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Quantum-Inspired Geometric Classification with Correlation Group Structures and VQC Decision Modeling
Nishikanta Mohanty, Arya Ansuman Priyadarshi, Bikash K. Behera, Badshah Mukherjee
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Summary
The paper introduces a geometry-driven, quantum-inspired classification framework that utilizes Correlation Group Structures (CGR) and compact SWAP-test-based overlap estimation. By mapping features into anchor-centered correlation neighborhoods, the method generates nonlinear representations evaluated against class medoids. It employs a hybrid approach: a lightweight, non-probabilistic 'Fusion Score' classifier for small-to-moderate datasets, and a Variational Quantum Classifier (VQC) refinement layer for large-scale, imbalanced regimes (e.g., Credit Card Fraud). The framework demonstrates competitive performance and interpretability across multiple benchmarks.
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Relation Signals (3)
SWAP-test → computes → similarity signals
confidence 95% · use compact SWAP-test-based overlap estimation as the primary mechanism to compute similarity signals
VQC → refines → Fusion Score
confidence 92% · train a variational quantum classifier (VQC) as a nonlinear refinement layer
CGR → enhances → Robustness
confidence 90% · CGR organizes features into anchor-centered correlation neighbourhoods, generating nonlinear, correlation-weighted representations that enhance robustness
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Abstract
Abstract:We propose a geometry-driven quantum-inspired classification framework that integrates Correlation Group Structures (CGR), compact SWAP-test-based overlap estimation, and selective variational quantum decision modelling. Rather than directly approximating class posteriors, the method adopts a geometry-first paradigm in which samples are evaluated relative to class medoids using overlap-derived Euclidean-like and angular similarity channels. CGR organizes features into anchor-centered correlation neighbourhoods, generating nonlinear, correlation-weighted representations that enhance robustness in heterogeneous tabular spaces. These geometric signals are fused through a non-probabilistic margin-based fusion score, serving as a lightweight and data-efficient primary classifier for small-to-moderate datasets. On Heart Disease, Breast Cancer, and Wine Quality datasets, the fusion-score classifier achieves 0.8478, 0.8881, and 0.9556 test accuracy respectively, with macro-F1 scores of 0.8463, 0.8703, and 0.9522, demonstrating competitive and stable performance relative to classical baselines. For large-scale and highly imbalanced regimes, we construct compact Delta-distance contrastive features and train a variational quantum classifier (VQC) as a nonlinear refinement layer. On the Credit Card Fraud dataset (0.17% prevalence), the Delta + VQC pipeline achieves approximately 0.85 minority recall at an alert rate of approximately 1.31%, with ROC-AUC 0.9249 and PR-AUC 0.3251 under full-dataset evaluation. These results highlight the importance of operating-point-aware assessment in rare-event detection and demonstrate that the proposed hybrid geometric-variational framework provides interpretable, scalable, and regime-adaptive classification across heterogeneous data settings.
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- Source: https://arxiv.org/abs/2604.01930v1
- Canonical: https://arxiv.org/abs/2604.01930v1
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[1] Ansuman [1] 1] for Quantum Software and Information, of Technology Sydney, 15 Broadway, Ultimo, , 2007, , 2] ’s Quantum (OPC) Pvt. Ltd., , , 741246, , Quantum-Inspired Geometric Classification with Correlation Group Structures and VQC Decision Modeling aryapriyadarshi@icloud.com nishikanta2000@gmx.com K. bikas.riki@gmail.com badshah.mukherjee@outlook.com [ [ Abstract We propose a geometry-driven quantum-inspired classification framework that integrates Correlation Group Structures (CGR), compact SWAP-test-based overlap estimation, and selective variational quantum decision modeling. Rather than directly approximating class posteriors, the method adopts a geometry-first paradigm in which samples are evaluated relative to class medoids using overlap-derived Euclidean-like and angular similarity channels. CGR organizes features into anchor-centered correlation neighborhoods, generating nonlinear, correlation-weighted representations that enhance robustness in heterogeneous tabular spaces. These geometric signals are fused through a non-probabilistic margin-based fusion score, serving as a lightweight and data-efficient primary classifier for small-to-moderate datasets. On Heart Disease, Breast Cancer, and Wine Quality datasets, the fusion-score classifier achieves 0.8478, 0.8881, and 0.9556 test accuracy respectively, with macro-F1 scores of 0.8463, 0.8703, and 0.9522, demonstrating competitive and stable performance relative to classical baselines. For large-scale and highly imbalanced regimes, we construct compact Δ -distance contrastive features and train a variational quantum classifier (VQC) as a nonlinear refinement layer. On the Credit Card Fraud dataset (0.17% prevalence), the Δ +VQC pipeline achieves ≈ 0.85 minority recall at an alert rate of ≈ 1.31%, with ROC-AUC 0.9249 and PR-AUC 0.3251 under full-dataset evaluation. These results highlight the importance of operating-point-aware assessment in rare-event detection and demonstrate that the proposed hybrid geometric–variational framework provides interpretable, scalable, and regime-adaptive classification across heterogeneous data settings. keywords: Compact Swap Test, Correlation Group Structures, Medoids, Coordinate Descent Optimisation, VQC Decision Modeling, Quantum Classification 1 Introduction Modern machine learning continues to face persistent challenges when operating within high-dimensional feature spaces and heterogeneous data regimes. In domains such as fraud detection, medical diagnostics, and risk screening, classification performance is frequently constrained by limited labeling, extreme class imbalance, and non-stationary (concept-drifting) distributions [1, 2]. Traditional supervised models often struggle when the decision boundary is governed by a complex mixture of global structure and localized sub-population patterns. These issues are exacerbated in high dimensions, where distance concentration and the broader “curse of dimensionality” can erode the discriminative contrast of standard similarity measures and nearest-neighbor style reasoning [3, 4]. A robust alternative is prototype-driven similarity representation, where samples are assessed based on their relationship to class representatives (e.g., medoids). Medoid-based clustering and classification are well established due to their interpretability and robustness properties, particularly compared to mean-based prototypes in the presence of outliers [5]. However, the efficacy of prototype-based approaches depends critically on the discriminative power of the similarity metrics used. In high-dimensional settings, standard metrics may become less informative and can conflate magnitude and directional components, motivating hybrid geometric representations that explicitly separate distance-like and angle-like structure [3]. To address this, we leverage quantum-inspired primitives, specifically SWAP-test-style overlap estimation, to compute similarity signals via interference-based routines [6]. Under amplitude-style encodings, overlap estimation provides a principled mechanism to access inner-product structure within a high-dimensional feature space using a number of qubits that scales logarithmically with the feature dimension (in the state representation), enabling the construction of geometry-aware similarity features [7, 8]. By mapping overlaps into distinct angular and distance-like components, we obtain a hybrid representation that captures complementary geometric information often missed by purely classical measures. Similarity signals alone, however, can be unstable when latent sub-population structure is ignored, particularly in imbalanced regimes where majority-class geometry can dominate the similarity landscape [1, 9]. We therefore propose a Correlation Group Structure mechanism a Cluster/Group-Responsive (CGR) construction to partition data into structured configurations that make similarity aggregation responsive to underlying clusters. Over this CGR configuration space, we apply coordinate-descent-driven model selection to identify the best-performing group-combination and calibration parameters for a target metric. Once the optimal CGR structure is fixed, we compute CGR-conditioned distance–angle deltas and train a lightweight variational quantum classifier (VQC) as the final decision model, aligning with established variational/quantum-feature-space learning paradigms [7, 8]. 2 Quantum Distance Classification Geometric classification versus probabilistic classification. Most supervised classification pipelines are presented in a probabilistic form: given an input x, a model produces class scores that are interpreted as an estimate of a posterior distribution p(y∣x)p(y x) (e.g., logistic regression, neural networks, and many calibrated ensemble methods). Decision boundaries in such models are typically learned by minimizing a likelihood-based objective (or a surrogate such as cross-entropy), and the learned parameters may not retain a direct geometric interpretation in the original feature space. By contrast, distance-based or geometric classification assigns labels based on the position of x relative to representative structures in the data (e.g., prototypes, medoids, centroids, or sub-manifolds). In its simplest form, a geometric classifier selects the label of the nearest prototype under a chosen metric, or uses a margin defined by the difference between distances to competing class representatives. This paradigm is often preferred when interpretability is required, when decision logic naturally depends on similarity (e.g., “is this case closer to known fraud patterns or normal patterns?”), or when data are heterogeneous and local geometry is more stable than a global parametric surface. Geometric Foundations of Quantum Classification A distinguishing advantage of quantum information processing is its role as a native toolbox for computing geometric relationships between data representations. Unlike classical architectures that may require significant computational overhead to determine high-dimensional similarities, quantum states exist within a Hilbert space where inner products, overlaps, and fidelities are fundamental, physically measurable quantities. Consequently, quantum routines are often inherently designed to estimate similarity measures—either directly through interference mechanisms, such as SWAP-test-style circuits, or indirectly through the expectation values of observables. This physical architecture aligns seamlessly with geometric classification paradigms, where decision boundaries are defined by distances, angles, margins, or kernel similarities. Within this framework, overlap estimation serves as a primitive that maps to distance-like and angular measures for normalized encodings. These can be further composed into prototype-based decision rules, effectively linking quantum measurements to interpretable geometric signals [6, 7, 8]. SWAP test and overlap estimation. A canonical quantum primitive for comparing two quantum states is the SWAP test, which estimates the squared overlap s(x,x′)=|⟨ψ(x)|ψ(x′)⟩|2,s(x,x )\;=\; | ψ(x)\,|\,ψ(x ) |^2, (1) via a simple interference circuit and a measurement on an ancilla qubit [6]. Overlap estimation underpins a range of quantum machine learning constructions, including quantum kernels and feature-space methods, where the overlap (or a function of it) plays the role of a similarity function in a Hilbert space [7, 8]. In distance-based quantum classification, overlap estimates can be mapped into geometric quantities such as angular separation or Euclidean-like dissimilarities for normalized vectors. For example, one may define an angle Θ(x,x′)=arccos(s(x,x′)), (x,x )\;=\; \! ( s(x,x ) ), (2) and a monotone distance-like mapping D(x,x′)=2(1−s(x,x′)),D(x,x )\;=\; 2 (1- s(x,x ) ), (3) Recent Approaches to Quantum Geometric Classification Building on these primitives, several approaches have formalized quantum formulations of distance- and similarity-based learning. Early investigations into quantum nearest-neighbor acceleration demonstrated that, provided efficient state preparation is feasible, the ability to compute inner products can significantly enhance similarity search and related classification tasks [10]. Subsequent proposals introduced explicit distance-based classifiers that utilize interference circuits to evaluate the proximity between a query state and class representatives, facilitating label assignment via nearest-prototype rules [11]. More broadly, the field has seen the rise of quantum kernel methods, which define a similarity function k(x,x′)=|⟨ϕ(x)|ϕ(x′)⟩|2k(x,x )=| φ(x)|φ(x ) |^2 within an implicit feature space. By employing classical learning algorithms on the resulting quantum-computed kernel matrix, these methods place overlap estimation at the center of the learning pipeline, bridging the gap between quantum state interference and established statistical learning theory [7, 8]. 3 Novelty and Contribution In this work, we propose a geometric classification paradigm and use compact SWAP-test-based overlap estimation as the primary mechanism to compute similarity signals between samples and representative prototypes. Unlike purely probabilistic classifiers that directly learn p(y∣x)p(y x) from raw features, our approach first constructs geometry-aware similarity features including both distance-like and angle-like components derived from overlaps and organizes them under correlation group structures (CGR) to improve robustness in heterogeneous feature regimes. Crucially, the proposed framework is hybrid and data-regime-aware. We introduce a Fusion Score as the primary, lightweight classifier that aggregates CGR group-wise geometric evidence into class scores.Fusion Score is a non-probabilistic classifier: it produces per-class margin-like dissimilarity scores from overlap-derived geometric evidence and predicts via an argmin decision rule, without explicitly modeling calibrated posteriors p(y∣x)p(y x). Empirically, Fusion Score is highly effective on small-to-moderate datasets (Heart Disease, Wine Quality, Breast Cancer), where the class structure is well captured by prototype distances and a parametric learner may struggle due to limited sample support. However, in large, highly imbalanced and behaviourally complex datasets (Fraud), Fusion Score alone can underfit because discrimination depends on non-linear interactions across groups and subtle margin patterns. To address this, we construct Δ -distance margin features from the CGR distance tensor and train a variational quantum classifier (VQC) on these Δ -features as a selective refinement (or residual decision model). In practice, the VQC is invoked when Fusion Score is not optimal, for example, under explicit imbalance/complexity regimes, or for samples exhibiting low fusion confidence (small best-vs-runner-up margin). This design retains interpretability and stability from geometric fusion in low-data settings, while enabling non-linear correction via VQC where large-scale, imbalanced structure demands a richer decision boundary. The overall process is divided into the following steps after initial data preparation and preprocessing: • Correlation Group structure formation. • Euclidean Medoid calculation as per available classes • Quantum distance calculation (Euclidean and Angular) by Compact Swaptest. • Finding the optimal configuration by Coordinate descent Optimisation • Measuring the classification performance by Fusion score (angular score and Euclidean score) • Calculating distance deltas based on the Optimal configuration • Train VQC on distance deltas as final classifier. The detailed implementation of these steps, including prototype construction, correlation group configuration, coordinate-descent model selection, and VQC training, is presented in Section 4. 4 Process Flow and algorithms This section presents the end-to-end methodology of the proposed classification process. The central objective of the framework is to construct a robust, geometry-driven decision pipeline in which quantum overlap estimation is used as a primary similarity primitive, while correlation-aware grouping and coordinate-descent selection ensure stability across heterogeneous data regimes. Unlike conventional probabilistic classifiers that learn a direct mapping from raw features to posteriors, our approach explicitly decomposes classification into (i) representation by prototype-referenced geometry and (i) decision modeling on a compact set of contrastive delta features. This design yields interpretability (via distances, angles, and margins), configurability (via group-structure selection), and a clear separation between similarity extraction and final decision learning. At a high level, the pipeline begins by forming Correlation Group Structures (CGR) that capture latent sub-population patterns and define a discrete configuration space of feasible group-combination “configs.” Within each config, class representatives are computed using Euclidean medoids, providing robust prototypes that are less sensitive to outliers than mean-based representatives. We then compute quantum-inspired geometric similarities between each sample and the medoids using a compact SWAP-test routine, which estimates state overlaps and maps them into two complementary channels: a distance-like (Euclidean) measure and an angular measure. These channels provide a richer geometric representation than either metric alone, particularly in high-dimensional settings where magnitude- and direction-based separations may differ. Given the CGR configuration space, we apply a coordinate-descent optimization procedure to identify the optimal configuration for a target performance metric. This stage simultaneously calibrates the relative contribution of Euclidean and angular similarity channels and selects the configuration that yields the best empirical performance under the chosen evaluation criterion. Classification performance is first measured through a fusion score, which combines the Euclidean and angular signals into a margin-like decision statistic and serves as a strong, lightweight geometric classifier in its own right. After selecting the optimal configuration, we compute distance deltas contrastive differences between class-referenced distances/angles under the selected CGR structure to obtain a compact and stable feature vector per sample. Finally, we train a variational quantum classifier (VQC) on these delta features as a refinement decision model, particularly suited to regimes where fusion scores alone are not optimal (e.g., large, highly imbalanced, and behaviourally complex datasets). In contrast, for small-to-moderate datasets where sample support is limited, the fusion score often provides the most reliable performance while a VQC may be data-constrained. The methodology is formalized in the remainder of this section through a sequence of algorithms and their associated mathematical definitions, presented in the order as per Section 3. 4.1 Correlation Group Structures and Nonlinear Feature Construction This subsection formally defines Correlation Group Structures (CGR) and introduces the nonlinear feature construction used throughout the proposed framework. The CGR mechanism partitions the original feature space into anchor-centered correlation neighborhoods and then induces nonlinear features by combining (i) correlation-weighted feature groups and (i) correlation group configurations. This yields a compact representation that preserves local dependency structure while enabling expressive, interaction-like behavior without requiring explicit polynomial expansion. Let X∈ℝN×MX ^N× M denote the dataset with feature index set ℱ=1,…,MF=\1,…,M\, and let R∈[−1,1]M×MR∈[-1,1]^M× M be the feature–feature correlation matrix computed on X (excluding the target). The Correlation Group Structure (CGR) mechanism constructs anchor-specific correlation neighborhoods and converts them into nonlinear, correlation-structured features. Correlation group membership. Each feature can act as an anchor a∈ℱa . For a fixed group size parameter m, we define the membership set of anchor a by ranking all other features by absolute correlation magnitude and selecting the top-m: ℳa=a∪TopM(|Ra,f|f∈ℱ∖a,m).M_a\;=\;\a\ (\|R_a,f|\_f \a\,\,m ). (4) The ranking in (4) is sign-agnostic; the sign of Ra,fR_a,f may still be retained as a weight if required. Anchor-wise nonlinear CGR feature (correlation-weighted ℓ2 _2 aggregation). Given a record x∈ℝMx ^M, we define the correlation-weighted subvector induced by anchor a as va(x)=(Ra,fxf)f∈ℳa∈ℝ|ℳa|,v_a(x)\;=\; (R_a,f\,x_f )_f _a ^|M_a|, (5) and compute the anchor-wise CGR feature as the ℓ2 _2 norm of this subvector: ϕa(x)=‖va(x)‖2=∑f∈ℳa(Ra,fxf)2. _a(x)\;=\;\|v_a(x)\|_2\;=\; _f _a (R_a,f\,x_f )^2. (6) Equation (6) is nonlinear in the original features due to the squaring and square-root operations, and it summarizes the joint magnitude of a correlation neighborhood around the anchor. If correlation weighting is not used in the aggregation, the unweighted alternative is ϕa(x)=‖xℳa‖2=∑f∈ℳaxf2. _a(x)\;=\;\|x_M_a\|_2\;=\; _f _ax_f^2. (7) CGR embedding. Stacking the anchor-wise features yields the CGR representation Φ(x)=[ϕ1(x),ϕ2(x),…,ϕM(x)]⊤∈ℝM,Φ(X)∈ℝN×M. (x)\;=\; [ _1(x), _2(x),…, _M(x) ] ^M, (X) ^N× M. (8) CGR-data multiplicative feature (correlation strength × group activation). In addition to the direct aggregation in (6)–(7), we also define a multiplicative nonlinear feature that explicitly separates correlation structure from record-level activation. First, we compute a dataset-level correlation-group strength for each anchor: ga=‖(|Ra,f|)f∈ℳa‖2=∑f∈ℳa|Ra,f|2.g_a\;=\; \| (|R_a,f| )_f _a \|_2\;=\; _f _a|R_a,f|^2. (9) Next, we compute the record-specific activation of the same variable set: ha(x)=‖xℳa‖2=∑f∈ℳaxf2.h_a(x)\;=\;\|x_M_a\|_2\;=\; _f _ax_f^2. (10) The multiplicative CGR feature is then za(x)=ga⋅ha(x),z_a(x)\;=\;g_a· h_a(x), (11) and stacking across anchors produces Z(x)=[z1(x),…,zM(x)]⊤Z(x)=[z_1(x),…,z_M(x)] and Z(X)∈ℝN×MZ(X) ^N× M. The transformation ϕa(x) _a(x) in (6) and the product feature za(x)z_a(x) in (11) both introduce nonlinearity through ℓ2 _2 aggregation and multiplicative coupling. The term gag_a is constant for a fixed dataset (derived from R), whereas ha(x)h_a(x) varies across records, so za(x)z_a(x) increases for records that strongly express a feature group that is also strongly correlated around anchor a. It presents the algorithms used to (i) construct CGR anchor memberships from a correlation matrix and (i) generate a CGR feature matrix for any chosen configuration of anchor-specific subsets. Algorithm 1 Build CGR Anchors, Membership Sets, and Correlation Weights 1:Correlation matrix R over features (and optionally target) with column set ℱ∪yF∪\y\; target name y; top-m parameter m 2:Ordered anchors A; membership map ℳaa∈\M_a\_a ; weight map ρa,f\ _a,f\ 3:ℱ←F← all columns in R excluding y 4:if y exists in R then 5: tf←|Rf,y|t_f←|R_f,y| for all f∈ℱf 6: ←A← features sorted by tft_f in descending order 7:else 8: ←ℱA ⊳ fallback ordering 9:end if 10:for each anchor a∈a do 11: cf←Ra,fc_f← R_a,f for all f∈ℱf ⊳ exclude target from membership 12: Sort features by |cf||c_f| descending 13: ℳ~← M← first m features in the sorted list 14: ℳa←[a]⊕[f∈ℳ~:f≠a]M_a←[a] [f∈ M:f≠ a] ⊳ anchor first 15: for each member f∈ℳaf _a do 16: if f=af=a then 17: ρa,f←1.0 _a,f← 1.0 18: else 19: ρa,f←Ra,f _a,f← R_a,f ⊳ if missing/NaN then set to 0 20: end if 21: end for 22:end for 23:return A, ℳa\M_a\, ρa,f\ _a,f\ Algorithm 2 Compute Anchor Feature Vector for a Given Anchor and Subset 1:Dataset X∈ℝN×MX ^N× M with feature columns; anchor a; subset κ(a)⊆ℱκ(a) ; weights ρa,f _a,f 2:Anchor feature vector ua∈ℝNu_a ^N 3:for i=1i=1 to N do 4: ua(i)←∑f∈κ(a)(ρa,fXi,f)2u_a(i)← _f∈κ(a) ( _a,f\,X_i,f )^2 5:end for 6:return uau_a Algorithm 3 Build CGR Feature Matrix for a Configuration 1:Dataset X∈ℝN×MX ^N× M; anchors =a1,…,aKA=\a_1,…,a_K\; configuration κ with subsets κ(a)κ(a); weights ρa,f _a,f 2:CGR feature matrix F∈ℝN×KF ^N× K 3:Initialize F←N×KF 0_N× K 4:for j=1j=1 to K do 5: a←aja← a_j 6: F:,j←AnchorFeatureVector(X,a,κ(a),ρ)F_:,j (X,a,κ(a),ρ) ⊳ Algorithm 2 7:end for 8:return F 4.1.1 Class-Wise Medoid Computation on CGR-Induced Nonlinear Features After constructing the CGR-induced nonlinear representation, we compute class-wise medoids to obtain robust prototype representatives for distance-based classification. Medoids are preferred over mean-based prototypes because they are selected from observed samples and are therefore less sensitive to outliers and heavy-tailed feature distributions. Let Z(X)∈ℝN×dZ(X) ^N× d denote the CGR-induced nonlinear feature matrix, where each record is mapped to a vector zi=Z(xi)∈ℝd,i=1,…,N,z_i\;=\;Z(x_i) ^d, i=1,…,N, (12) and d is the embedding dimension (typically d=Md=M when using one anchor per original feature, but may be smaller if anchors are subsampled). For each class c∈0,1,…,C−1c∈\0,1,…,C-1\, define the index set of samples in that class as ℐc=i∈1,…,N:yi=c,Nc=|ℐc|.I_c\;=\;\\,i∈\1,…,N\\;:\;y_i=c\,\, N_c=|I_c|. (13) Let c=zi:i∈ℐcZ_c=\z_i:i _c\ denote the set of nonlinear feature vectors belonging to class c. Euclidean medoid objective. The medoid of class c is defined as the sample (in nonlinear feature space) that minimizes the total within-class Euclidean dissimilarity: mc=zjc∗,jc∗=argminj∈ℐc∑i∈ℐc‖zi−zj‖2.m_c\;=\;z_j^*_c, j^*_c\;=\; _j _c\; _i _c \|z_i-z_j \|_2. (14) The vector mc∈ℝdm_c ^d serves as the prototype representative for class c in subsequent quantum similarity computations. If a class exhibits multi-modal structure, the framework can be extended to K-medoids per class by selecting a set mc,1,…,mc,K\m_c,1,…,m_c,K\ that minimizes the standard K-medoids objective. The remainder of the pipeline remains unchanged, with quantum similarity computed against each medoid and aggregated according to the selected configuration. All medoid computations are performed after transforming the original data into the CGR-induced nonlinear space. This ensures that class prototypes reflect correlation-structured group activations rather than raw feature coordinates, improving robustness in heterogeneous datasets where local dependency structure is predictive of class membership. Algorithm 4 Approximate Euclidean Medoid via Subsampling 1:Point set P=pii=1nP=\p_i\_i=1^n, pi∈ℝdp_i ^d; max subsample size mmaxm_ ; random seed s 2:Approximate medoid μ∈ℝdμ ^d 3:n←|P|n←|P| 4:if n≤mmaxn≤ m_ then 5: S←PS← P 6:else 7: Sample index set ⊂1,…,nJ⊂\1,…,n\ with ||=mmax|J|=m_ uniformly without replacement using seed s 8: S←pj:j∈S←\p_j:j \ 9:end if 10:Compute pairwise distance matrix Δ∈ℝ|S|×|S| ^|S|×|S| where Δij=‖Si−Sj‖2 _ij=\|S_i-S_j\|_2 11:Compute row-sums ui←∑j=1|S|Δiju_i← _j=1^|S| _ij for i=1,…,|S|i=1,…,|S| 12:i⋆←argminiuii ← _iu_i 13:μ←Si⋆μ← S_i 14:return μ Algorithm 5 Fit Class-Wise Medoids in CGR Feature Space 1:Training feature matrix F∈ℝN×dF ^N× d (CGR-induced nonlinear features); labels y∈0,…,C−1Ny∈\0,…,C-1\^N; mmaxm_ ; seed s 2:Medoid dictionary μcc∈\ _c\_c 3:←unique(y)C (y) 4:for each class c∈c do 5: Fc←Fi:yi=cF_c←\F_i:y_i=c\ ⊳ rows of F belonging to class c 6: μc←EuclideanMedoidSubsample(Fc,mmax,s) _c (F_c,m_ ,s) ⊳ Algorithm 4 7:end for 8:return μcc∈\ _c\_c 4.1.2 Compact SWAP Test for Quantum Distance and Angular Similarity The compact SWAP-test routine used in this work is inherited from our Quantum-SMOTE feature-similarity module, where it was introduced as a resource-efficient overlap estimator for high-dimensional vectors under amplitude-style preparation. The same primitive applies directly to geometric classification because both settings require a stable estimate of the overlap between two data-derived quantum states. We retain the compact construction in which one state (|ϕ⟩|φ ) is prepared on a single qubit and the second state (|ψ⟩|ψ ) is prepared on a multi-qubit register. Compared to symmetric two-register SWAP-test implementations, this reduces circuit width and state-preparation overhead while preserving an ancilla-measurement statistic that maps to a squared-overlap estimate [12, 13]. In this manuscript, the overlap estimate is transformed into two geometric signals: an angular distance (via cosine similarity) and a Euclidean distance reconstruction, which form the inputs to CGR configuration selection and downstream decision modeling. Given x,y∈ℝMx,y ^M, we compute (i) an angular distance capturing directional similarity and (i) a Euclidean distance capturing magnitude separation. Under amplitude-style encoding, vectors are mapped to normalized quantum states and similarity is expressed through state overlaps [14, 8]. If ‖x‖2=0\|x\|_2=0 or ‖y‖2=0\|y\|_2=0, state preparation is undefined and we use a classical fallback. Degenerate handling (classical fallback). Let nx=‖x‖2n_x=\|x\|_2 and ny=‖y‖2n_y=\|y\|_2. If nx=0n_x=0 or ny=0n_y=0, we set D(x,y)=‖x−y‖2,D(x,y)=\|x-y\|_2, (15) and Θ(x,y)=arccos(x⊤y‖x‖2‖y‖2),‖x‖2>0,‖y‖2>0,0,otherwise, (x,y)= cases \! ( x y\|x\|_2\,\|y\|_2 ),&\|x\|_2>0,\ \|y\|_2>0,\\[8.0pt] 0,&otherwise, cases (16) with clipping of the cosine argument to [−1,1][-1,1] for numerical stability. Compact state preparation. Assume nx>0n_x>0 and ny>0n_y>0 and define Z=‖x‖22+‖y‖22.Z=\|x\|_2^2+\|y\|_2^2. (17) We prepare a single-qubit state encoding relative norms, |ϕ⟩=1Z(‖x‖2|0⟩−‖y‖2|1⟩),|φ = 1 Z (\|x\|_2\,|0 -\|y\|_2\,|1 ), (18) and a multi-qubit state by interleaving normalized components, |ψ⟩=12∑k=1M(xk‖x‖2|2k−2⟩+yk‖y‖2|2k−1⟩),|ψ = 1 2 _k=1^M ( x_k\|x\|_2\,|2k-2 + y_k\|y\|_2\,|2k-1 ), (19) corresponding to an amplitude vector of length 2M2M, padded to the nearest power-of-two for amplitude initialization [14]. Overlap estimate. The SWAP test estimates overlaps via ancilla-controlled interference and measurement [12, 13]. Using ancilla a, a single qubit for |ϕ⟩|φ , and a register for |ψ⟩|ψ , we apply Ha→CSWAP(a;ϕ-qubit,ψ-qubit0)→Ha,H_a\;→\;CSWAP(a;\,φ-qubit,\,ψ-qubit_0)\;→\;H_a, (20) where the controlled-SWAP acts only between the ϕφ qubit and the first ψ qubit. Let p0=Pr[a=0]p_0= [a=0]. The squared-overlap estimate is [12] s(x,y)= 2p0−1,s(x,y)∈[0,1]after clipping.s(x,y)\;=\;2p_0-1, s(x,y)∈[0,1]\ after clipping. (21) We then define the cosine similarity as cos(x,y)=s(x,y). (x,y)= s(x,y). (22) Overlap estimation is a standard primitive in distance-based quantum classification and quantum feature-space methods [11, 10, 7, 8]. Angular and Euclidean distances. The angular distance is Θ(x,y)=arccos(cos(x,y)), (x,y)= \! ( (x,y) ), (23) and the Euclidean distance is reconstructed as D(x,y)=2Zs(x,y).D(x,y)= 2Z\,s(x,y). (24) The pair (D(x,y),Θ(x,y))(D(x,y), (x,y)) provides complementary magnitude- and direction-based geometric signals used in subsequent configuration selection, fusion scoring, and delta construction. Algorithm 6 Compact SWAP Test for Distance and Angular Similarity 1:Vectors x,y∈ℝMx,y ^M 2:Distance D(x,y)D(x,y) and angular distance Θ(x,y) (x,y) 3:nx←‖x‖2n_x←\|x\|_2, ny←‖y‖2n_y←\|y\|_2 4:if nx=0n_x=0 or ny=0n_y=0 then 5: D←‖x−y‖2D←\|x-y\|_2 6: Θ←0 ← 0 7: if nx>0n_x>0 and ny>0n_y>0 then 8: Θ←arccos(clip(x⊤ynxny,−1,1)) ← \! (clip ( x yn_xn_y,-1,1 ) ) 9: end if 10: return (D,Θ)(D, ) 11:end if 12:Z←nx2+ny2Z← n_x^2+n_y^2 13:Prepare |ϕ⟩|φ on one qubit using amplitudes [nxZ,−nyZ] [ n_x Z,\ - n_y Z ] 14:Prepare |ψ⟩|ψ on a multi-qubit register with amplitudes ψ2k−2=xknx2 _2k-2= x_kn_x 2, ψ2k−1=ykny2 _2k-1= y_kn_y 2 for k=1,…,Mk=1,…,M 15:Pad ψ with zeros to the next power-of-two length 16:Apply H to ancilla a 17:Apply CSWAP(a;ϕ-qubit,ψ-register qubit 0)CSWAP(a;φ-qubit,ψ-register qubit 0) 18:Apply H to ancilla a 19:Measure ancilla and obtain p0=Pr[a=0]p_0= [a=0] 20:s←clip(2p0−1,0,1)s (2p_0-1,0,1) 21:cos←s ← s 22:Θ←arccos(clip(cos,−1,1)) ← \! (clip( ,-1,1) ) 23:D←max(2Zs,0)D← (2Z\,s,0) 24:return (D,Θ)(D, ) 4.2 Coordinate-descent optimization over the CGR configuration space Given anchors =a1,…,aKA=\a_1,…,a_K\ and membership lists ℳaa∈\M_a\_a , a CGR configuration κ assigns each anchor a a subset κ(a)κ(a) of fixed cardinality k with a∈κ(a)a∈κ(a). For each configuration, we construct CGR features (Algorithms 1-3), fit class medoids (Algorithms 4-5), and evaluate classification performance using the fusion-score classifier (Algorithm 8) under a target metric (Algorithm 9). We identify κ⋆κ by maximizing validation macro-F1: κ⋆=argmaxκ∈ΩkF1macro(y(val),y^(val)(κ)).κ \;=\; _κ∈ _k\;F1_macro\! (y^(val), y^(val)(κ) ). (25) Since Ωk _k is combinatorial, we apply coordinate descent by updating one anchor subset κ(a)κ(a) at a time. We additionally employ progressive initialization across subset sizes k∈[kmin,kmax]k∈[k_ ,k_ ] by using the best configuration at k to initialize the search at k+1k+1. Algorithm 7 Coordinate descent with progressive initialization for CGR configuration selection (objective: macro-F1) 1:Training split (X(tr),y(tr))(X^(tr),y^(tr)); validation split (X(val),y(val))(X^(val),y^(val)); anchors A; membership map ℳa\M_a\; weight map ρa,f\ _a,f\; subset-size range k∈[kmin,kmax]k∈[k_ ,k_ ]; maximum passes P; candidate budget M per anchor (optional); seed s; fusion hyperparameters α,βα,β with α+β=1α+β=1; angular flag use_angular; class-weight mode (e.g., inv_sqrt). 2:Best configuration record rec[k]=κk⋆,F1k⋆,metricsk⋆rec[k]=\ _k ,F1_k ,metrics_k \ for each k 3:κprev←∅ _prev← 4:for k=kmink=k_ to kmaxk_ do 5: Initialize κ: 6: if κprev≠∅ _prev≠ then 7: κ←NormalizeConfig(κprev,k,,ℳa)κ← NormalizeConfig( _prev,k,A,\M_a\) ⊳ truncate/pad; ensure a∈κ(a)a∈κ(a) 8: else 9: for each a∈a do 10: κ(a)←[a]⊕κ(a)←[a] first (k−1)(k-1) elements of ℳa∖aM_a \a\ 11: end for 12: end if 13: (F1⋆,metrics⋆)←ScoreConfig(κ)(F1 ,metrics )← ScoreConfig(κ) 14: for p=1p=1 to P do 15: improved←falseimproved 16: for each a∈a do 17: a←CandidateSubsets(a,ℳa,k)C_a← CandidateSubsets(a,M_a,k) ⊳ anchor-first subsets 18: a←DedupSubsets(a)C_a← DedupSubsets(C_a) ⊳ order-invariant after anchor 19: if M is specified and |a|>M|C_a|>M then 20: a←SelectSubset(a,M;s)C_a← SelectSubset(C_a,M;s) 21: end if 22: κabest←κ(a) _a^best←κ(a); F1abest←F1⋆F1_a^best 1 23: for each S∈aS _a do 24: if S=κ(a)S=κ(a) then 25: continue 26: end if 27: κ′←κ ←κ with κ′(a)←Sκ (a)← S 28: (F1′,metrics′)←ScoreConfig(κ′)(F1 ,metrics )← ScoreConfig(κ ) 29: if F1′>F1abestF1 >F1_a^best then 30: F1abest←F1′F1_a^best 1 ; κabest←S _a^best← S; metricsabest←metrics′metrics_a^best 31: end if 32: end for 33: if κabest≠κ(a) _a^best≠κ(a) then 34: κ(a)←κabestκ(a)← _a^best; F1⋆←F1abestF1 1_a^best; metrics⋆←metricsabestmetrics _a^best 35: improved←trueimproved 36: end if 37: end for 38: if improved=falseimproved=false then 39: break 40: end if 41: end for 42: rec[k]←κ,F1⋆,metrics⋆rec[k]←\κ,F1 ,metrics \ 43: κprev←κ _prev←κ 44:end for 45:return recrec 46:Subroutine ScoreConfig(κ)(κ): 47:Build F(tr)=F(X(tr);κ)F^(tr)=F(X^(tr);κ) and F(val)=F(X(val);κ)F^(val)=F(X^(val);κ); standardize using training statistics; fit class medoids on F(tr)F^(tr); construct class weights from priors (optional); infer y^(val) y^(val) using Algorithm 8; evaluate macro-F1 using Algorithm 9. 4.3 Fusion score inference from SWAP-test distance and angular channels For each standardized CGR feature vector v∈ℝd′v ^d and class medoid μc _c, Algorithm 6 yields the pair (Dc,Θc)(D_c, _c), where DcD_c is an Euclidean-like distance and Θc _c is an angular distance. We emphasize that fusion-score inference is non-probabilistic: the quantities (Dc,Θc)(D_c, _c) are transformed into a per-class score scs_c that acts as a margin-like statistic, and the predicted label is obtained by the minimum-score rule rather than by estimating calibrated probabilities. We convert these channels into a single margin-like statistic by normalizing within the sample: D~c=Dc∑c′=1CDc′+ε,Θ~c=Θc∑c′=1CΘc′+ε, D_c\;=\; D_c _c =1^CD_c + , _c\;=\; _c _c =1^C _c + , (26) and fusing them via a convex mixture sc=αD~c+βΘ~c,if the angular channel is enabled,D~c,otherwise,s_c\;=\; casesα\, D_c+β\, _c,&if the angular channel is enabled,\\ D_c,&otherwise, cases (27) where α,β∈[0,1]α,β∈[0,1] with α+β=1α+β=1. The predicted class is obtained by the minimum-score rule y^=argminc∈1,…,Csc. y\;=\; _c∈\1,…,C\s_c. (28) Optionally, a class-weight map wcw_c may be applied to scs_c to moderate imbalance effects. Algorithm 8 Fusion score inference using compact SWAP-test distances and angles 1:Standardized feature matrix F∈ℝN×d′F ^N× d ; class medoids μcc=1C\ _c\_c=1^C; use_angular; fusion weights (α,β)(α,β) with α+β=1α+β=1; ε>0 >0; optional class weights wc\w_c\ 2:Predicted labels y^∈1,…,CN y∈\1,…,C\^N 3:for i=1i=1 to N do 4: v←Fi,:v← F_i,: 5: for c=1c=1 to C do 6: (Dc,Θc)←CompactSwapDistAngle(v,μc)(D_c, _c) (v, _c) ⊳ Algorithm 6 7: end for 8: D~c←Dc/(∑c′Dc′+ε) D_c← D_c/( _c D_c + ) for all c 9: if use_angular then 10: Θ~c←Θc/(∑c′Θc′+ε) _c← _c/( _c _c + ) for all c 11: sc←αD~c+βΘ~cs_c←α D_c+β _c for all c 12: else 13: sc←D~cs_c← D_c for all c 14: end if 15: if class weights wcw_c are provided then 16: sc←wc⋅scs_c← w_c· s_c for all c 17: end if 18: y^i←argmincsc y_i← _cs_c 19:end for 20:return y y In practice, we tune α on the validation set using a finite grid, while setting β=1−αβ=1-α to preserve a convex fusion of the two channels. Separately, we report FβF_ _F over a grid of βF _F values to assess precision–recall sensitivity; this reporting parameter is distinct from the fusion mixing weights. 4.4 Evaluation under macro-F1 and related classification metrics To evaluate fusion-based predictions during configuration selection, we optimize macro-F1, which is robust to class imbalance. For each class c, let precision and recall be Pc=TPcTPc+FPcP_c= TP_cTP_c+FP_c and Rc=TPcTPc+FNcR_c= TP_cTP_c+FN_c, and define F1c=2PcRcPc+Rc.F1_c\;=\; 2P_cR_cP_c+R_c. (29) Macro-F1 averages equally across classes: F1macro=1C∑c=1CF1c.F1_macro\;=\; 1C _c=1^CF1_c. (30) In addition to accuracy and macro-F1, we report macro-precision and macro-recall, and optionally FβF_β (e.g., β=2β=2) to emphasize recall. Algorithm 9 Metric evaluation for configuration selection (macro-F1 target) 1:True labels y; predicted labels y y; averaging mode (macro or weighted); optional β>0β>0 2:Metric dictionary: Accuracy, F1, Precision, Recall (and optionally FβF_β) 3:Compute Accuracy←1N∑i=1N[yi=y^i]Accuracy← 1N _i=1^NI[y_i= y_i] 4:for c=1c=1 to C do 5: Compute TPc,FPc,FNcTP_c,FP_c,FN_c from (y,y^)(y, y) 6: Pc←TPc/(TPc+FPc)P_c← TP_c/(TP_c+FP_c) with safe handling for 0 denominators 7: Rc←TPc/(TPc+FNc)R_c← TP_c/(TP_c+FN_c) with safe handling for 0 denominators 8: F1c←2PcRc/(Pc+Rc)F1_c← 2P_cR_c/(P_c+R_c) 9: if β is provided then 10: Fβ,c←(1+β2)PcRcβ2Pc+RcF_β,c←(1+β^2) P_cR_cβ^2P_c+R_c 11: end if 12:end for 13:if macro averaging then 14: F1macro←1C∑cF1cF1_macro← 1C _cF1_c; similarly compute macro-precision and macro-recall 15: if β is provided then 16: Fβ,macro←1C∑cFβ,cF_β,macro← 1C _cF_β,c 17: end if 18:else 19: Weight each class by support πc=nc/N _c=n_c/N and compute weighted metrics 20:end if 21:return metrics 4.5 Calibration of fusion mixing and metric sensitivity sweeps After selecting an optimal CGR configuration κ⋆κ (Algorithm 7) and fixing the evaluation protocol (Section 4.4), we perform a lightweight calibration stage to (i) tune the fusion mixing weight and (i) report precision–recall sensitivity through an FβF_ _F sweep. Fusion mixing calibration. When the angular channel is enabled, the fusion score combines Euclidean-like and angular channels through a convex mixture: sc(α)=αD~c+(1−α)Θ~c,α∈[0,1],s_c(α)\;=\;α\, D_c\;+\;(1-α)\, _c, α∈[0,1], (31) where α is tuned on the validation split using a finite grid α⊂[0,1]G_α⊂[0,1] to maximize macro-F1. In this implementation, the angular weight is not tuned independently; it is defined as 1−α1-α to preserve a convex fusion. Metric sensitivity sweep. Separately, we compute FβF_ _F over a grid βFG_ _F to assess how performance varies as recall is emphasized (βF>1 _F>1) or precision is emphasized (βF<1 _F<1). This parameter βF _F is a reporting parameter for the metric and is distinct from the fusion mixing weight α. Algorithm 10 Grid calibration of fusion mixing weight α under fixed configuration κ⋆κ (objective: macro-F1) 1:Best configuration κ⋆κ ; training split (X(tr),y(tr))(X^(tr),y^(tr)); validation split (X(val),y(val))(X^(val),y^(val)); grid αG_α; use_angular; ε>0 >0; optional class weights wc\w_c\. 2:Selected α⋆α and validation record. 3:Build CGR features under κ⋆κ : F(tr)=F(X(tr);κ⋆)F^(tr)=F(X^(tr);κ ) and F(val)=F(X(val);κ⋆)F^(val)=F(X^(val);κ ). 4:Standardize using training statistics. 5:Fit class medoids μc\ _c\ on F(tr)F^(tr) (Algorithms 4–5). 6:if use_angular then 7: for each α∈α _α do 8: Predict y^(val)(α) y^(val)(α) using Algorithm 8 with (α,1−α)(α,1-α). 9: Compute F1macro(α)F1_macro(α) using Algorithm 9. 10: end for 11: α⋆←argmaxα∈αF1macro(α)α ← _α _αF1_macro(α) 12:else 13: α⋆←1α ← 1 14:end if 15:return α⋆α Algorithm 11 FβF_ _F reporting sweep under fixed predictions (metric sensitivity) 1:Validation labels y(val)y^(val); fixed predictions y^(val) y^(val) (obtained using α⋆α ); grid βFG_ _F. 2:Reporting table (βF,FβF)\( _F,F_ _F)\. 3:for each βF∈βF _F _ _F do 4: Compute macro-FβF_ _F on (y(val),y^(val))(y^(val), y^(val)) using Algorithm 9 with parameter β←βFβ← _F. 5:end for 6:return (βF,FβF)\( _F,F_ _F)\ 4.6 Fusion model packaging, persistence, and scoring of new records For practical deployment and reproducibility, we package the fusion-score classifier obtained under the selected configuration and calibrated mixing weight as a self-contained model artifact. A fusion model for a given subset size k is defined as: ℱk=κk⋆,Scalerk,μk,cc=1C,wk,cc=1C,αk⋆,use_angular,F_k\;=\;\ _k ,\;Scaler_k,\;\ _k,c\_c=1^C,\;\w_k,c\_c=1^C,\; _k ,\; use\_angular\, where ScalerkScaler_k denotes the standardization transform fitted on training CGR features, μk,c _k,c are class medoids, wk,cw_k,c are optional class weights, and αk⋆ _k is the calibrated fusion mixing weight (Section 4.5). We additionally retain validation/test summaries (macro-F1, accuracy, confusion matrices) for each stored artifact. In the full pipeline, we persist a small set of top-performing fusion models (e.g., top-2 over k) so that (i) fusion-only inference is directly available for small-to-moderate datasets where it is optimal, and (i) a consistent fusion baseline is available when computing Δ -distance features for the VQC refinement stage. Algorithm 12 Persisting top-r fusion models and scoring new records 1:Candidate records rec[k]=κk⋆,F1k⋆,metricsk⋆rec[k]=\ _k ,F1_k ,metrics_k \ from Algorithm 7; training/validation/test splits; grid αG_α; top-r; use_angular; ε>0 >0; class-weight mode. 2:Persisted fusion models ℱkjj=1r\F_k_j\_j=1^r and scoring routine for new samples. 3:Select top←K_top← the r values of k with highest validation macro-F1 in rec[k]rec[k]. 4:for each k∈topk _top do 5: κ⋆←κk⋆κ ← _k from rec[k]rec[k]. 6: Build CGR features F(tr),F(val),F(te)F^(tr),F^(val),F^(te) under κ⋆κ ; fit ScalerkScaler_k on F(tr)F^(tr) and transform all splits. 7: Fit class medoids μk,c\ _k,c\ on standardized F(tr)F^(tr). 8: Calibrate αk⋆ _k on validation using Algorithm 10. 9: Predict y^(val) y^(val) and y^(te) y^(te) using Algorithm 8 with (αk⋆,1−αk⋆)( _k ,1- _k ). 10: Compute and store metrics + confusion matrices for validation/test using Algorithm 9. 11: Persist fusion artifact ℱk=κ⋆,Scalerk,μk,c,wk,c,αk⋆,use_angularF_k=\κ ,Scaler_k,\ _k,c\,\w_k,c\, _k , use\_angular\. 12:end for 13:Scoring new record x with a persisted model ℱkF_k: 14:Build CGR features f(x;κ⋆)f(x;κ ); transform via ScalerkScaler_k; compute (Dc,Θc)(D_c, _c) to each μk,c _k,c via Algorithm 6; fuse using (αk⋆,1−αk⋆)( _k ,1- _k ) and infer y^(x) y(x) via Algorithm 8. 4.7 Δ -distance construction under the selected CGR–fusion model The fusion score (Section 4.3) provides an effective and interpretable geometric classifier in small-to-moderate regimes. However, for large, highly imbalanced, and behaviourally complex datasets (e.g., fraud), fusion alone can be sub-optimal because discrimination often depends on non-linear interactions across CGR groups and subtle class-contrastive margins. To expose these interactions in a compact form, we construct Δ -distance features from the class-referenced distance/angle channels under the selected CGR configuration κ⋆κ . Let (Dc,Θc)(D_c, _c) be the Euclidean-like and angular distances from a sample to class medoid μc _c produced by Algorithm 6. After within-sample normalization (Eq. 26), define fused class scores scs_c (Eq. 27). We then form a contrastive margin feature vector Δ(x)∈ℝm (x) ^m by taking differences between competing classes. For binary classification (C=2C=2) we use: ΔD(x)=D~1(x)−D~0(x),ΔΘ(x)=Θ~1(x)−Θ~0(x), _D(x)\;=\; D_1(x)- D_0(x), _ (x)\;=\; _1(x)- _0(x), (32) and optionally the fused margin Δs(x)=s1(x)−s0(x) _s(x)=s_1(x)-s_0(x). For C>2C>2, we construct a best-vs-runner-up margin using c⋆=argmincscc = _cs_c and c(2)=argminc≠c⋆scc^(2)= _c≠ c s_c: ΔD(x)=D~c(2)(x)−D~c⋆(x),ΔΘ(x)=Θ~c(2)(x)−Θ~c⋆(x). _D(x)\;=\; D_c^(2)(x)- D_c (x), _ (x)\;=\; _c^(2)(x)- _c (x). (33) In practice, we concatenate the selected deltas into a compact vector and standardize it using training statistics to obtain a stable input representation for the subsequent VQC refinement. Algorithm 13 Δ -distance construction (contrastive margins) under fixed κ⋆κ and calibrated fusion 1:Standardized CGR features F∈ℝN×d′F ^N× d ; class medoids μc\ _c\; use_angular; fusion weights (α,1−α)(α,1-α); ε>0 >0. 2:Delta feature matrix Δ∈ℝN×m ^N× m. 3:for i=1i=1 to N do 4: v←Fi,:v← F_i,: 5: for c=1c=1 to C do 6: (Dc,Θc)←CompactSwapDistAngle(v,μc)(D_c, _c) (v, _c) 7: end for 8: Compute D~c,Θ~c D_c, _c using Eq. 26 9: Compute fused scores scs_c using Eq. 27 10: if C=2C=2 then 11: ΔD←D~1−D~0 _D← D_1- D_0; ΔΘ←Θ~1−Θ~0 _ ← _1- _0 12: Δi←[ΔD,ΔΘ] _i←[ _D, _ ] ⊳ optionally add Δs=s1−s0 _s=s_1-s_0 13: else 14: c⋆←argmincscc ← _cs_c; c(2)←argminc≠c⋆scc^(2)← _c≠ c s_c 15: ΔD←D~c(2)−D~c⋆ _D← D_c^(2)- D_c ; ΔΘ←Θ~c(2)−Θ~c⋆ _ ← _c^(2)- _c 16: Δi←[ΔD,ΔΘ] _i←[ _D, _ ] ⊳ optionally augment with additional class-pair deltas 17: end if 18:end for 19:Standardize Δ using training-fold statistics. 20:return Δ 4.8 Variational quantum classifier (VQC) on Δ -distance features The VQC stage is introduced as a selective refinement mechanism and is invoked only when the geometric fusion classifier (Section 4.3) is empirically sub-optimal under complex, large-scale, and imbalanced regimes (e.g., fraud). In small datasets, we observed that fusion scoring often attains stable performance while the VQC may be data-constrained. Conversely, for high-volume and highly imbalanced datasets, the Δ -distance representation (Section 4.7) provides a compact, class-contrastive feature space in which the VQC can learn non-linear interactions that are not captured by a convex fusion alone. Let Z∈ℝN×mZ ^N× m denote the Δ -feature matrix, where m=2m=2 in the fraud setting (ΔD,ΔΘ _D, _ ). We encode Z into rotation angles, construct a low-depth VQC ansatz with optional data re-uploading, and train parameters using SPSA with minibatching. The complete VQC flow is formalized in Algorithms 14–16. 4.8.1 Angle mapping and feature scaling Before quantum encoding, Δ -features are standardized using training statistics and clipped to a bounded range to stabilize training. The clipped values are then mapped to angles in [−π,π][-π,π]. We additionally allow per-feature scaling coefficients λj _j (implemented as fixed multipliers in the encoding layer) to control the relative influence of each Δ channel. Algorithm 14 Standardize–clip–angle map for Δ -features 1:Δ -feature matrix Z∈ℝN×mZ ^N× m; scaler S (fit on train only); clip bound zmax>0z_ >0; feature scales λ∈ℝ+mλ ^m_+. 2:Angle matrix A∈[−π,π]N×mA∈[-π,π]^N× m and fitted scaler S. 3:Fit scaler S on training Z; compute standardized Zs←(Z)Z_s (Z). 4:Clip Zc←clip(Zs,−zmax,zmax)Z_c (Z_s,-z_ ,z_ ) elementwise. 5:Map to angles A←(π/zmax)⋅ZcA←(π/z_ )· Z_c. 6:Apply feature scaling A:,j←λj⋅A:,jA_:,j← _j· A_:,j for j=1,…,mj=1,…,m. 7:return (A,)(A,S). 4.8.2 Circuit construction with optional data re-uploading Given m encoded angles and n qubits, we define an L-repetition ansatz. Each repetition optionally re-uploads data (angle encoding) and then applies a fixed entangling pattern followed by trainable single-qubit rotations. Algorithm 15 VQC circuit construction with data re-uploading 1:Number of qubits n; number of inputs m; repetitions L; re-upload flag reupload. 2:Parameterized circuit U(θ;x)U(θ;x) with input parameters x∈ℝmx ^m and trainable parameters θ. 3:Initialize circuit U on n qubits. 4:for r=1r=1 to L do 5: if r=1r=1 or reupload is enabled then 6: for j=1j=1 to m do 7: Apply Ry(xj)R_y(x_j) to qubit q←(j−1)modnq←(j-1) n. 8: end for 9: end if 10: if n≥2n≥ 2 then 11: Apply entangling chain: CX(q,q+1)CX(q,q+1) for q=0,…,n−2q=0,…,n-2. 12: Apply phase chain: CZ(q,q+1)CZ(q,q+1) for q=0,…,n−2q=0,…,n-2. 13: end if 14: for q=0q=0 to n−1n-1 do 15: Apply trainable rotations Ry(θr,q(1))R_y( _r,q^(1)) and Rz(θr,q(2))R_z( _r,q^(2)). 16: end for 17:end for 18:return U(θ;x)U(θ;x). 4.8.3 K-fold selection, retraining, and artifact persistence For each selected k (typically top-ranked by fusion validation performance), we train a VQC on the corresponding Δ -dataset. We select hyperparameters via stratified K-fold cross-validation and then retrain on the full training split before evaluating on validation and test splits. For binary classification, we additionally tune a decision threshold on validation probabilities. Algorithm 16 VQC per-k training with stratified K-fold selection, threshold tuning, and artifact saving 1:Selected k values K; Δ datasets (Zktr,yktr),(Zkval,ykval),(Zkte,ykte)\(Z_k^tr,y_k^tr),(Z_k^val,y_k^val),(Z_k^te,y_k^te)\; HP space ℋH (reps, reupload, SPSA iters, batch size); K-fold splitter SKFSKF; clip bound zmaxz_ ; feature scales λ; target metric (e.g., macro-F1); threshold grid T (binary only). 2:Best VQC artifacts per k and an overall best alias. 3:for each k∈k do 4: best_hp←∅best\_hp← ; best_cv←−∞best\_cv←-∞; best_thr←0.5best\_thr← 0.5. 5: for each hp∈ℋhp do 6: scores←[]scores←[\ ]. 7: for each fold (ℐtr,ℐva)(I_tr,I_va) from SKF(Zktr,yktr)SKF(Z_k^tr,y_k^tr) do 8: Map angles: (Atr,)←(A_tr,S)← Algorithm 14 on Zktr[ℐtr]Z_k^tr[I_tr]; Ava←A_va← Algorithm 14 using same scaler S on Zktr[ℐva]Z_k^tr[I_va]. 9: Build circuit via Algorithm 15 using hphp. 10: Train SPSA via Algorithm 18 on (Atr,ytr)(A_tr,y_tr) to obtain θ⋆θ . 11: Compute probabilities PvaP_va via Algorithm 17. 12: if binary classification then 13: Choose threshold τ⋆∈τ maximizing the target metric on (yva,Pva)(y_va,P_va). 14: Append fold score (target metric under τ⋆τ ) to scoresscores. 15: else 16: Append fold score using argmaxcPva,c _cP_va,c to scoresscores. 17: end if 18: end for 19: cv_mean←mean(scores)cv\_mean (scores). 20: if cv_mean>best_cvcv\_mean>best\_cv then 21: best_cv←cv_meanbest\_cv← cv\_mean; best_hp←hpbest\_hp← hp; store best_thrbest\_thr if binary. 22: end if 23: end for 24: Retrain with best_hpbest\_hp on full training split: map (Zktr,Zkval,Zkte)→(Atr,Aval,Ate)(Z_k^tr,Z_k^val,Z_k^te)→(A^tr,A^val,A^te) using Algorithm 14. 25: Train final θ⋆θ with Algorithm 18 on (Atr,ytr)(A^tr,y^tr). 26: Evaluate on validation/test: obtain Pval,PteP^val,P^te, compute metrics and confusion matrices. 27: Save artifacts: circuit, weights, scaler, and metadata (Appendix algorithm / implementation). 28:end for 29:Save an overall best alias (e.g., best validation macro-F1 across k). 5 Experimental Setup, data gathering, and statistics We evaluate the proposed geometry-driven classification pipeline on four tabular datasets spanning two regimes: (i) small-to-moderate, relatively balanced benchmarks and (i) large-scale, highly imbalanced rare-event detection. This separation is intentional and reflects the hybrid decision policy of our method. Specifically, the pipeline is designed to operate as a two-stage classifier: a lightweight geometric decision rule based on the fusion score is used when data are limited and prototype-based similarity is sufficient, while a variational quantum classifier (VQC) is introduced when the dataset is larger and the class structure is more heterogeneous and/or severely imbalanced, where additional nonlinear decision capacity becomes beneficial. 5.1 Small / moderately balanced regime: Heart Disease, Breast Cancer, Wine Quality For Heart Disease, Breast Cancer, and Wine Quality, we report results using the fusion score classifier constructed from compact SWAP-test overlap estimation under the CGR structure. In this regime, the fusion score serves as the primary model because it is data-efficient, geometry-interpretable, and empirically stable without requiring the higher-capacity VQC stage, which can be sensitive when the number of training instances is limited. Evaluation follows a fixed train/test protocol, with the validation split used only for configuration selection (e.g., CGR configuration and fusion-channel settings). We report standard classification metrics (accuracy, macro-F1, class-wise precision/recall/F1) and confusion matrices, and compare against classical baselines trained on the same raw feature space. ROC-AUC and PR-AUC are omitted for Fusion since it is a non-probabilistic scoring rule and does not output calibrated probabilities; AUC metrics are reported for probabilistic baselines and for the VQC stage on Fraud. Table 1: Test-set comparison: Fusion-score (ours) vs. classical baselines across Heart Disease, Breast Cancer, and Wine Quality. Metrics are rows and models are columns. For Fusion, the reported setting is the validation-selected best-k and α⋆α per dataset. Dataset Metric Fusion (ours) LogReg LinearSVM RBF_SVM RandomForest Heart Disease Accuracy 0.8478 0.8565 0.8565 0.8870 0.8913 Macro-F1 0.8463 0.8551 0.8551 0.8861 0.8898 Weighted-F1 0.8479 0.8566 0.8566 0.8871 0.8911 ROC-AUC – 0.9021 0.8997 0.9372 0.9408 PR-AUC – 0.8976 0.8908 0.9435 0.9437 Breast Cancer Accuracy 0.8881 0.9860 0.9580 0.9720 0.9790 Macro-F1 0.8703 0.9849 0.9547 0.9698 0.9772 Weighted-F1 0.8827 0.9860 0.9579 0.9719 0.9789 ROC-AUC – 0.9983 0.9881 0.9962 0.9979 PR-AUC – 0.9974 0.9834 0.9946 0.9964 Wine Quality (3-class) Accuracy 0.9556 1.0000 1.0000 1.0000 1.0000 Macro-F1 0.9522 1.0000 1.0000 1.0000 1.0000 Weighted-F1 0.9547 1.0000 1.0000 1.0000 1.0000 5.1.1 Interpretation: Fusion-score results on small benchmarks Across Heart Disease, Breast Cancer, and Wine Quality, the fusion-score classifier provides a strong and stable decision rule when the dataset size is limited and class structure can be captured through class prototypes. The key observation is that the overlap-derived similarity channels (Euclidean-like distance and optional angular separation) preserve sufficient geometric signal for discrimination once organized under the CGR feature grouping. In this regime, the fusion score acts as a low-variance estimator: it requires no high-capacity parameterization beyond the configuration search and the fusion weights, and it therefore avoids the instability that can arise from training heavier decision models on limited samples. Empirically, we observe that the fusion-score stage is competitive with classical baselines on these datasets, and the resulting confusion matrices indicate that most errors arise from borderline cases where samples lie close to the class medoid boundaries. Overall, these results support the intended role of fusion scoring as the primary classifier for small-to-moderate benchmarks, delivering geometry-interpretable decisions with minimal training overhead. 5.2 Large / highly imbalanced regime: Credit Card Fraud Fraud classification is treated separately due to extreme class imbalance and deployment-style operating constraints. Here, the full pipeline is executed: after selecting the best CGR configuration and fusion setup, we construct Δ -distance features (contrastive differences between class-referenced similarity channels) and train a VQC on these compact geometric features. Results are reported via full-dataset scoring at the fixed operating threshold used for our VQC inference stage (τ=0.3τ=0.3), together with metrics appropriate for rare-event detection, including PR-AUC/ROC-AUC and confusion matrices. For classical comparisons on Fraud, baseline models are trained on the raw feature representation (not on fusion or Δ features, which are native to our method). To support operationally meaningful comparisons, we primarily report an alert-rate matched operating point (thresholds calibrated on validation to match the predicted-positive rate) and include fixed-threshold results (e.g., 0.5) as a secondary reference, since default thresholds can be overly conservative under extreme imbalance and are sensitive to probability calibration and class-prior shift. Table 2: Fraud classification on the full dataset (raw-feature baselines) under an alert-rate matched operating point. The proposed model is scored using the fixed VQC threshold τ=0.3τ=0.3. Metric Ours (Δ +VQC, τ=0.3) LogReg_raw LinearSVM_raw RF_raw N evaluated 284,807 284,807 284,807 284,807 Prevalence 0.0017 0.0017 0.0017 0.0017 Alert rate 0.0131 0.0130 0.0133 0.0128 Precision (fraud) 0.1124 0.1188 0.1157 0.1277 Recall (fraud) 0.8496 0.8963 0.8923 0.9492 F1 (fraud) 0.1985 0.2098 0.2049 0.2251 PR-AUC 0.3251 0.7392 0.7287 0.9269 ROC-AUC 0.9249 0.9852 0.9852 0.9808 Macro-F1 0.5963 0.6019 0.5994 0.6097 Accuracy 0.9881 0.9883 0.9880 0.9887 TP 418 441 439 467 FP 3,301 3,272 3,355 3,190 FN 74 51 53 25 TN 281,014 281,043 280,960 281,125 Table 3: Fraud classification on the full dataset (raw-feature baselines with SMOTE applied on the training split) under an alert-rate matched operating point. The proposed model is scored using the fixed VQC threshold τ=0.3τ=0.3. Metric Ours (Δ +VQC, τ=0.3) LogReg_SMOTE LinearSVM_SMOTE RF_SMOTE N evaluated 284,807 284,807 284,807 284,807 Prevalence 0.0017 0.0017 0.0017 0.0017 Alert rate 0.0131 0.0130 0.0130 0.0090 Precision (fraud) 0.1124 0.1192 0.1193 0.1835 Recall (fraud) 0.8496 0.8984 0.9004 0.9512 F1 (fraud) 0.1985 0.2105 0.2107 0.3076 PR-AUC 0.3251 0.7521 0.7609 0.9393 ROC-AUC 0.9249 0.9830 0.9832 0.9915 Macro-F1 0.5963 0.6023 0.6024 0.6519 Accuracy 0.9881 0.9884 0.9883 0.9926 TP 418 442 443 468 FP 3,301 3,265 3,270 2,083 FN 74 50 49 24 TN 281,014 281,050 281,045 282,232 Table 4: Fraud classification on the full dataset at a fixed classical decision threshold of 0.5 (raw-feature baselines). The proposed model is shown for reference using τ=0.3τ=0.3. Metric Ours (Δ +VQC, τ=0.3) LogReg_raw LinearSVM_raw_cal RF_raw N evaluated 284,807 284,807 284,807 284,807 Prevalence 0.0017 0.0017 0.0017 0.0017 Alert rate 0.0131 0.0242 0.0012 0.0016 Precision (fraud) 0.1124 0.0658 0.8663 0.9516 Recall (fraud) 0.8496 0.9228 0.5793 0.8801 F1 (fraud) 0.1985 0.1229 0.6943 0.9145 PR-AUC 0.3251 0.7401 0.7242 0.9269 ROC-AUC 0.9249 0.9851 0.9848 0.9808 Macro-F1 0.5963 0.5557 0.8469 0.9572 Accuracy 0.9881 0.9772 0.9991 0.9997 TP 418 454 285 433 FP 3,301 6,442 44 22 FN 74 38 207 59 TN 281,014 277,873 284,271 284,293 Table 5: Fraud classification on the full dataset at a fixed classical decision threshold of 0.5 (raw-feature baselines with SMOTE applied on the training split). The proposed model is shown for reference using τ=0.3τ=0.3. Metric Ours (Δ +VQC, τ=0.3) LogReg_SMOTE LinearSVM_SMOTE_cal RF_SMOTE N evaluated 284,807 284,807 284,807 284,807 Prevalence 0.0017 0.0017 0.0017 0.0017 Alert rate 0.0131 0.0046 0.0049 0.0017 Precision (fraud) 0.1124 0.3279 0.3059 0.9433 Recall (fraud) 0.8496 0.8659 0.8679 0.9126 F1 (fraud) 0.1985 0.4757 0.4523 0.9277 PR-AUC 0.3251 0.7521 0.7589 0.9393 ROC-AUC 0.9249 0.9830 0.9832 0.9915 Macro-F1 0.5963 0.7370 0.7253 0.9638 Accuracy 0.9881 0.9967 0.9964 0.9998 TP 418 426 427 449 FP 3,301 873 969 27 FN 74 66 65 43 TN 281,014 283,442 283,346 284,288 5.2.1 Interpretation: Fraud results under extreme imbalance The Credit Card Fraud dataset constitutes a qualitatively different evaluation regime, combining extreme class imbalance (fraud prevalence ≈0.17%≈ 0.17\%) with a deployment-style requirement to interpret performance at an explicit operating point. In this setting, overall accuracy is dominated by the majority class and is therefore not a meaningful indicator of detection quality. Instead, performance must be interpreted through the precision–recall trade-off under a controlled alert budget, i.e., recovering a large fraction of minority events while limiting false positives to a level that is operationally reviewable. Under full-dataset scoring, the proposed Δ +VQC pipeline evaluated at τ=0.3τ=0.3 produces an alert rate of ≈1.31%≈ 1.31\% and attains a minority recall of ≈0.85≈ 0.85 (Table 2). This operating point surfaces the majority of fraudulent transactions while maintaining a bounded alert volume, but naturally yields modest minority precision due to the ultra-imbalanced base rate. To contextualize these results, we compare against classical baselines trained directly on the raw feature space (not on fusion or Δ features, which are native to our method). When baseline thresholds are calibrated to match a comparable alert-rate budget (primary comparison), tree-ensemble and margin-based models achieve substantially higher PR-AUC and, in some cases, higher recall at similar alert rates, consistent with the strong separability of the benchmark feature representation. We additionally report fixed-threshold (0.5) results as a secondary reference (Tables 4-5). These results illustrate that under extreme imbalance, adopting a default probability threshold can materially change the effective operating point, shifting the alert rate and consequently the precision–recall balance. Some models become highly conservative and sacrifice minority recall, whereas others recover recall by emitting more alerts. For this reason, we treat alert-rate matched evaluation as the principal criterion for rare-event detection and interpret the proposed pipeline as a quantum-compatible geometric alternative whose behavior is most meaningfully characterized under explicit operating constraints rather than at an arbitrary default threshold.threshold. 6 Final Thoughts and Conclusion In this work, we introduced a geometry-first classification paradigm in which compact SWAP-test-based overlap estimation is employed to extract interpretable similarity evidence with respect to representative class prototypes. By deriving Euclidean-like and angular similarity channels from overlap measurements, the framework grounds classification decisions in explicit geometric relations between data points and class medoids, providing both interpretability and structural clarity. To enhance robustness in heterogeneous tabular feature spaces, we incorporated a correlation-group rotation (CGR) structuring mechanism that organizes similarity computation into correlation-consistent feature groups. This design improves numerical stability and preserves semantic coherence across feature subsets, thereby strengthening interpretability while mitigating instability in high-variance dimensions. The proposed fusion-score stage operates as a non-probabilistic decision rule that aggregates overlap-derived similarity channels into a per-class margin-like score and predicts via an argmin criterion. As a lightweight and data-efficient classifier, this fusion mechanism is particularly suitable for small-to-moderate datasets, where stable primary decision rules are preferable to heavily parameterized models. For more complex regimes where direct fusion scoring may become suboptimal, we introduced a Δ -distance construction that forms compact, contrastive class-referenced difference features. These Δ -features preserve geometric separability while reducing dimensionality, enabling a more concise representation of discriminative information. Building upon this compact representation, we integrated a variational quantum classifier (VQC) as an on-demand decision layer. Trained on Δ -features, the VQC acts as a selective nonlinear discriminator for large-scale and highly imbalanced settings, while maintaining the geometry-driven front-end of the pipeline. Empirically, across Heart Disease, Breast Cancer, and Wine Quality datasets, the fusion-score classifier achieves competitive performance relative to classical baselines, consistent with its intended role as a stable and interpretable primary classifier under limited data conditions. For the Credit Card Fraud dataset, full-dataset evaluation underscores the importance of operating-point analysis—particularly alert rate and minority recall/precision trade-offs. In this extreme imbalance regime, the Δ +VQC pipeline demonstrates the ability to recover high minority recall while maintaining a controlled alert budget. We further emphasize that operational evaluation is essential in rare-event detection. Default decision thresholds can misrepresent performance under extreme imbalance; therefore, alert-rate-matched comparisons are treated as the principal evaluation criterion in fraud detection scenarios, with fixed-threshold metrics serving only as secondary references. To support reproducibility and deployment-style evaluation, the complete pipeline is accompanied by practical artifacts, including fitted scalers, medoids, tuned fusion parameters, Δ -feature construction procedures, and trained VQC checkpoints, enabling consistent scoring of new records. Looking forward, several directions merit further investigation. These include calibration under class-prior shift for the VQC stage; more principled regularization and stability control mechanisms for CGR configuration and fusion-weight tuning; hardware-executed overlap estimation and VQC training with noise-aware performance analysis; and broader benchmarking across diverse non-PCA tabular datasets, including scenarios with distribution drift and evolving subpopulation prototypes. Collectively, these extensions aim to strengthen the theoretical grounding, practical robustness, and real-world deployability of the proposed geometry-driven quantum-classical hybrid framework. Acknowledgment The authors express gratitude to the IBM Quantum Experience platform and its team for creating the Qiskit platform and granting free access to their simulators for executing quantum circuits and conducting the experiments detailed below. The authors express appreciation for the Centre for Quantum Software and Information (CQSI) . 7 Statements and Declarations Competing Interests: The authors have no financial or non-financial competing interests. Authors’ contributions: The authors confirm their contribution to the paper as follows: Study conception and design: A.A.P., N.M., B.K.B., B.M.; Data collection: A.A.P., N.M.; Analysis and interpretation of results: A.A.P., N.M., B.K.B., B.M.; Draft manuscript preparation: A.A.P., N.M., B.K.B.; All authors reviewed the results and approved the final version of the manuscript. Funding: Authors declare that there has been no external funding. Availability of data and materials: All the data provided in this manuscript is generated during the simulation and can be provided upon reasonable request. References He and Garcia [2009] He H, Garcia EA. Learning from Imbalanced Data. IEEE Transactions on Knowledge and Data Engineering. 2009;21(9):1263–1284. 10.1109/TKDE.2008.239. 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Supervised Learning with Quantum-Enhanced Feature Spaces. Nature. 2019;567:209–212. 10.1038/s41586-019-0980-2. Schuld and Killoran [2019] Schuld M, Killoran N. Quantum Machine Learning in Feature Hilbert Spaces. Physical Review Letters. 2019;122(4):040504. 10.1103/PhysRevLett.122.040504. Chawla et al. [2002] Chawla NV, Bowyer KW, Hall LO, Kegelmeyer WP. SMOTE: Synthetic Minority Over-sampling Technique. Journal of Artificial Intelligence Research. 2002;16:321–357. Wiebe et al. [2015] Wiebe N, Kapoor A, Svore KM. Quantum Algorithms for Nearest-Neighbor Methods for Supervised and Unsupervised Learning. Quantum Information & Computation. 2015;15(3–4):318–358. Schuld et al. [2017] Schuld M, Sinayskiy I, Petruccione F. Implementing a Distance-Based Classifier with a Quantum Interference Circuit. EPL (Europhysics Letters). 2017;119(6):60002. 10.1209/0295-5075/119/60002. Nielsen and Chuang [2000] Nielsen MA, Chuang IL. Quantum Computation and Quantum Information. Cambridge University Press; 2000. García-Escartín and Chamorro-Posada [2013] García-Escartín JC, Chamorro-Posada P. The Swap Test and the Hong–Ou–Mandel Effect Are Equivalent. Physical Review A. 2013;87(5):052330. 10.1103/PhysRevA.87.052330. Schuld and Petruccione [2018] Schuld M, Petruccione F. Supervised Learning with Quantum Computers. Springer; 2018. 8 Appendix 8.1 VQC additional artifcats This section details the end-to-end VQC training and inference pipeline, including forward probability evaluation, SPSA-based optimization with minibatching and stabilization strategies, and artifact persistence for reproducible deployment. Together, these components ensure numerically stable training, consistent class decoding, and reliable inference aligned with the geometry-driven front-end of the framework. 8.1.1 Forward probability evaluation and class decoding We evaluate class probabilities from measurement outcomes. In our experiments we use statevector simulation to compute exact probabilities;in shot-based settings, the same procedure applies with empirical frequencies. Algorithm 17 computes class probability distributions by measuring the parameterized quantum circuit and decoding basis-state outcomes via direct or parity-based mapping. Algorithm 17 Forward pass: class probabilities from a parameterized VQC 1:Circuit U(θ;x)U(θ;x); inputs A∈ℝN×mA ^N× m; number of classes C; mapping mode direct or parity; n qubits. 2:Probability matrix P∈[0,1]N×CP∈[0,1]^N× C. 3:for i=1i=1 to N do 4: Prepare state |ψi⟩=U(θ;Ai)|0⟩⊗n _i=U(θ;A_i) 0 n. 5: Obtain basis probabilities p(s)p(s) for s∈0,…,2n−1s∈\0,…,2^n-1\. 6: if C=2C=2 and parity mapping then 7: Pi,0←∑s:parity(s)=0p(s)P_i,0← _s:\,parity(s)=0p(s); Pi,1←∑s:parity(s)=1p(s)P_i,1← _s:\,parity(s)=1p(s). 8: else⊳ direct mapping 9: Pi,c←p(c)P_i,c← p(c) for c=0,…,C−1c=0,…,C-1 and renormalize over c. 10: end if 11:end for 12:return P. 8.1.2 SPSA optimization with minibatching, gradient clipping, and early stopping We optimize VQC parameters using SPSA, requiring only two loss evaluations per iteration. Minibatching reduces computational cost in large datasets while preserving stable updates. Algorithm 18 trains VQC parameters using stochastic perturbation-based gradient estimation with minibatching, gradient clipping, and early stopping for stable and efficient optimization. Algorithm 18 SPSA + minibatch VQC training (statevector forward) 1:Training angles A(tr)A^(tr) and labels y(tr)y^(tr); circuit builder Algorithm 15; reps L; steps T; batch size B; SPSA constants (a,c,α,γ)(a,c,α,γ); clip norm GmaxG_ ; early-stop patience p; tolerance δ. 2:Trained parameters θ⋆θ and training record. 3:Initialize θ∼(0,σ2)θ (0,σ^2). 4:best_loss←+∞best\_loss←+∞; no_improve←0no\_improve← 0. 5:for t=1t=1 to T do 6: at←a/tαa_t← a/t^α; ct←c/tγc_t← c/t^γ. 7: Sample minibatch indices ℬB of size B (without replacement). 8: Draw Rademacher perturbation Δ∈−1,+1|θ| ∈\-1,+1\^|θ|. 9: θ+←θ+ctΔθ^+←θ+c_t ; θ−←θ−ctΔθ^-←θ-c_t . 10: Compute f+←CE(θ+;ℬ)f^+ (θ^+;B) and f−←CE(θ−;ℬ)f^- (θ^-;B) using Algorithm 17. 11: SPSA gradient estimate: g^←f+−f−2ct⋅Δ−1 g← f^+-f^-2c_t· ^-1 (elementwise). 12: Clip g^←g^⋅min1,Gmax/‖g^‖2 g← g· \1,G_ /\| g\|_2\. 13: Update θ←θ−atg^θ←θ-a_t g. 14: mb_loss←(f++f−)/2mb\_loss←(f^++f^-)/2. 15: if mb_loss<best_loss−δmb\_loss<best\_loss-δ then 16: best_loss←mb_lossbest\_loss← mb\_loss; no_improve←0no\_improve← 0. 17: else 18: no_improve←no_improve+1no\_improve← no\_improve+1. 19: end if 20: if no_improve≥pno\_improve≥ p then 21: break 22: end if 23:end for 24:Compute final loss on an evaluation subset of training samples. 25:return θ⋆←θ ←θ and training record. 8.1.3 VQC artifact persistence and reproducible inference To ensure that VQC results are reproducible and that the trained model can be applied consistently to new records, we persist a complete artifact bundle after training. The saved bundle includes: (i) the finalized circuit structure and learned parameters θ⋆θ , (i) the preprocessing objects required to reconstruct the angle-encoded inputs (training-fitted scaler, clipping bound zmaxz_ , and feature scaling coefficients λ), (i) the decoding configuration (class mapping mode and, for binary tasks, the selected decision threshold τ), and (iv) evaluation summaries (validation/test metrics and confusion matrices). This packaging mirrors the inference-time pipeline exactly and prevents train–test leakage by enforcing that all transforms are derived from training splits only. The persistence procedure is formalized in Algorithm 19. Algorithm 19 VQC artifact saving for reproducible scoring 1:Trained model U(θ⋆;x),θ⋆\U(θ ;x),θ \; scaler S; feature list; clip bound zmaxz_ ; optional threshold τ; metadata (k, hp, scores). 2:Persisted artifact bundle. 3:Save circuit serialization (e.g., QPY); save θ⋆θ vector; save scaler object. 4:Write metadata JSON: feature columns, zmaxz_ , τ, n qubits, reps, reupload, final loss, validation/test metrics. 5:Optionally write an alias pointer for the best model across k. 8.2 Fusion Score Evaluation of Test datasets Additional tables for Fusion score evaluation. Table 6: Dataset characteristics and Fusion-score performance (best-k selected by coordinate descent; α tuned via α-sweep on validation). Dataset N d #classes Class counts Split (train/val/test) best-k α⋆α Val Acc Val F1macro_macro Heart Disease 918 11 2 (0:410, 1:508) 516/172/230 3 1.00 0.8198 0.8184 Breast Cancer (Wisconsin) 569 30 2 (B:357, M:212) 319/107/143 3 0.90 0.9159 0.9052 Wine Quality (3-class) 178 13 3 (1:59, 2:71, 3:48) 99/34/45 4 0.85 0.9412 0.9428 Table 7: Test-set Fusion-score performance at the selected best-k and α⋆α . Dataset Test Acc Test F1macro_macro Heart Disease 0.8478 0.8463 Breast Cancer (Wisconsin) 0.8881 0.8703 Wine Quality (3-class) 0.9556 0.9522 Table 8: Fusion-score confusion matrices (rows=true, cols=pred) for the selected best-k per dataset. Dataset Split Confusion matrix Heart Disease Val [63141778] bmatrix63&14\\ 17&78 bmatrix Heart Disease Test [861718109] bmatrix86&17\\ 18&109 bmatrix Breast Cancer (Wisconsin) Val [670931] bmatrix67&0\\ 9&31 bmatrix Breast Cancer (Wisconsin) Test [9001637] bmatrix90&0\\ 16&37 bmatrix Wine Quality (3-class) Val [10100131009] bmatrix10&1&0\\ 0&13&1\\ 0&0&9 bmatrix Wine Quality (3-class) Test [150001800210] bmatrix15&0&0\\ 0&18&0\\ 0&2&10 bmatrix