−I)=(1±r±r1),r∈[0,1). _±=I± r(J-I)= pmatrix1&± r\\ ± r&1 pmatrix, r∈[0,1). (74) If w=(0,0)w=(0,0) then both classical and ideal QRAQ row risks vanish, so Δi=0 _i=0. We may therefore assume w≠0w≠ 0. Without loss of generality assume w1w2≥0w_1w_2≥ 0; the case w1w2<0w_1w_2<0 is symmetric under swapping the two context labels, as we make explicit at the end. Step 1: per-context quadratic forms. For b=(b1,b2)∈±12b=(b_1,b_2)∈\± 1\^2, direct computation gives bΣ±b⊤ b _±b =b12+b22±2rb1b2=2±2rb1b2, =b_1^2+b_2^2± 2rb_1b_2=2± 2rb_1b_2, (75) bΣ±w⊤ b _±w =b1w1+b2w2±r(b1w2+b2w1)=(b1±rb2)w1+(b2±rb1)w2, =b_1w_1+b_2w_2± r(b_1w_2+b_2w_1)=(b_1± rb_2)w_1+(b_2± rb_1)w_2, (76) wΣ±w⊤ w _±w =w12+w22±2rw1w2. =w_1^2+w_2^2± 2rw_1w_2. (77) Hence by Eq. (61), J±⋆(b)=wΣ±w⊤−((b1±rb2)w1+(b2±rb1)w2)22±2rb1b2.J_± (b)=w _±w - ((b_1± rb_2)w_1+(b_2± rb_1)w_2 )^22± 2rb_1b_2. (78) Step 2: enumeration modulo global sign. Since Jτ⋆(b)=Jτ⋆(−b)J_τ (b)=J_τ (-b), the four sign vectors in ±12\± 1\^2 reduce to two representatives modulo the global sign: b+=(+,+)b_+=(+,+) and b−=(+,−)b_-=(+,-). We evaluate Eq. (78) at b±b_± and both contexts. Sub-step 2a: b=b+b=b_+. b1b2=+1b_1b_2=+1, so 2±2rb1b2=2±2r2± 2rb_1b_2=2± 2r and bΣ±w⊤=(1±r)(w1+w2)b _±w =(1± r)(w_1+w_2). Hence J+⋆(b+)=(w12+w22+2rw1w2)−(1+r)(w1+w2)22=(1−r)(w1−w2)22,J_+ (b_+)=(w_1^2+w_2^2+2rw_1w_2)- (1+r)(w_1+w_2)^22= (1-r)(w_1-w_2)^22, (79) where we used (w1+w2)2=w12+w22+2w1w2(w_1+w_2)^2=w_1^2+w_2^2+2w_1w_2. Similarly, J−⋆(b+)=(1+r)(w1−w2)22.J_- (b_+)= (1+r)(w_1-w_2)^22. (80) Sub-step 2b: b=b−b=b_-. b1b2=−1b_1b_2=-1, so 2±2rb1b2=2∓2r2± 2rb_1b_2=2∓ 2r and bΣ±w⊤=(1∓r)(w1−w2)b _±w =(1∓ r)(w_1-w_2). Hence J+⋆(b−)=(1+r)(w1+w2)22,J−⋆(b−)=(1−r)(w1+w2)22.J_+ (b_-)= (1+r)(w_1+w_2)^22, J_- (b_-)= (1-r)(w_1+w_2)^22. (81) These clean forms admit an eigendecomposition interpretation: Σ± _± has eigenvalue (1±r)(1± r) along (1,1)(1,1) and (1∓r)(1∓ r) along (1,−1)(1,-1) (verified by direct multiplication), so fitting αb±α b_± under Στ _τ removes the component along b±b_± and leaves a residual weighted by the eigenvalue along the orthogonal direction b∓b_∓. Step 3: classical per-row minimum. Set A≔w12+w22>0A w_1^2+w_2^2>0 (since w≠0w≠ 0) and P≔w1w2P w_1w_2, so (w1±w2)2=A±2P(w_1± w_2)^2=A± 2P. The closed forms of Step 2 become J+⋆(b+) J_+ (b_+) =(1−r)(A−2P)2,J−⋆(b+)=(1+r)(A−2P)2, = (1-r)(A-2P)2, J_- (b_+)= (1+r)(A-2P)2, (82) J+⋆(b−) J_+ (b_-) =(1+r)(A+2P)2,J−⋆(b−)=(1−r)(A+2P)2. = (1+r)(A+2P)2, J_- (b_-)= (1-r)(A+2P)2. Hence with F(c)=12[J+⋆(c)+J−⋆(c)]F(c)= 12 [J_+ (c)+J_- (c) ], F(b+)=A−2P2=(w1−w2)22,F(b−)=A+2P2=(w1+w2)22.F(b_+)= A-2P2= (w_1-w_2)^22, F(b_-)= A+2P2= (w_1+w_2)^22. (83) Under w1w2≥0w_1w_2≥ 0 we have P≥0P≥ 0 and F(b+)≤F(b−)F(b_+)≤ F(b_-), so ℰC,irow±=F(b+)=(w1−w2)22=A−2P2.E^row±_C,i=F(b_+)= (w_1-w_2)^22= A-2P2. (84) Step 4: per-context QRAQ argmin. Direct subtraction gives J+⋆(b+)−J+⋆(b−)=−rA−2P,J−⋆(b+)−J−⋆(b−)=rA−2P.J_+ (b_+)-J_+ (b_-)=-rA-2P, J_- (b_+)-J_- (b_-)=rA-2P. (85) Under P≥0P≥ 0 the first difference is ≤0≤ 0 for every r∈[0,1)r∈[0,1), so minb⁡J+⋆(b)=J+⋆(b+) _bJ_+ (b)=J_+ (b_+). The second difference has sign sgn(rA−2P)sgn(rA-2P), i.e., the threshold r0≔2P/A=2|w1w2|/(w12+w22)∈[0,1]r_0 2P/A=2|w_1w_2|/(w_1^2+w_2^2)∈[0,1]: for r>r0r>r_0, minb⁡J−⋆(b)=J−⋆(b−) _bJ_- (b)=J_- (b_-); for r≤r0r≤ r_0, minb⁡J−⋆(b)=J−⋆(b+) _bJ_- (b)=J_- (b_+). Step 5: closed-form QRAQ minimum and gap. Case (I): r>r0r>r_0. Using the minimisers from Step 4, ℰQ,irow,∞=12J+⋆(b+)+12J−⋆(b−)=14[(1−r)(A−2P)+(1−r)(A+2P)]=(1−r)A2.E^row,∞_Q,i= 12J_+ (b_+)+ 12J_- (b_-)= 14 [(1-r)(A-2P)+(1-r)(A+2P) ]= (1-r)A2. (86) Subtracting from ℰC,irow±=(A−2P)/2E^row±_C,i=(A-2P)/2, Δi∞=A−2P2−(1−r)A2=rA−2P2>0. _i^∞= A-2P2- (1-r)A2= rA-2P2>0. (87) Case (I): 0≤r≤r00≤ r≤ r_0. Both contexts are minimised at b+b_+, so ℰQ,irow,∞=F(b+)=ℰC,irow±E^row,∞_Q,i=F(b_+)=E^row±_C,i and Δi∞=0 _i^∞=0. Case w1w2<0w_1w_2<0. Symmetric under b+↔b−b_+ b_- (and equivalently under swapping the two context labels): the classical minimiser becomes b−b_-, the QRAQ context-++ minimiser flips to b−b_- whenever r<2|P|/Ar<2|P|/A (and to b+b_+ whenever r>2|P|/Ar>2|P|/A), and the analogous computation yields Δi∞=(rA−2|P|)/2 _i^∞=(rA-2|P|)/2 above the threshold r0=2|P|/Ar_0=2|P|/A. Combining the two cases produces Eq. (73) with |w1w2||w_1w_2| in both the numerator and the indicator. Monotonicity above threshold follows from ∂Δi∞/∂r=A/2>0∂ _i^∞/∂ r=A/2>0. ∎ Corollary C.10 (Resource-fair closed-form advantage condition). Under the hypotheses of Corollary C.9, set η=1η=1 and S=1S=1. Then Δi(S)>0 _i(S)>0 if and only if −(1−r0)+(1+r0)(5+r0)2<r< 1, -(1-r_0)+ (1+r_0)(5+r_0)2\;<\;r\;<\;1, (88) where r0=2|w1w2|/(w12+w22)r_0=2|w_1w_2|/(w_1^2+w_2^2) is the ideal-regime threshold defined in Eq. (73). Proof of Corollary C.10. We work with general S≥1S≥ 1 and η∈(0,1]η∈(0,1] and specialise at the end. Finite-shot optimal scale. Under signed per-row scales, the per-row QRAQ risk (Eq. (10)) restricted to row i is ℰQ,irow(S)=minb(τ),μτ∑τ=1Kπτ[(w−μτb(τ))Στ(w−μτb(τ))⊤+νK(η)Sμτ2Tτ],E^row_Q,i(S)= _\b^(τ)\,\,\ _τ\ _τ=1^K _τ [(w- _τb^(τ))\, _τ\,(w- _τb^(τ)) + _K(η)S\, _τ^2\,T_τ ], (89) where w=Wi,:w=W_i,: and Tτ=∑j(Στ)jjT_τ= _j( _τ)_j. Because the variance term is quadratic in μτ _τ, the objective remains a quadratic in μτ _τ for each fixed b(τ)b^(τ). Minimising over μτ _τ yields the finite-shot optimal scale and the corresponding minimum Jτ⋆(b;S)=wΣτw⊤−(bΣτw⊤)2bΣτb⊤+νK(η)Tτ/S,J _τ(b;\,S)\;=\;w\, _τ\,w - (b\, _τ\,w )^2b\, _τ\,b + _K(η)\,T_τ/S, (90) which reduces to Jτ⋆(b)J _τ(b) of Eq. (61) when S→∞S→∞. Specialisation to M=K=2M=K=2, Σ±=I±r(J−I) _±=I± r(J-I). As in Corollary C.9, write w=(w1,w2)≠0w=(w_1,w_2)≠ 0 and assume w1w2≥0w_1w_2≥ 0 without loss of generality. Since T±=tr⁡(Σ±)=2T_±=tr( _±)=2, the finite-shot row risk for each context and sign vector b±b_± is J±⋆(b;S)=wΣ±w⊤−((b1±rb2)w1+(b2±rb1)w2)22±2rb1b2+2νK(η)/S.J _±(b;\,S)=w\, _±\,w - ((b_1± r\,b_2)\,w_1+(b_2± r\,b_1)\,w_2 )^22± 2r\,b_1b_2+2 _K(η)/S. (91) Plus context: b+b_+ is always optimal. We claim J+⋆(b+;S)≤J+⋆(b−;S)J _+(b_+;\,S)≤ J _+(b_-;\,S). Using Eq. (91), this is equivalent to (1+r)(w1+w2)22+2νK(η)S(1+r)≥(1−r)(w1−w2)22+2νK(η)S(1−r). (1+r)(w_1+w_2)^22+ 2 _K(η)S(1+r)\;≥\; (1-r)(w_1-w_2)^22+ 2 _K(η)S(1-r). (92) Under w1w2≥0w_1w_2≥ 0, the left-hand side has a larger numerator ((1+r)(w1+w2)2≥(1−r)(w1−w2)2(1+r)(w_1+w_2)^2≥(1-r)(w_1-w_2)^2) and a smaller denominator (11+r≤11−r 11+r≤ 11-r), so the inequality holds. Minus context and necessity. Recall from Eq. (84) that the classical optimum chooses b+b_+ common to both contexts. If J−⋆(b+;S)≤J−⋆(b−;S)J _-(b_+;\,S)≤ J _-(b_-;\,S), then the quantum quantizer also chooses b+b_+ in both contexts, so its risk satisfies ℰQ,irow(S)=12(J+⋆(b+;S)+J−⋆(b+;S))>12(J+⋆(b+;∞)+J−⋆(b+;∞))=ℰC,irow±,E^row_Q,i(S)= 12 (J _+(b_+;\,S)+J _-(b_+;\,S) )> 12 (J _+(b_+;\,∞)+J _-(b_+;\,∞) )=E^row±_C,i, (93) where the strict inequality follows from νK(η)/S>0 _K(η)/S>0 in the denominator. Hence quantum advantage requires J−⋆(b+;S)>J−⋆(b−;S)J _-(b_+;\,S)>J _-(b_-;\,S), and in that case ℰQ,irow(S)=12(J+⋆(b+;S)+J−⋆(b−;S))=A2(2−(1+r)2(1+r)+νK(η)/S).E^row_Q,i(S)= 12 (J _+(b_+;\,S)+J _-(b_-;\,S) )= A2 (2- (1+r)^2(1+r)+ _K(η)/S ). (94) Setting S=1S=1 and η=1η=1 (so that ν2(1)=1 _2(1)=1) gives ℰQ,irow(1)=A(3−r2)/(2(2+r))E^row_Q,i(1)=A(3-r^2)/\! (2(2+r) ). Comparing with ℰC,irow±=(A−2|P|)/2E^row±_C,i=(A-2|P|)/2 and writing r0=2|P|/Ar_0=2|P|/A yields r2+(1−r0)r−(1+2r0)>0,r^2+(1-r_0)\,r-(1+2r_0)>0, (95) whose positive root gives the lower threshold in Eq. (88). Sufficiency. Conversely, assume Eq. (88) holds. Then the preceding calculation gives 12(J+⋆(b+; 1)+J−⋆(b−; 1))<ℰC,irow± 12(J _+(b_+;\,1)+J _-(b_-;\,1))<E^row±_C,i. Since ℰQ,irow(1)=12(J+⋆(b+; 1)+min⁡J−⋆(b+; 1),J−⋆(b−; 1))≤12(J+⋆(b+; 1)+J−⋆(b−; 1)),E^row_Q,i(1)= 12 (J _+(b_+;\,1)+ \J _-(b_+;\,1),\,J _-(b_-;\,1)\ )≤ 12 (J _+(b_+;\,1)+J _-(b_-;\,1) ), (96) quantum advantage follows. The sufficient condition of Corollary C.8, applied with S=1S=1, requires 1>S0(i)=A/(rA−2|P|)1>S_0^(i)=A/(rA-2|P|), i.e. rA−2|P|>ArA-2|P|>A, which fails for every r∈[0,1)r∈[0,1). Thus, in the two-context symmetric covariance family of Corollary C.9, Corollary C.8 can never certify quantum advantage at S=1S=1. This is because Corollary C.8 evaluates the finite-shot QRAQ risk at the scale μτ⋆μ _τ that is optimal in the ideal S=∞S=∞ regime (Eq. (61)), yielding only an upper bound on the true finite-shot risk. Corollary C.10 circumvents this looseness by optimising the scale jointly with the shot-noise term. ∎ C.9 Formal row-additivity corollary Corollary C.11 (Row additivity). Under the assumptions of Theorem C.7 and for every S∈ℕ∪∞S∈N∪\∞\, ℰCrow±−ℰQrow(S)=∑i=1NΔi(S).E^row±_C-E^row_Q(S)= _i=1^N _i(S). (97) In the ideal limit, every summand is nonnegative and the total gap is strictly positive if and only if at least one row satisfies the sign-disagreement condition of Theorem C.7. At finite S, a strict total advantage holds if and only if ∑iΔi(S)>0 _i _i(S)>0; sufficient conditions are that all finite-shot row gaps are nonnegative and at least one is positive. Proof of Corollary C.11. Step 1: row-wise decomposition. The trace form Tr⁡[(W−W^τ)Στ(W−W^τ)⊤]Tr[(W- W_τ) _τ(W- W_τ) ] in Eq. (3) decomposes additively across rows of W−W^τW- W_τ, and the variance contribution to the QRAQ risk also decomposes additively across rows by the diagonality of Cov⁡(W^τQ)Cov( W^Q_τ) (Lemma C.2(i)). Both the classical row-wise minimum Eq. (56) and the QRAQ row-wise minimum are therefore independent across i: ℰCrow±=∑i=1NℰC,irow±,ℰQrow(S)=∑i=1NℰQ,irow(S),S∈ℕ∪∞.E^row±_C= _i=1^NE^row±_C,i, ^row_Q(S)= _i=1^NE^row_Q,i(S), S∈N∪\∞\. (98) This yields the row-additive identity ℰCrow±−ℰQrow(S)=∑i=1NΔi(S)E^row±_C-E^row_Q(S)= _i=1^N _i(S). Step 2: ideal limit. At S=∞S=∞, Theorem C.7(a) gives Δi∞≥0 _i^∞≥ 0 row by row, so ℰCrow±−ℰQrow(∞)=∑i=1NΔi∞≥0E^row±_C-E^row_Q(∞)= _i=1^N _i^∞≥ 0, with strict positivity if and only if at least one row satisfies (D1) (by Step 5 of the proof of Theorem C.7). If N0N_0 rows satisfy (D1) with common gap lower bound Δ0>0 _0>0, the ideal total gap is at least N0Δ0N_0 _0. Step 3: finite-S regime. For finite S, on a row of zero ideal gap the QRAQ risk satisfies ℰQ,irow(S)≥ℰQ,irow,∞=ℰC,irow±E^row_Q,i(S) ^row,∞_Q,i=E^row±_C,i (Step 6 of the proof of Theorem C.7), so Δi(S)≤0 _i(S)≤ 0 on those rows. Hence non-negativity of ∑i=1NΔi(S) _i=1^N _i(S) is not automatic: a strict total advantage requires that per-row gaps on (D1)-rows compensate the shot-noise loss on the remaining rows. The stated sufficiency is immediate: if Δi(S)≥0 _i(S)≥ 0 for every i and Δi(S)>0 _i(S)>0 for at least one i, then ∑i=1NΔi(S)>0 _i=1^N _i(S)>0. By Theorem C.7(b), every (D1)-row achieves Δi(S)>0 _i(S)>0 for S>S0(i)S>S_0^(i). The remaining rows must be controlled directly because finite-shot regularization can make their gaps negative. They are harmless whenever their finite-shot gaps are nonnegative; for example, this holds on a zero-gap row if it admits a zero-scale QRAQ optimizer, in which case the variance term in Eq. (69) vanishes. ∎ C.10 Formal scale-granularity boundary Theorem C.12 (Scale-granularity boundary). Under the assumptions of Theorem C.7, the following hold. (a) If the feasible classical set of a scale class S is contained in row±S_row±, for example ∈tensor±,tensor+,row+,row±S∈\S_tensor±,S_tensor+,S_row+,S_row±\, then ℰC−ℰQrow(∞)≥ℰCrow±−ℰQrow(∞)=∑i=1NΔi∞≥0.E^S_C-E^row_Q(∞) ^row±_C-E^row_Q(∞)= _i=1^N _i^∞≥ 0. (99) The ideal separation is strict whenever at least one row satisfies the sign-disagreement condition. At finite S, a strict separation over S holds whenever the row-summed ideal margin in Eq. (99) exceeds the row-summed finite-shot inflation. In particular, if every row has a positive ideal gap and S>maxi⁡S0(i)S> _iS_0^(i), then the finite-shot separation persists. (b) If S is not contained in row±S_row±, for example col±S_col±, row×col±S_row× col±, or group(g)±S_group(g)±, no universal separation follows from the row-wise theorem. The necessary and sufficient ideal condition is the direct margin δ≔ℰC−ℰQrow(∞)>0. _S ^S_C-E^row_Q(∞)>0. (100) At finite S, QRAQ retains a strict advantage over S whenever δ>ℰQrow(S)−ℰQrow(∞) _S>E^row_Q(S)-E^row_Q(∞). (c) For entry±=entryS_entry±=S_entry, the classical model can realize W exactly by choosing entry-wise scales, so the classical risk is zero and no positive reconstruction-risk advantage is possible. Proof of Theorem C.12. Part (a): coarser-than-row classes. Let S be a scale class whose feasible classical set is contained in the signed per-row feasible set, e.g. ∈tensor±,tensor+,row+,row±S∈\S_tensor±,S_tensor+,S_row+,S_row±\. The classical optimum is taken over a smaller feasible set, so ℰC≥ℰCrow±.E^S_C ^row±_C. (101) The stated inclusions hold by definition: per-tensor is per-row with constant scale, and signed scales contain non-negative scales. Substituting Theorem C.7(a) on the right yields Eq. (99), with strict inequality whenever at least one row satisfies (D1). For finite S, combine Eq. (99) with the bias inflation bound Eq. (70): the strict separation persists at any S>maxi⁡S0(i)S> _iS_0^(i) such that the (row-summed) bias inflation is dominated by the ideal gap. Part (b): scale classes incomparable to or finer than row±S_row±. The classes col±,row×col±,group(g)±S_col±,S_row× col±,S_group(g)± are not contained in row±S_row± in general (e.g., a per-column signed scale ατ,j _τ,j is not expressible as a per-row signed scale unless it is constant in j). Consequently, the inclusion argument of part (a) does not apply, and the classical risk in these classes can in principle drop below the per-row signed minimum on structured instances where a per-context column-sign reassignment is absorbed into the column scale. For any finite-dimensional problem instance (W,Στ,πτ)(W,\ _τ, _τ\), however, both ℰCE^S_C and ℰQrow(∞)E^row_Q(∞) are well defined, so δ _S in Eq. (100) is computable from the data; by definition, ℰQrow(∞)<ℰC⇔δ>0.E^row_Q(∞)<E^S_C _S>0. (102) The finite-shot statement follows from ℰQrow(S)=ℰQrow(∞)+(ℰQrow(S)−ℰQrow(∞))E^row_Q(S)=E^row_Q(∞)+ (E^row_Q(S)-E^row_Q(∞) ) (103) combined with Eq. (70) summed over rows: QRAQ retains a strict advantage over S as soon as δ _S exceeds the (row-summed) bias inflation. We do not claim a closed-form sufficient criterion on W and Στ\ _τ\ alone; in the experiments reported in Appendix E, we verify the margin condition numerically. Part (c): per-entry classes. With entry±=entryS_entry±=S_entry, every weight entry has a free real scale, so the codebook contains W itself and the classical risk is identically zero, precluding any positive reconstruction-risk advantage. ∎ C.11 Arbitrary priors Theorem C.13 (Arbitrary priors). Let K≥2K≥ 2, let πτ≥0 _τ≥ 0 with ∑τ=1Kπτ=1 _τ=1^K _τ=1, and let π≔τ∈[K]:πτ>0T_π \τ∈[K]: _τ>0\ (104) be the active context set. Under the scale class row±S_row± and the assumptions of Theorem C.7 restricted to πT_π, the ideal per-row gap is Δi∞=minc∈±1M∑τ∈πτJτ⋆(c)−∑τ∈πτminb∈±1M⁡Jτ⋆(b)≥0. _i^∞= _c∈\± 1\^M _τ∈T_π _τJ_τ (c)- _τ∈T_π _τ _b∈\± 1\^MJ_τ (b)≥ 0. (105) Strict positivity, Δi∞>0 _i^∞>0, holds if and only if ⋂τ∈πℳτ=∅ _τ∈T_πM_τ= modulo the global sign symmetry b∼−b -b, where ℳτ=argminb∈±1M⁡Jτ⋆(b)M_τ= *arg\,min_b∈\± 1\^MJ_τ (b). The finite-shot threshold of Theorem C.7(b) extends with π-weighted sums restricted to πT_π. Proof. The contexts τ∉πτ∉T_π contribute zero to both classical and QRAQ risks (both prefactors πτ _τ vanish in the corresponding summands of Eq. (62) and Eq. (64)), so without loss of generality we may assume π=[K]T_π=[K] with πτ>0 _τ>0 for every τ∈[K]τ∈[K]. Under this assumption, Eq. (105) reduces to Eq. (65); non-negativity Δi∞≥0 _i^∞≥ 0 follows from Step 4 of the proof of Theorem C.7, and the strict-positivity criterion follows from Step 5 of the same proof. The finite-shot extension is identical to Steps 7–8 of the proof of Theorem C.7 with all sums over τ restricted to πT_π. ∎ The closed-form family of Corollary C.9 extends, in the K-context case, to the diagonal-plus-rank-one covariance structure Στ=I+rτ(uu⊤−I) _τ=I+r_τ(u -I) for a common direction u∈ℝMu∈R^M with ∥u∥2=1 u _2=1 and per-context anisotropy rτr_τ; the row-by-row Jτ⋆J_τ algebra parallels Step 1 of Section C.8 and we omit the verbatim restatement. C.12 Non-anticommuting HS-orthogonal observables via Gram bound Theorem C.14 (Gram-bounded HS-orthogonal observables). Let Aτ=1K\A_τ\_τ=1^K be self-adjoint operators on ℂdC^d satisfying (i) Aτ2=IA_τ^2=I for every τ; (i) tracelessness Tr⁡(Aτ)=0Tr(A_τ)=0 for every τ; (i) Hilbert–Schmidt orthogonality Tr⁡(Aτ′Aτ)=0Tr(A_τ A_τ)=0 for every τ≠τ′τ≠τ ; and assume there exists a constant κ>0κ>0 such that the Gram operator bound (∑τ=1KbτAτ)2⪯κK⋅Ifor every b∈±1K ( _τ=1^Kb_τA_τ )^2 κ K· I every b∈\± 1\^K (106) holds. Set cG≔1/κKc_G 1/ κ K and ρb(G)≔1d(I+cG∑τ=1KbτAτ) _b^(G) 1d (I+c_G _τ=1^Kb_τA_τ ). Then: (a) ρb(G) _b^(G) is a valid density matrix (Hermitian, unit trace, positive semidefinite) for every b∈±1Kb∈\± 1\^K; (b) the QRAQ risk Eq. (10) holds with νK(η) _K(η) replaced by νG(η)=κK/η2−1 _G(η)=κ K/η^2-1; (c) the separation of Theorem C.7(b) is preserved with νK _K replaced by νG _G. When Aτ\A_τ\ pairwise anticommute, (i)–(i) are automatic (Lemma C.1) and κ=1κ=1, so Theorem C.7 is recovered verbatim. The spectral bound Eq. (106) is implied by the pointwise operator inequality λmax(∑τ,τ′bτbτ′Aτ,Aτ′/2)≤κK _ ( _τ,τ b_τb_τ \A_τ,A_τ \/2 )≤κ K, which for any unit vector |ψ⟩|ψ reduces to ⟨ψ|Sb2|ψ⟩≤κ∥b∥22 ψ|S_b^2|ψ ≤κ b _2^2, i.e., the Gram matrix of Aτ|ψ⟩τ=1K\A_τ|ψ \_τ=1^K has spectral radius at most κ for every |ψ⟩|ψ . Proof. Step 1: Hermiticity, unit trace, positivity. Tracelessness (assumption (i)) gives Tr⁡(Sb)=∑τbτTr⁡(Aτ)=0Tr(S_b)= _τb_τTr(A_τ)=0, so Tr⁡(ρb(G))=Tr⁡(I)/d=1Tr( _b^(G))=Tr(I)/d=1. Since each AτA_τ is self-adjoint, SbS_b and ρb(G) _b^(G) are Hermitian. By Eq. (106), Sb2⪯κK⋅IS_b^2 κ K· I, which means every eigenvalue of SbS_b lies in [−κK,κK][- κ K, κ K]. Hence I+cGSb⪰(1−cGκK)I=0,cG=1κK,I+c_GS_b (1-c_G κ K )I=0, c_G= 1 κ K, (107) so ρb(G)⪰0 _b^(G) 0. This proves part (a). Step 2: per-shot mean and variance. By the observable expectation formula Eq. (5) and HS orthogonality (i) together with Tr⁡(Aτ2)=dTr(A_τ^2)=d (from Aτ2=IA_τ^2=I), Tr⁡(ρb(G)Aτ)=1dTr⁡(Aτ)+cGd∑τ′=1Kbτ′Tr⁡(Aτ′Aτ)=cGdbτTr⁡(Aτ2)=cGbτ,Tr ( _b^(G)A_τ )= 1dTr(A_τ)+ c_Gd _τ =1^Kb_τ Tr(A_τ A_τ)= c_Gdb_τTr(A_τ^2)=c_Gb_τ, (108) where we used (i) and (i). Since the matched binary outcome m(τ)∈±1m^(τ)∈\± 1\ satisfies (m(τ))2=1(m^(τ))^2=1 and bτ2=1b_τ^2=1, Var⁡(m(τ))=1−(cGbτ)2=1−cG2.Var (m^(τ) )=1-(c_Gb_τ)^2=1-c_G^2. (109) Step 3: depolarizing channel and S-shot averaging. For η(ρ)=ηρ+(1−η)I/dN_η(ρ)=ηρ+(1-η)I/d, the same argument as in Step 3 of Lemma C.3 (using only Tr⁡(Aτ)=0Tr(A_τ)=0, which holds by (i)) shows that the per-shot mean becomes ηcGbτη c_Gb_τ and the per-shot variance becomes 1−(ηcG)21-(η c_G)^2; the S-shot empirical mean satisfies [m¯(τ)]=ηcGbτ,Var⁡(m¯(τ))=1−η2cG2S.E [ m^(τ) ]=η c_Gb_τ, ( m^(τ) )= 1-η^2c_G^2S. (110) Step 4: noise coefficient. Define μτ,g≔ηcGγτ,g _τ,g η c_G _τ,g. The proof of Lemma C.3, Step 5, applies verbatim with cKc_K replaced by cGc_G and yields Var⁡(W^τ,ijQ)=μτ,g(i,j)2S(1η2cG2−1)=μτ,g(i,j)2SνG(η),νG(η)=κKη2−1.Var ( W^Q_τ,ij )= _τ,g(i,j)^2S ( 1η^2c_G^2-1 )= _τ,g(i,j)^2S _G(η), _G(η)= κ Kη^2-1. (111) This proves part (b). Step 5: separation. The proof of Theorem C.7 uses only the per-slot mean and variance from Lemma C.3, the diagonality of Cov⁡(W^τQ)Cov( W^Q_τ) (Lemma C.2), and the bias-variance decomposition (Lemma C.4). Substituting Step 4 into Steps 7–8 of that proof, with νK(η) _K(η) replaced by νG(η) _G(η) everywhere, gives the analogous threshold S>S~0(i)=νG(η)∑τπτTτ(μτ⋆)2/Δi∞S> S_0^(i)= _G(η) _τ _τT_τ( _τ )^2/ _i^∞ under (D1), and the corresponding finite-shot strict separation. This proves part (c). ∎ C.13 Multi-qubit QRAC and storage-noise tradeoff Theorem C.15 (Multi-qubit QRAC). Let n≥1n≥ 1, K≤2n+1K≤ 2n+1, and Aτ=1K\A_τ\_τ=1^K a pairwise anticommuting set of self-inverse observables on n qubits (e.g. the Jordan–Wigner family of Appendix A.4). Then: (a) the Bloch radius c=1/Kc=1/ K is tight (Lemma C.1); (b) the formal separation and scale-granularity results, Theorems C.7 and C.12, hold with the noise coefficient νK(η)=K/η2−1 _K(η)=K/η^2-1; (c) the minimal Hilbert-space dimension 2n2^n supporting K pairwise anticommuting self-inverse self-adjoint operators satisfies 2n≥2⌈(K−1)/2⌉2^n≥ 2 (K-1)/2 , i.e., n≥⌈(K−1)/2⌉n≥ (K-1)/2 . Saturating the Jordan–Wigner bound K=2n+1K=2n+1 gives a logical storage-cell reduction of factor (2n+1)/n→2(2n+1)/n→ 2 as n→∞n→∞ relative to a K-context classical scheme that stores K separate sign bits per weight. This is a logical comparison; physical copies for shot averaging and control overhead must be accounted for separately. The statistical cost is a K/η2−1K/η^2-1 factor inflation in shot-noise variance. Proof. Step 1: existence (Jordan–Wigner construction). The Jordan–Wigner family of Appendix A.4 consists of 2n+12n+1 self-adjoint operators on ℂ2nC^2^n, each squaring to I, that pairwise anticommute (Pauli string verification). Hence for any K≤2n+1K≤ 2n+1 we may take Aτ=1K\A_τ\_τ=1^K as the first K such operators, and Lemma C.1 applies with d=2nd=2^n, giving parts (a) and (b) below. Step 2: tightness of c=1/Kc=1/ K and parameters. Lemma C.1(i) directly gives part (a). Step 3: separation under νK(η) _K(η). The proof of Theorem C.7 uses only Lemmas C.1, C.2, C.3, and C.4, all of which hold for any K≥2K≥ 2 on d=2n≥2d=2^n≥ 2. Substituting these lemmas into Steps 1–8 of that proof shows that Theorems C.7 and C.12 hold with the noise coefficient νK(η)=K/η2−1 _K(η)=K/η^2-1 unchanged in form. This is part (b). Step 4: minimal dimension lower bound. Suppose Aτ=1K\A_τ\_τ=1^K is a pairwise anticommuting set of self-inverse self-adjoint operators on ℂdC^d. These operators satisfy the defining anticommutation relations of the complex Clifford system with K generators. The standard classification of finite-dimensional complex representations of these relations implies that every such representation has dimension at least 2k2^k for K=2kK=2k and at least 2k2^k for K=2k+1K=2k+1: for even K this is the irreducible representation of ℂℓ2k≃M2k(ℂ)C _2k M_2^k(C), and for odd K the algebra splits as ℂℓ2k+1≃M2k(ℂ)⊕M2k(ℂ)C _2k+1 M_2^k(C) M_2^k(C) while the generator relations are already realized on one irreducible summand. Hence d≥2⌈(K−1)/2⌉d≥ 2 (K-1)/2 and, for d=2nd=2^n, n≥⌈(K−1)/2⌉n≥ (K-1)/2 . Step 5: storage–noise tradeoff. Saturating K=2n+1K=2n+1 in part (c) gives logical storage cost n qubits per weight versus K=2n+1K=2n+1 separate classical sign bits, i.e., reduction factor (2n+1)/n→2(2n+1)/n→ 2 as n→∞n→∞. This comparison ignores the extra physical copies or re-preparations used for shot averaging; those costs enter through the explicit finite-shot threshold. The inflation in shot-noise variance is νK(η)=K/η2−1 _K(η)=K/η^2-1 from part (b). ∎ C.14 Correlated weight noise Theorem C.16 (Correlated weight noise). Suppose Assumption 4.1 holds and the M qubits of a single output channel i are correlated, while qubits in distinct output channels remain independent. Let Λτ,i⪰0 _τ,i 0 be the M×M× M covariance matrix of the i-th row W^τ,i,:Q W^Q_τ,i,: in context τ. Then the QRAQ risk Eq. (10) generalises to ℰQ(W,B(τ),μτ,Λτ,i)=∑τ=1Kπτ[Rτ(W,Qτ)+∑i=1NTr⁡(Λτ,iΣτ)],E_Q (W,\B^(τ), _τ,\ _τ,i\\ )= _τ=1^K _τ [R_τ(W,Q_τ)+ _i=1^NTr ( _τ,i _τ ) ], (112) with Qτ,ij=μτ,g(i,j)Bij(τ)Q_τ,ij= _τ,g(i,j)B^(τ)_ij. Moreover, Theorem C.7 holds with the finite-shot threshold condition (D2) of that theorem replaced by the per-row spectral condition ∑τ=1Kπτλmax(Λτ,i)Tr⁡(Στ)<Δi∞ _τ=1^K _τ _ ( _τ,i)Tr( _τ)< _i^∞ (113) for every row i of positive ideal gap. Proof. Step 1: row-wise risk decomposition under correlated noise. The within-row covariance of W^τ,i,:Q W^Q_τ,i,: is, by hypothesis, the M×M× M matrix Λτ,i _τ,i (no diagonality assumption on Λτ,i _τ,i), while distinct rows remain independent. Apply Lemma C.4 with Cov⁡(Uτ,i,:)=Λτ,iCov(U_τ,i,:)= _τ,i: Tr⁡[(W−W^τ)Στ(W−W^τ)⊤]=Rτ(W,Qτ)+∑i=1NTr⁡(Λτ,iΣτ).ETr [(W- W_τ) _τ(W- W_τ) ]=R_τ(W,Q_τ)+ _i=1^NTr ( _τ,i _τ ). (114) Multiplying by πτ _τ and summing over τ gives Eq. (112). Step 2: per-row variance bound. For any positive semidefinite matrices A,B∈ℝM×MA,B∈R^M× M with A⪰0A 0, B⪰0B 0, Tr⁡(AB)≤λmax(A)Tr⁡(B),Tr(AB)≤ _ (A)Tr(B), (115) since Tr⁡(AB)=Tr⁡(A1/2BA1/2)≤λmax(A)Tr⁡(B)Tr(AB)=Tr(A^1/2BA^1/2)≤ _ (A)Tr(B) via the Loewner ordering A⪯λmax(A)IA _ (A)I. Applying this with A=Λτ,iA= _τ,i, B=ΣτB= _τ, Tr⁡(Λτ,iΣτ)≤λmax(Λτ,i)Tr⁡(Στ).Tr ( _τ,i _τ )≤ _ ( _τ,i)Tr( _τ). (116) Step 3: per-row separation. Following Step 7 of the proof of Theorem C.7, but replacing the diagonal noise contribution (νK(η)/S)μτ2Tτ( _K(η)/S) _τ^2T_τ in Eq. (69) by Tr⁡(Λτ,iΣτ)Tr( _τ,i _τ), the per-row QRAQ risk satisfies ℰQ,irow≤∑τ=1KπτJτ(Bτ⋆,μτ⋆)+∑τ=1KπτTr⁡(Λτ,iΣτ)≤ℰQ,irow,∞+∑τ=1Kπτλmax(Λτ,i)Tr⁡(Στ),E^row_Q,i≤ _τ=1^K _τJ_τ(B_τ , _τ )+ _τ=1^K _τTr ( _τ,i _τ ) ^row,∞_Q,i+ _τ=1^K _τ _ ( _τ,i)Tr( _τ), (117) where the second inequality uses Eq. (116). Subtracting from ℰC,irow±E^row±_C,i and recalling Δi∞=ℰC,irow±−ℰQ,irow,∞ _i^∞=E^row±_C,i-E^row,∞_Q,i, Δi≥Δi∞−∑τ=1Kπτλmax(Λτ,i)Tr⁡(Στ), _i≥ _i^∞- _τ=1^K _τ _ ( _τ,i)Tr( _τ), (118) which is strictly positive whenever Eq. (113) holds. The remaining parts of Theorem C.7 (parts (a) on non-negativity and (D1)⇔ strict positivity at the ideal level) carry over verbatim because they only use the bias part of the risk, which is independent of Λτ,i\ _τ,i\. ∎ C.15 General Pauli noise channels Theorem C.17 (Pauli-noise calibration). Let (ρ)=∑P∈I,X,Y,ZpPPρPN_ p(ρ)= _P∈\I,X,Y,Z\p_PPρ P be a single-qubit Pauli-diagonal channel with pP≥0p_P≥ 0 and ∑PpP=1 _Pp_P=1 (so p is admissible in the sense of Eq. (23)), applied identically to the qubit before every measurement. Let the QRAC observables be Aτ∈X,Y,ZA_τ∈\X,Y,Z\ for τ=1,…,Kτ=1,…,K (K≤3K≤ 3). Then the channel acts on the observable side via its adjoint †(A)=∑PpPPAPN _ p(A)= _Pp_PPAP (which coincides with N_ p because Pauli-diagonal channels are self-adjoint with respect to the Hilbert–Schmidt inner product), and †(Aτ)=ητAτ,ητ≔∑PpPϵτ,P∈[−1,1],N _ p(A_τ)= _τA_τ, _τ _Pp_P _τ,P∈[-1,1], (119) where ϵτ,P=+1 _τ,P=+1 if P commutes with AτA_τ and −1-1 otherwise (so ϵτ,I=+1 _τ,I=+1 always). Assuming ητ∈(0,1] _τ∈(0,1] for all τ, the per-context QRAQ shot-noise statistics of Eq. (9) hold with η replaced by ητ _τ, giving the per-context noise coefficient ντasy()=1ητ2cK2−1=Kητ2−1. _τ^asy( p)= 1 _τ^2c_K^2-1= K _τ^2-1. (120) The per-row QRAQ risk Eq. (69) generalizes to the weighted finite-shot form ℰQ,irow(S;)=minB(τ),μτ∑τπτ[Jτ(B(τ),μτ)+ντasy()Sμτ2Tτ],E^row_Q,i(S; p)= _\B^(τ)\,\ _τ\ _τ _τ [J_τ(B^(τ), _τ)+ _τ^asy( p)S _τ^2T_τ ], (121) and Theorem C.7 continues to hold with Eq. (57) replaced by S>(∑τπτντasy()Tτμτ⋆2)/Δi∞S> ( _τ _τ _τ^asy( p)T_τ _τ 2 )/ _i^∞. Proof. Step 1: action of Pauli conjugation on AτA_τ. For any two Pauli operators P,P′∈I,X,Y,ZP,P ∈\I,X,Y,Z\, Pauli conjugation acts as P′PP′⁣†=(−1)[P,P′]P,[P,P′]=0if P,P′ commute,1if P,P′ anticommute,P P =(-1)^[P,P ]P, [P,P ]= cases0&if P,P commute,\\ 1&if P,P anticommute, cases (122) since P′PP′⁣†=P′PP′⁣−1P P =P P -1 for unitary P′P and Pauli conjugation flips sign exactly on anticommuting pairs. Hence P′AτP′⁣†=ϵτ,P′AτP A_τP = _τ,P A_τ for each P′∈I,X,Y,ZP ∈\I,X,Y,Z\, where ϵτ,P′=+1 _τ,P =+1 if P′P commutes with AτA_τ and −1-1 otherwise. By linearity, †(Aτ)=∑P∈I,X,Y,ZpPPAτP=(∑PpPϵτ,P)Aτ=ητAτ,N _ p(A_τ)= _P∈\I,X,Y,Z\p_PPA_τP= ( _Pp_P _τ,P )A_τ= _τA_τ, (123) which proves the ητ _τ identity in the theorem. Step 2: post-channel measurement statistics. By the trace identity Tr⁡((ρ)A)=Tr⁡(ρ†(A))Tr (N_ p(ρ)A )=Tr (ρN _ p(A) ) (which holds because the Pauli-diagonal channel is self-adjoint with respect to the Hilbert–Schmidt inner product) and Step 1, Tr⁡((ρb)Aτ)=Tr⁡(ρbητAτ)=ητTr⁡(ρbAτ)=ητcKbτ,Tr (N_ p( _b)A_τ )=Tr ( _b _τA_τ )= _τTr( _bA_τ)= _τc_Kb_τ, (124) using Lemma C.3 (noiseless mean) for the last equality. Hence the per-shot mean of measuring AτA_τ on (ρb)N_ p( _b) is ητcKbτ _τc_Kb_τ, and the per-shot variance is 1−(ητcK)21-( _τc_K)^2 since the outcome remains in ±1\± 1\. Step 3: per-context noise coefficient. Repeating Step 5 of the proof of Lemma C.3 verbatim with η replaced by ητ _τ (which depends on the context through Step 1), the calibrated scale μτ≔ητcKγτ _τ _τc_K _τ yields Var⁡(W^τ,ijQ)=μτ2S(1ητ2cK2−1)=μτ2Sντasy(),ντasy()=Kητ2−1.Var ( W^Q_τ,ij )= _τ^2S ( 1 _τ^2c_K^2-1 )= _τ^2S _τ^asy( p), _τ^asy( p)= K _τ^2-1. (125) Step 4: separation under per-context noise. The bias-variance decomposition (Lemma C.4) and per-row minimisation (Steps 1–7 of the proof of Theorem C.7) are τ-separable: each νK(η) _K(η) entering the τ-th summand is replaced by ντasy() _τ^asy( p). Step 8 of the proof of Theorem C.7 then gives the modified threshold S>(∑τπτντasy()Tτ(μτ⋆)2)/Δi∞S> ( _τ _τ _τ^asy( p)T_τ( _τ )^2 )/ _i^∞ for strict per-row advantage, as claimed. ∎ C.16 Formal finite-sample certificate Theorem C.18 (Finite-sample certificate for the ideal row-wise gap). Assume per-token bounded activations ∥xτ(s)∥2≤Bx x_τ^(s) _2≤ B_x almost surely and the uniform denominator lower bound minτ∈[K],b∈±1M⁡bΣτb⊤≥λ0>0 _τ∈[K],b∈\± 1\^Mb _τb ≥ _0>0. Set Nmin≔minτ⁡Nτ,εΣ≔MBx22log⁡(2KM2/δ)Nmin,N_ _τN_τ, _ MB_x^2 2 (2KM^2/δ)N_ , (126) and assume MεΣ≤λ0/2M _ ≤ _0/2. Let Δ∞ ^∞ and Δ^∞ ^∞ denote the population and empirical ideal gaps. With probability at least 1−δ1-δ, |Δ^∞−Δ∞|≤2K(1+2MBx2λ0)2∥W∥F2εΣ. | ^∞- ^∞ |≤ 2K (1+ 2MB_x^2 _0 )^2 W _F^2 _ . (127) Thus an empirical ideal gap exceeding the right-hand side certifies a positive population ideal gap. A finite-shot certificate subtracts the finite-shot inflation bound of Theorem C.7. Proof of Theorem C.18. Let Σ^τ≔1Nτ∑s=1Nτxτ(s)(xτ(s))⊤ _τ 1N_τ _s=1^N_τx_τ^(s) (x_τ^(s) ) (128) denote the empirical covariance computed from NτN_τ i.i.d. samples with ∥xτ(s)∥2≤Bx x_τ^(s) _2≤ B_x a.s. Step 1: entrywise concentration of Σ^τ _τ. Each scalar entry (Σ^τ−Στ)jk( _τ- _τ)_jk is a zero-mean average of NτN_τ i.i.d. random variables bounded in [−Bx2,Bx2][-B_x^2,B_x^2] (since |xjxk|≤∥x∥22≤Bx2|x_jx_k|≤ x _2^2≤ B_x^2). Hoeffding’s inequality [Hoeffding, 1963] gives, for fixed τ,j,kτ,j,k and any t>0t>0, Pr⁡(|(Σ^τ−Στ)jk|≥t)≤2exp⁡(−Nτt22Bx4). ( |( _τ- _τ)_jk |≥ t )≤ 2 (- N_τt^22B_x^4 ). (129) Setting t=Bx22log⁡(2KM2/δ)/Nτt=B_x^2 2 (2KM^2/δ)/N_τ and applying a union bound over the M2M^2 entries and K contexts gives, with probability at least 1−δ1-δ, maxτ,j,k⁡|(Σ^τ−Στ)jk|≤Bx22log⁡(2KM2/δ)Nmin. _τ,j,k |( _τ- _τ)_jk |≤ B_x^2 2 (2KM^2/δ)N_ . (130) Since ∥A∥F≤Mmaxj,k⁡|Ajk| A _F≤ M _j,k|A_jk| for A∈ℝM×MA∈R^M× M, on the same event, ∥Σ^τ−Στ∥F≤εΣ≔MBx22log⁡(2KM2/δ)Nminfor every τ. _τ- _τ _F≤ _ MB_x^2 2 (2KM^2/δ)N_ every τ. (131) We henceforth condition on this concentration event. The hypothesis MεΣ≤λ0/2M _ ≤ _0/2 of Theorem C.18, used in Step 2 below, is the minimal-sample-size condition MεΣ≤λ02.M _ ≤ _02. (132) Step 2: empirical denominator control. For every b∈±1Mb∈\± 1\^M and every τ, |b(Σ^τ−Στ)b⊤|≤∥bb⊤∥F∥Σ^τ−Στ∥F=MεΣ, |b( _τ- _τ)b |≤ b _F _τ- _τ _F=M _ , (133) since ∥bb⊤∥F=Tr⁡(bb⊤bb⊤)=M2=M b _F= Tr(b b )= M^2=M. Combined with the population lower bound bΣτb⊤≥λ0b _τb ≥ _0, bΣ^τb⊤≥bΣτb⊤−MεΣ≥λ0−MεΣ≥Eq.(132)λ02.b _τb ≥ b _τb -M _ ≥ _0-M _ Eq.~ eq:sample-size-cond≥ _02. (134) Both the population and empirical optimisers ατ⋆(b) _τ (b) from Eq. (61) are therefore well defined, and Jτ⋆(b;Σ)J_τ (b; ) is jointly continuous in Σ on the closed set Σ:bΣb⊤≥λ0/2\ :b b ≥ _0/2\ for every b. Step 3: Lipschitz continuity of the per-row functional. Fix a row w=Wi,:w=W_i,: and b∈±1Mb∈\± 1\^M. The ideal per-context quadratic Jτ⋆(b;Σ)=wΣw⊤−(bΣw⊤)2/(bΣb⊤)J_τ (b; )=w w -(b w )^2/(b b ) is continuously differentiable in Σ on the open half-space bΣb⊤>0\b b >0\. Under the bounded-activation assumption, ∥Σ∥op≤Bx2 _op≤ B_x^2 on every Σ in the convex hull of Στ,Σ^τ _τ, _τ on the concentration event, and the lower bound bΣb⊤≥λ0/2b b ≥ _0/2 from Step 2 holds throughout this convex hull. A direct gradient computation (treating w,bw,b as 1×M1× M row vectors so that w⊤w,b⊤w,b⊤b∈ℝM×Mw w,b w,b b∈R^M× M and bΣw⊤,bΣb⊤∈ℝb w ,b b ∈R) gives ∂Jτ⋆(b;Σ)∂Σ=w⊤w−2(bΣw⊤)(b⊤w)bΣb⊤+(bΣw⊤)2(b⊤b)(bΣb⊤)2, ∂ J_τ (b; )∂ =w w- 2(b w )(b w)b b + (b w )^2(b b)(b b )^2, (135) whose Frobenius norm is bounded above by ∥w⊤w∥F+2|bΣw⊤|∥b⊤w∥FbΣb⊤+(bΣw⊤)2∥b⊤b∥F(bΣb⊤)2. w w _F+ 2|b w | b w _Fb b + (b w )^2 b b _F(b b )^2. (136) Using |bΣw⊤|≤M∥Σ∥op∥w∥2≤MBx2∥w∥2|b w |≤ M _op w _2≤ MB_x^2 w _2, ∥b⊤w∥F=∥b∥2∥w∥2=M∥w∥2 b w _F= b _2 w _2= M w _2, ∥b⊤b∥F=M b b _F=M, ∥w⊤w∥F=∥w∥22 w w _F= w _2^2, and bΣb⊤≥λ0/2b b ≥ _0/2, this is at most ∥w∥22+4MBx2∥w∥22λ0+4M2Bx4∥w∥22λ02≤(1+2MBx2λ0)2∥w∥22≕L0(w). w _2^2+ 4MB_x^2 w _2^2 _0+ 4M^2B_x^4 w _2^2 _0^2≤ (1+ 2MB_x^2 _0 )^2 w _2^2 L_0(w). (137) The mean-value theorem along the segment Σt=Στ+t(Σ^τ−Στ) _t= _τ+t( _τ- _τ), t∈[0,1]t∈[0,1], then yields |Jτ⋆(b;Σ^τ)−Jτ⋆(b;Στ)|≤L0(w)∥Σ^τ−Στ∥F. |J_τ (b; _τ)-J_τ (b; _τ) |≤ L_0(w) _τ- _τ _F. (138) Min and max of equi-Lipschitz finite families remain Lipschitz with the same constant, so both ℰC,irow±E^row±_C,i and ℰQ,irow,∞E^row,∞_Q,i, viewed as functions of (Σ1,…,ΣK)( _1,…, _K), are L0(wi)L_0(w_i)-Lipschitz in ∑τ∥Σ^τ−Στ∥F _τ _τ- _τ _F. Step 4: row sum and certification. The per-row gap Δi∞=ℰC,irow±−ℰQ,irow,∞ _i^∞=E^row±_C,i-E^row,∞_Q,i is a difference of two L0(wi)L_0(w_i)-Lipschitz quantities, so |Δ^i∞−Δi∞|≤2L0(wi)∑τ=1K∥Σ^τ−Στ∥F. | _i^∞- _i^∞ |≤ 2L_0(w_i) _τ=1^K _τ- _τ _F. (139) Summing over i and using ∑i∥wi∥22=∥W∥F2 _i w_i _2^2= W _F^2, |Δ^∞−Δ∞|≤2(1+2MBx2λ0)2∥W∥F2∑τ=1K∥Σ^τ−Στ∥F≤2K(1+2MBx2λ0)2∥W∥F2εΣ, | ^∞- ^∞ |≤ 2 (1+ 2MB_x^2 _0 )^2 W _F^2 _τ=1^K _τ- _τ _F≤ 2K (1+ 2MB_x^2 _0 )^2 W _F^2 _ , (140) on the concentration event, which has probability at least 1−δ1-δ. This establishes Eq. (127). The certification statement then follows from Δ∞≥Δ^∞−|Δ^∞−Δ∞|>0 ^∞≥ ^∞-| ^∞- ^∞|>0 whenever Δ^∞>2K(1+2MBx2/λ0)2∥W∥F2εΣ ^∞>2K(1+2MB_x^2/ _0)^2 W _F^2 _ . ∎ C.17 Proof of Theorem C.19 Theorem C.19 (Quantum-activation fixed-POVM floor). Let K≥2K≥ 2 and let A1,…,AKA_1,…,A_K be traceless Hilbert–Schmidt orthogonal self-adjoint involutions on ℂdC^d: Aτ=Aτ†,Aτ2=I,Tr⁡(Aτ)=0,Tr⁡(AτAτ′)=dτ=τ′.A_τ=A_τ , A_τ^2=I, (A_τ)=0, (A_τA_τ )=d1\τ=τ \. (141) Assume additionally that ρb=1d(I+1K∑τ=1KbτAτ) _b= 1d (I+ 1 K _τ=1^Kb_τA_τ ) (142) is positive semidefinite for every b∈±1Kb∈\± 1\^K; equivalently, ‖∑τbτAτ‖op≤K \| _τb_τA_τ \|_op≤ K for every b. This validity condition is automatic when the AτA_τ pairwise anticommute. Let the context τ and label b∈±1Kb∈\± 1\^K be uniform. Any single-copy fixed-POVM receiver that first applies one POVM Eyy∈\E_y\_y∈Y independent of τ and then uses a context-dependent deterministic decoder fτ:→ℝf_τ:Y→R satisfies 1K∑τ=1Kby∣b[(fτ(y)−bτ)2]≥1−1K. 1K _τ=1^KE_bE_y b [(f_τ(y)-b_τ)^2 ]≥ 1- 1K. (143) Consequently, a scalar weight w has context-averaged weight-output MSE at least (1−1/K)w2(1-1/K)w^2, and independent scalar slots with squared weights summing to ∥W∥F2 W _F^2 have context-averaged total MSE at least (1−1/K)∥W∥F2(1-1/K) W _F^2. A context-matched quantum receiver using S independent fresh copies of ρb _b and measuring AτA_τ attains MSE (K−1)w2/S(K-1)w^2/S in the scalar case and (K−1)∥W∥F2/S(K-1) W _F^2/S in the matrix case. Proof of Theorem C.19. Setup. The observables Aτ=1K\A_τ\_τ=1^K live on a finite-dimensional Hilbert space. In the pairwise-anticommuting case, the Jordan–Wigner family of Appendix A.4 realizes any K≤2n+1K≤ 2n+1 on n qubits and automatically satisfies both HS orthogonality and the validity condition ρb⪰0 _b 0. In the more general HS-orthogonal case, positivity of every ρb _b is an explicit hypothesis of the theorem. The label bτ∈±1b_τ∈\± 1\ plays the role of the classical bit of context τ carried by the activation; the weight w∈ℝw∈R is classical. Step 1: setting up the average risk. A single-copy classical receiver applies the fixed POVM Eyy∈\E_y\_y∈Y on ℂdC^d and outputs b^τ=fτ(y)∈ℝ b_τ=f_τ(y)∈R from a context-dependent deterministic decoder fτf_τ. The conditional MSE at fixed (τ,b)(τ,b) is y∣b[(fτ(y)−bτ)2]=y∣b[fτ(y)2]−2y∣b[fτ(y)bτ]+1,E_y b [(f_τ(y)-b_τ)^2 ]=E_y b [f_τ(y)^2 ]-2E_y b [f_τ(y)b_τ ]+1, (144) using bτ2=1b_τ^2=1. Averaging over uniform τ∼U[K]τ [K] and uniform b∼U(±1K)b (\± 1\^K), ℳ¯≔1K∑τ=1Kby∣b[(fτ(y)−bτ)2]=¯−¯+1, M 1K _τ=1^KE_bE_y b [(f_τ(y)-b_τ)^2 ]= T- C+1, (145) where the second-moment and cross-correlation averages are ¯≔1K∑τ=1Kby∣b[fτ(y)2],¯≔2K∑τ=1Kby∣b[fτ(y)bτ]. T 1K _τ=1^KE_bE_y b [f_τ(y)^2 ], C 2K _τ=1^KE_bE_y b [f_τ(y)b_τ ]. (146) Step 2: state-averaged operator identities. Define the Hermitian observables Gτ≔∑y∈fτ(y)Ey,Hτ≔∑y∈fτ(y)2Ey.G_τ _y∈Yf_τ(y)E_y, H_τ _y∈Yf_τ(y)^2E_y. (147) By the Born rule, y∣b[ϕ(y)]=Tr⁡(ρb∑yϕ(y)Ey)E_y b[φ(y)]=Tr ( _b _yφ(y)E_y ) for every real ϕφ. Using ρb=(I+cK∑τ′bτ′Aτ′)/d _b=(I+c_K _τ b_τ A_τ )/d with cK=1/Kc_K=1/ K and the elementary identities b[bτ′]=0E_b[b_τ ]=0 and b[bτbτ′]=δττ′E_b[b_τb_τ ]= _τ , b[ρb] E_b [ _b ] =Id, = Id, (148) by∣b[fτ(y)2] E_bE_y b [f_τ(y)^2 ] =Tr⁡(b[ρb]Hτ)=1dTr⁡(Hτ), =Tr (E_b[ _b]H_τ )= 1dTr(H_τ), (149) by∣b[fτ(y)bτ] E_bE_y b [f_τ(y)b_τ ] =b[bτTr⁡(ρbGτ)]=cKd∑τ′b[bτbτ′]Tr⁡(Aτ′Gτ)=cKdTr⁡(AτGτ), =E_b [b_τTr( _bG_τ) ]= c_Kd _τ E_b[b_τb_τ ]Tr(A_τ G_τ)= c_KdTr(A_τG_τ), (150) where in Eq. (150) we used Tr⁡(Aτ)=0Tr(A_τ)=0 to cancel the contribution of the identity term in ρb _b. Substituting Eq. (149) and Eq. (150) into Eq. (145), ¯=1Kd∑τ=1KTr⁡(Hτ),¯=2cKKd∑τ=1KTr⁡(AτGτ). T= 1Kd _τ=1^KTr(H_τ), C= 2c_KKd _τ=1^KTr(A_τG_τ). (151) Step 3: Pauli–Parseval expansion. Choose a Hilbert–Schmidt orthonormal Hermitian basis Bss=0d2−1\B_s\_s=0^d^2-1 of ℂd×dC^d× d with B0=I/dB_0=I/ d and Tr⁡(BsBs′)=δss′Tr(B_sB_s )= _s , including the orthonormal traceless self-adjoint operators Aτ/dτ=1K\A_τ/ d\_τ=1^K as basis elements (possible by Eq. (141)). Re-indexing so that B[τ]=Aτ/dB_[τ]=A_τ/ d, expand Gτ=∑s=0d2−1gs(τ)BsG_τ= _s=0^d^2-1g_s^(τ)B_s with gs(τ)∈ℝg_s^(τ)∈R. Then Tr⁡(AτGτ)=dg[τ](τ),Tr⁡(Gτ2)=∑s=0d2−1(gs(τ))2,Tr(A_τG_τ)= dg_[τ]^(τ), (G_τ^2)= _s=0^d^2-1 (g_s^(τ) )^2, (152) by orthonormality of Bs\B_s\. Hence ∑τ=1K(g[τ](τ))2≤∑τ=1K∑s=0d2−1(gs(τ))2=∑τ=1KTr⁡(Gτ2). _τ=1^K (g_[τ]^(τ) )^2≤ _τ=1^K _s=0^d^2-1 (g_s^(τ) )^2= _τ=1^KTr(G_τ^2). (153) Step 4: operator Cauchy–Schwarz Gτ2⪯HτG_τ^2 H_τ. By Naimark’s theorem, lift the POVM Ey\E_y\ to a projective measurement Πy\ _y\ on a larger Hilbert space, satisfying Ey=V†ΠyVE_y=V _yV for an isometry V. Then (∑yfτ(y)Πy)2=∑yfτ(y)2Πy ( _yf_τ(y) _y )^2= _yf_τ(y)^2 _y since the Πy _y are mutually orthogonal projectors. Compressing back to the original space and using the operator Jensen inequality (or equivalently V†AVV†AV⪯V†A2V AVV AV V A^2V for self-adjoint A when VV†⪯IVV I), one obtains Gτ2=(∑yfτ(y)Ey)2⪯∑yfτ(y)2Ey=Hτ.G_τ^2= ( _yf_τ(y)E_y )^2 _yf_τ(y)^2E_y=H_τ. (154) Taking trace, Tr⁡(Gτ2)≤Tr⁡(Hτ).Tr(G_τ^2) (H_τ). (155) Step 5: bounding ¯ C in terms of ¯ T. Combining Eq. (151), the basis identity Eq. (152), scalar Cauchy–Schwarz on g[τ](τ)τ=1K\g_[τ]^(τ)\_τ=1^K, Eq. (153), and Eq. (155), |¯| | C| =|2cKKd∑τ=1KTr⁡(AτGτ)|=2cKdKd|∑τ=1Kg[τ](τ)| = | 2c_KKd _τ=1^KTr(A_τG_τ) |= 2c_K dKd | _τ=1^Kg_[τ]^(τ) | ≤2cKdKdK(∑τ=1K(g[τ](τ))2)1/2≤2cKdKdK(∑τ=1KTr⁡(Gτ2))1/2 ≤ 2c_K dKd K ( _τ=1^K (g_[τ]^(τ) )^2 )^1/2≤ 2c_K dKd K ( _τ=1^KTr(G_τ^2) )^1/2 ≤2cKdKdK(∑τ=1KTr⁡(Hτ))1/2=2cKdKdKKd¯ ≤ 2c_K dKd K ( _τ=1^KTr(H_τ) )^1/2= 2c_K dKd K Kd T =2K¯, = 2 K T, (156) where we used cK=1/Kc_K=1/ K and ∑τTr⁡(Hτ)=Kd¯ _τTr(H_τ)=Kd T from Eq. (151). Step 6: AM–GM and the floor. By the elementary inequality 2ab≤a2+b22ab≤ a^2+b^2 (i.e., (a−b)2≥0(a-b)^2≥ 0) with a=¯a= T and b=1/Kb=1/ K, |¯|≤2K¯≤¯+1K.| C|≤ 2 K T≤ T+ 1K. (157) Substituting into Eq. (145), ℳ¯=¯−¯+1≥¯−(¯+1K)+1=1−1K, M= T- C+1≥ T- ( T+ 1K )+1=1- 1K, (158) which is Eq. (143). Multiplying by w2w^2 gives the scalar weight floor (1−1/K)w2(1-1/K)w^2; the matrix case follows by direct sum over ∥W∥F2 W _F^2 independent slots. The bound is intentionally stated as a conservative single-copy fixed-readout floor; it is sufficient for our use here and is not a lower bound against arbitrary collective measurements on multiple independent copies. Step 7: quantum-matched upper bound. Given S independent identically prepared copies of ρb _b (Assumption 4.1), the quantum receiver measures the matched observable AτA_τ on each copy and averages. By Eq. (40) (with cK=1/Kc_K=1/ K), each per-shot outcome has mean cKbτc_Kb_τ and variance 1−cK2=1−1/K1-c_K^2=1-1/K. Independence of the shots yields the empirical mean m¯ m with [m¯]=cKbτE[ m]=c_Kb_τ and Var⁡(m¯)=(1−1/K)/SVar( m)=(1-1/K)/S. The unbiased affine decoder b^τ≔m¯/cK=Km¯ b_τ m/c_K= K m satisfies [b^τ]=bτE[ b_τ]=b_τ and Var⁡(b^τ)=(K−1)/SVar( b_τ)=(K-1)/S, so the per-context weight-output MSE is w2(K−1)/S=O(1/S)w^2(K-1)/S=O(1/S). ∎ Appendix D Pseudocode This appendix collects the two algorithmic primitives of QRAQ: calibration, which solves the per-context sign matrices and scales from calibration data; and inference, which prepares the qubit states and measures them. Algorithm 1 QRAQ calibration under signed per-row scales 1:Full-precision weight W∈ℝN×MW∈R^N× M; per-context calibration samples Xτ(s)\X_τ^(s)\; context prior π; noise coefficient νK(η) _K(η); shot count S. 2:Per-context sign matrices B(τ)\B^(τ)\ and signed row scales μτ,i\ _τ,i\. 3:Form empirical covariances Σ^τ=1Nτ∑sXτ(s)(Xτ(s))⊤ _τ= 1N_τ _sX_τ^(s)(X_τ^(s)) . 4:for i=1,…,Ni=1,…,N do 5: for τ=1,…,Kτ=1,…,K do 6: Solve the row-wise binary quadratic subproblem Bi,:(τ)←argminb∈±1M⁡minμ∈ℝ⁡πτ[(Wi,:−μb)Σ^τ(Wi,:−μb)⊤+νK(η)Sμ2∑j(Σ^τ)jj].B^(τ)_i,:← *arg\,min_b∈\± 1\^M _μ∈R _τ [(W_i,:-μ b) _τ(W_i,:-μ b) + _K(η)Sμ^2 _j( _τ)_j ]. (159) 7: Set μτ,i←Bi,:(τ)Σ^τWi,:⊤Bi,:(τ)Σ^τ(Bi,:(τ))⊤+(νK(η)/S)∑j(Σ^τ)jj _τ,i← B^(τ)_i,: _τW_i,: B^(τ)_i,: _τ(B^(τ)_i,:) +( _K(η)/S) _j( _τ)_j. 8: end for 9:end for 10:Return B(τ)τ=1K\B^(τ)\_τ=1^K and μτ,iτ,i\ _τ,i\_τ,i. For tensor, column, group, or row-times-column scale classes, Algorithm 1 should be replaced by the corresponding grouped calibration problem. The row-wise loop above is mathematically exact only for signed per-row scales, which is the main regime of Theorem C.7. Algorithm 2 QRAQ inference under context τ 1:Pre-calibrated B(τ)\B^(τ)\ and μτ\ _τ\; context label τ∈[K]τ∈[K]; input activation X∈ℝM×TX∈R^M× T; one logical QRAC register per weight entry (one qubit for K≤3K≤ 3, n=⌈(K−1)/2⌉n= (K-1)/2 qubits in general) prepared as Eq. (6) or Eq. (8); shot count S; depolarizing parameter η∈(0,1]η∈(0,1]; matched observable AτA_τ. 2:Quantized output Y^τ∈ℝN×T Y_τ∈R^N× T. 3:for (i,j)∈[N]×[M](i,j)∈[N]×[M] do 4: Measure logical register (i,j)(i,j) in observable AτA_τ for S independent shots; average the outcomes to m¯ij(τ)∈[−1,1] m^(τ)_ij∈[-1,1]. 5: Set W^τ,ijQ←(μτ,i/(ηcK))m¯ij(τ) W^Q_τ,ij← ( _τ,i/(η c_K) ) m^(τ)_ij. 6:end for 7:Return Y^τ=W^τQX Y_τ= W^Q_τX. Algorithm 1 is a finite-dimensional combinatorial problem of the same shape as classical one-bit calibration plus a scalar noise regulariser; it can be solved either exactly for small M, via its closed form in Corollary C.9, or via standard integer-programming heuristics for large M. Algorithm 2 is essentially the classical one-bit inference pipeline with a Pauli measurement in place of a bit read. Appendix E Experiment details This appendix gives the simulator details behind Section 6. The experiments are designed to isolate one theoretical prediction at a time: sign-disagreement margins, finite-shot variance, multi-context QRAC scaling, and noise robustness. The simulator used for the figures is stand-alone Python and includes theorem-to-code consistency tests for the closed-form gap, row additivity, finite-shot risk, and Pauli-noise coefficients. Setup. Weights are i.i.d. standard Gaussians, W∈ℝN×MW∈R^N× M, Wij∼(0,1)W_ij N(0,1). The K per-context activation covariances follow the shared-factor Wishart model Gτ=ρF0+1−ρ2Fτ,F0,F1,…,FK∼iid(0,IM×M),Στ=1MGτGτ⊤+ϵIM,G_τ=ρ F_0+ 1-ρ^2F_τ, F_0,F_1,…,F_K iid N(0,I_M× M), _τ= 1MG_τG_τ +ε I_M, (160) with eigenvalue floor ϵ=10−3ε=10^-3 and uniform prior πτ=1/K _τ=1/K. At |ρ|=1|ρ|=1, all Στ _τ collapse to the same matrix, so the sign-disagreement condition fails by construction. The deterministic covariance family Σ±=I±r(J−I) _±=I± r(J-I) is used only for the closed-form unit test of Corollary C.9. Measurement and baselines. The encoder Eq. (8) is implemented for arbitrary K using the Jordan–Wigner observables of Appendix A.4, with n=⌈(K−1)/2⌉n= (K-1)/2 qubits per weight. Depolarizing noise is evaluated through the analytic variance coefficient in Eq. (9); Monte-Carlo runs draw S independent Pauli outcomes from the exact binary probabilities. The classical baseline is the optimal signed per-row one-bit quantizer Eq. (56), solved by exhaustive enumeration over ±1M\± 1\^M for M≤12M≤ 12. Metrics. The primary metric is the relative population gap (ℰCrow±−ℰQrow)/ℰCrow±(E^row±_C-E^row_Q)/E^row±_C. Population-gap experiments evaluate the covariance-level objective at S=∞S=∞ and η=1η=1. Finite-shot experiments report both analytic risks from Eq. (69) and Monte-Carlo averages over Pauli outcomes. The transformer-head diagnostic reports the entrywise classical-sign agreement rate, which tracks but does not replace the row-wise (D1) certificate. Figure 5: Additional scaling sweeps. Panels show the (K,ρ)(K,ρ) landscape, the (N,M)(N,M) landscape, and transformer-head sign agreement versus relative gap. Results. Figure 5 extends the main experiments. The (K,ρ)(K,ρ) sweep shows that the relative population gap vanishes at |ρ|=1|ρ|=1 and grows with K when contexts remain distinct; the K=7K=7 row reaches about 42%42\% at ρ=0ρ=0. The (N,M)(N,M) sweep is essentially flat along N and increases with M, matching row additivity and the larger space of possible sign disagreements. The transformer-head diagnostic shows the gap falling as sign agreement rises, with Spearman rank correlation rs=−0.99r_s=-0.99 in the reported sweep. The closed-form sanity test instantiates five hand-computed triples (w1,w2,r)(w_1,w_2,r) and matches Eq. (73) to tolerance 10−1210^-12. E.1 Additional robustness experiments The following audits change one assumption at a time while keeping the signed per-row baseline and reconstruction metric fixed. Universality across weight distributions. Theorem C.7 treats W as deterministic, so Gaussian weights are not required. We replay the canonical K=4K=4, ρ=0ρ=0, (N,M)=(24,6)(N,M)=(24,6) configuration with unit-variance Gaussian, Laplace, uniform, and rescaled Student-t3t_3 weights. The relative population gap remains positive for all draws, with mean gaps 38.8%38.8\%, 44.6%44.6\%, 32.5%32.5\%, and 43.8%43.8\% over 1010 seeds. Finite-shot gaps at S=512S=512 track these population values within sampling error. The interpretation is simple: heavier-tailed rows tend to amplify the context-specific alignment term w⊤Στbw _τb, while the strict separation itself does not rely on a weight-distribution assumption. Per-context channel anisotropy. Theorem C.17 replaces the common depolarizing coefficient with per-context coefficients ντ=K/ητ2−1 _τ=K/ _τ^2-1. To emphasize this calibration, we fix K=4K=4, S=512S=512, and the mean fidelity η¯=0.65 η=0.65, then sweep a linear spread such that the largest setting is (ητ)=(0.35,0.55,0.75,0.95)( _τ)=(0.35,0.55,0.75,0.95). The asymmetric protocol agrees with the symmetric reference at zero spread and remains positive over the full sweep, with at most a four-percentage-point loss. This matches the convex dependence of ντ _τ on ητ _τ: spreading fidelities at fixed arithmetic mean increase average shot noise, but the tested margins remain above zero. Figure 6: Large-K scaling on the shared-factor Wishart model. The relative population gap grows from K=2K=2 to K=15K=15 and then begins to saturate in the tested regime. Large-K scaling. Theorem C.15 guarantees representability for K≤2n+1K≤ 2n+1, but it does not imply that the gap must increase with K on every instance. In the shared-factor Wishart model, Figure 6 shows monotone growth and saturation: for ρ=0ρ=0 and (N,M)=(24,4)(N,M)=(24,4), the mean relative gap rises from 24.0%±7.1%24.0\%± 7.1\% at K=2K=2 to 43.6%±3.0%43.6\%± 3.0\% at K=15K=15. The standard deviation shrinks as K grows, consistent with averaging over more independent context covariances. At K=15K=15, the logical QRAC register uses 77 qubits per weight rather than 1515 separate context-specific sign bits, excluding the physical preparations required for finite-shot readout.

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