Quantum-Inspired Geometric Classification with Correlation Group Structures and VQC Decision Modeling

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.

• The paper presents a hybrid classification pipeline that combines correlation group structures with a SWAP-test similarity estimator to address high-dimensional and imbalanced data challenges.
• It implements a two-level decision mechanism where a fusion-margin classifier is selectively enhanced by a Variational Quantum Classifier, improving nonlinear refinement.
• Experimental results show competitive accuracy on standard benchmarks and robust minority recall in ultra-imbalanced regimes, highlighting its practical deployment potential.

Quantum-Inspired Geometric Classification: Correlation Group Structures and VQC Decision Modeling

Overview

The paper "Quantum-Inspired Geometric Classification with Correlation Group Structures and VQC Decision Modeling" (2604.01930) presents a quantum-inspired, geometry-driven classification methodology tailored for heterogeneous tabular data across both balanced and imbalanced regimes. The key innovations include: (1) Correlation Group Structures (CGR) for feature grouping, (2) a compact SWAP-test-based similarity estimator, and (3) a hybrid decision mechanism in which a data-efficient non-probabilistic fusion-margin classifier is augmented by a selective Variational Quantum Classifier (VQC) operating on prototypical margin deltas.

Problem Setting and Motivation

Classical classifiers, especially in high dimensions or under severe class imbalance, often suffer from distance concentration, unreliability of standard metrics, and non-stationary data distributions. Prototype-driven schemes (notably medoid-based) offer robustness and interpretability by evaluating samples with respect to representative class prototypes, but are highly sensitive to the metric and feature space geometry. The authors approach these challenges by:

• Leveraging quantum-inspired primitives (notably the SWAP test) to derive overlap-based geometric channels (distance and angle).
• Introducing a feature grouping protocol (CGR) that encodes latent correlation structure and facilitates robust, nonlinear groupwise representation.
• Embedding these primitives into a two-level decision pipeline that remains interpretable and data-efficient for small data regimes, and scalable to large, complex, and highly imbalanced settings.

Methodology

Correlation Group Structures (CGR)

CGR encodes intra-feature dependencies by partitioning features into anchor-centric, correlation-weighted neighborhoods. Each anchor $a$ forms a group by selecting its top-$m$ correlated features (on a sign-agnostic basis), allowing feature representation through nonlinear, correlation-amplified $\ell_2$ aggregates. This process yields a nonlinear embedding retaining interpretable local interaction effects and mitigating distortions from unstructured high-dimensional spaces.

Quantum-Inspired Similarity via Compact SWAP Test

CGR embeddings are compared to class medoids using a resource-efficient SWAP-test circuit, yielding both Euclidean-like distances and angular (cosine-derived) separations. The quantum feature preparation is designed so that both magnitude and directionality components are available—crucial in capturing the heterogeneous signal structure of tabular data. When normalization fails, classical fallback ensures the method’s robustness.

Fusion Score: Geometric Non-Probabilistic Classifier

Dissimilarity signals to all class medoids are normalized and aggregated via a convex fusion of Euclidean and angular terms (with convex weights $\alpha, \beta$), producing per-class margin-like scores. Classification proceeds via $\arg\min$s, eschewing probability calibration. This strictly geometric margin-based inference is both highly interpretable and robust, particularly suited for small or moderate-size datasets, where overparameterization or deep capacity models are unwarranted.

Variational Quantum Classifier (VQC) and ∆-Distance Refinement

For complex, large-scale, and highly imbalanced cases (e.g., the Credit Card Fraud domain), the fusion-only regime underfits due to nonlinear interactions and subtle geometric separability. Here, the methodology constructs $\Delta$-distance features: class-contrastive, normalized differences in the geometric similarity channels. The VQC is trained on these low-dimensional compact features, learning a nonlinear refinement that corrects for fusion underfitting in complex margin regimes. Quantum data encoding is performed via angle mapping (with scaling and clipping), and a low-depth, optionally data-reuploaded circuit is optimized with minibatched SPSA.

Experimental Results

Empirical evaluation spans both standard benchmarks (Heart Disease, Breast Cancer, Wine Quality) and a large, ultra-imbalanced rare-event dataset (Credit Card Fraud).

• On small/moderate datasets: The fusion score is competitive with classical baselines, achieving $0.8478$, $0.8881$, $0.9556$ accuracy and $0.8463$, $m$0, $m$1 macro-F1 on Heart Disease, Breast Cancer, and Wine Quality, respectively. The method’s performance is stable across class partitions.
• On Credit Card Fraud (rare-event regime, 0.17% prevalence): The ∆+VQC pipeline attains $m$2 minority recall at a $m$3 alert rate, ROC-AUC of $m$4, and PR-AUC of $m$5. Despite not matching tree ensembles or heavily tuned probabilistic baselines in precision-recall AUC, the quantum-inspired framework maintains strong recall under operational alert constraints, validating its practical relevance as a geometric rare-event detector.

The authors emphasize evaluation under deployment-driven alert rate matching rather than default-threshold accuracy or conventional probability cutoffs.

Theoretical and Practical Implications

• Interpretability and Robustness: The fusion approach grounds decisions in explicit, class-referenced geometric quantities, improving transparency in domains sensitive to black-box prediction (e.g., healthcare, risk).
• Data Regime Adaptivity: The decoupling of primary geometric fusion from nonlinear quantum refinement enables regime-aware deployment—using only the necessary model complexity as dictated by class structure and data scale.
• Quantum Compatibility: By structuring the pipeline so that the only truly quantum steps are compact, potentially log-dimension, and low-depth (SWAP tests and VQC), the proposal is amenable to near-term quantum hardware, pending further advances.
• Calibration and Operational Evaluation: The methodology foregrounds operating-point-aware analysis—essential in rare-event detection—highlighting the risks of superficial threshold-based evaluation metrics.

Future Directions

The work motivates several lines for further research:

• Quantum hardware execution with noise-aware analysis, especially for the SWAP-based similarity engine and the VQC.
• More principled regularization and stability controls for CGR configuration and fusion-channel calibration, potentially leveraging Bayesian or information-theoretic criteria.
• Handling class-prior shift and distribution drift, especially for VQC training under evolving datasets.
• Expansion to broader non-PCA tabular and real-world heterogeneous datasets, further benchmarking the hybrid geometric paradigm’s generality.

Conclusion

This paper establishes a regime-adaptive, quantum-inspired geometric classification pipeline combining interpretable prototype-based similarity with correlation-encoded feature groupings and selective quantum nonlinear refinement. The framework achieves robust, competitive accuracy in small-to-moderate regimes—even under severe class imbalance—and provides a modular, interpretable alternative for deployment in risk-sensitive tabular AI tasks. The hybrid approach, integrating efficient quantum-inspired primitives and a regime-selective decision process, extends the practical plausibility of geometric quantum machine learning for heterogeneous, real-world data (2604.01930).