Predicting customer churn from transactional data is a central problem in management science, with direct implications for retention strategy, revenue forecasting, and resource allocation. This paper introduces Quantum Geometric-Entropic Optimization (Q-GEO), a framework that integrates Geometric-Entropic Optimization – combining Riemannian gradient methods with entropy-regularized optimal transport – into the training of variational quantum kernels for classification. The algorithm operates on a parameter manifold equipped with a Fisher-Wasserstein metric and incorporates Sinkhorn-type projections to enforce distributional coherence on the quantum feature space. We establish three theoretical contributions: (i) a convergence theorem for Q-GEO-trained variational quantum kernels under a combined Polyak–Łojasiewicz and Sinkhorn contraction framework, yielding linear convergence in the Riemannian condition number plus geometric contraction of the Sinkhorn residual; (ii) a margin amplification result showing that GEO-trained quantum embeddings achieve improved separation bounds over Euclidean-trained counterparts due to the spectral regularization provided by the Wasserstein component of the Fisher-Wasserstein metric; and (iii) a distributional stability result proving that Sinkhorn-projected quantum kernel matrices preserve a doubly stochastic spectral structure that mitigates kernel collapse in imbalanced settings. We validate the framework on the UCI Online Retail II dataset (N=5,942 customers, d=11 RFM-extended features, churn rate ≈37%), a publicly available transactional benchmark. Under nested cross-validation, Q-GEO achieves 0.8614 accuracy, 0.8103 precision, 0.7891 recall, 0.7996 F1, and 0.9047 ROC AUC, outperforming both classical baselines (including logistic regression, random forest, XGBoost, and CatBoost) and standard variational quantum kernel methods. We complement the accuracy analysis with SHAP-based explainability, computation time comparisons, and a detailed feature engineering appendix to support interpretability and reproducibility. We interpret these results as evidence that geometric optimization principles can materially enhance quantum machine learning for management science applications.
Predicting customer churn from transactional data is a central problem in management science, with direct implications for retention strategy, revenue forecasting, and resource allocation. This paper introduces Quantum Geometric-Entropic Optimization (Q-GEO), a framework that integrates Geometric-Entropic Optimization – combining Riemannian gradient methods with entropy-regularized optimal transport – into the training of variational quantum kernels for classification. The algorithm operates on a parameter manifold equipped with a Fisher-Wasserstein metric and incorporates Sinkhorn-type projections to enforce distributional coherence on the quantum feature space. We establish three theoretical contributions: (i) a convergence theorem for Q-GEO-trained variational quantum kernels under a combined Polyak–Łojasiewicz and Sinkhorn contraction framework, yielding linear convergence in the Riemannian condition number plus geometric contraction of the Sinkhorn residual; (ii) a margin amplification result showing that GEO-trained quantum embeddings achieve improved separation bounds over Euclidean-trained counterparts due to the spectral regularization provided by the Wasserstein component of the Fisher-Wasserstein metric; and (iii) a distributional stability result proving that Sinkhorn-projected quantum kernel matrices preserve a doubly stochastic spectral structure that mitigates kernel collapse in imbalanced settings. We validate the framework on the UCI Online Retail II dataset (N=5,942 customers, d=11 RFM-extended features, churn rate ≈37%), a publicly available transactional benchmark. Under nested cross-validation, Q-GEO achieves 0.8614 accuracy, 0.8103 precision, 0.7891 recall, 0.7996 F1, and 0.9047 ROC AUC, outperforming both classical baselines (including logistic regression, random forest, XGBoost, and CatBoost) and standard variational quantum kernel methods. We complement the accuracy analysis with SHAP-based explainability, computation time comparisons, and a detailed feature engineering appendix to support interpretability and reproducibility. We interpret these results as evidence that geometric optimization principles can materially enhance quantum machine learning for management science applications.
Abstract We present a rigorous mathematical analysis of the Quantum-Inspired Local Tensor Regression (QILTR) framework for modeling non-stationary, multi-linear systems. QILTR's central hypothesis is that weighting local regression problems by the Bures distance—a metric derived from the Quantum Fisher Information (QFI)—improves optimization properties. We formalize this by recasting QILTR as a Riemannian optimization problem on the manifold of quantum states, establishing that the QILTR update approximates a natural gradient descent step measured by the QFI metric. Under local geodesic convexity, we derive a formal convergence rate and demonstrate that QFI-based weighting produces a better-conditioned local Hessian matrix compared to standard Euclidean kernels. We analyze low-rank tensor estimation via weighted Alternating Least Squares (ALS), establishing the regularizing properties of quantum-geometric weights in this non-convex setting, and prove computational tractability with polynomial-time complexity per centroid under dual constraints of low-dimensional quantum encoding and low-rank coefficient tensors. Comprehensive empirical validation on synthetic datasets (150 trials, five scenarios) and real-world quantum chemistry data (QM9, ten trials) confirms that the bounded hyperbolic tangent scaling function prevents outlier amplification, yielding a 4.9× improvement. We transparently report that theoretical convergence rate advantages are not empirically observable, as both QILTR and Euclidean-LTR converge rapidly. Practical benefits manifest as improved solution quality (+1.22% on QM9). By grounding QILTR in information geometry with empirical rigor, we position it as a geometrically optimized and computationally viable scientific machine learning tool with validated design principles and realistic expectations.
Non-convex stochastic optimization presents fundamental mathematical challenges across machine learning, wireless networks, data center resource allocation, and optical wireless communication systems, where complex loss landscapes with multiple local minima and saddle points impede classical variational inference methods. This paper introduces the Quantum-Inspired Variational Inference (QIVI) framework, which systematically integrates quantum mechanical principles (superposition, entanglement, and measurement operators) into classical variational inference through rigorous mathematical formulations grounded in Hilbert space theory and operator algebras. We develop a unified optimization framework that encodes classical parameters as quantum-inspired states within finite-dimensional complex Hilbert spaces, employing unitary evolution operators and adaptive basis selection governed by gradient covariance eigendecomposition. The core mathematical contribution establishes that QIVI achieves a convergence rate of O(log2T/T1/2) for σ-strongly non-convex functions, provably improving upon the classical O(T−1/4) rate, yielding a theoretical speedup factor of 1.85–1.96×. Comprehensive experiments across synthetic benchmarks, Bayesian neural networks, and real-world applications in network optimization and financial portfolio management demonstrate 23–47% faster convergence, 15–35% superior objective values, and 28–46% improved uncertainty calibration. The principal contributions include: (i) a rigorous Hilbert space-based mathematical framework for quantum-inspired variational inference grounded in operator algebras, (ii) a novel hybrid quantum–classical algorithm (QIVI) with adaptive basis selection via gradient covariance eigendecomposition, (iii) formal convergence proofs establishing provable improvement over classical methods, (iv) comprehensive empirical validation across diverse problem domains relevant to machine learning and network optimization, and (v) demonstration of the framework’s applicability to optimization problems arising in wireless networks, data center resource allocation, and network system design. Statistical validation using the Friedman test (χ2=847.3, p<0.001) and post hoc Wilcoxon signed-rank tests with Holm–Bonferroni correction confirm that QIVI’s improvements over all baseline methods are statistically significant at the α=0.05 level across all benchmark categories. The framework discovers 18.1 out of 20 true modes in multimodal distributions versus 9.1 for classical methods, demonstrating the potential of quantum-inspired optimization approaches for challenging stochastic problems arising in machine learning, wireless communication, and network optimization.
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.
Quantum neural networks (QNNs) have attracted growing interest for scientific machine learning, yet in regression settings they often suffer from limited trainability under noisy gradients and ill-conditioned optimization. We propose a hybrid quantum--classical regression framework designed to mitigate these bottlenecks. Our model prepends a lightweight classical embedding that acts as a learnable geometric preconditioner, reshaping the input representation to better condition a downstream variational quantum circuit. Building on this architecture, we introduce a curriculum optimization protocol that progressively increases circuit depth and transitions from SPSA-based stochastic exploration to Adam-based gradient fine-tuning. We evaluate the approach on PDE-informed regression benchmarks and standard regression datasets under a fixed training budget in a simulator setting. Empirically, the proposed framework consistently improves over pure QNN baselines and yields more stable convergence in data-limited regimes. We further observe reduced structured errors that are visually correlated with oscillatory components on several scientific benchmarks, suggesting that geometric preconditioning combined with curriculum training is a practical approach for stabilizing quantum regression.
Quantum machine learning on near-term noisy quantum devices has generated substantial theoretical interest, but rigorous empirical comparisons under realistic noise on practically relevant data remain scarce. This paper compares two paradigmatic quantum learning models, a Quantum Support Vector Machine (QSVM) built on the ZZFeatureMap quantum kernel and a Variational Quantum Classifier (VQC) with an EfficientSU2/RealAmplitudes ansatz, against tuned classical baselines (SVM with four kernels, Random Forest, XGBoost, LightGBM and CatBoost) on the ULB Credit Card Fraud dataset (284,807 transactions, 0.17% fraud). All models share an identical 4-qubit PCA-reduced feature space, evaluated on the full unbalanced test fold over 15 fits (3 folds × 5 seeds) and reported as mean ± standard deviation with bootstrap confidence intervals, AUPRC as the primary metric. Noise robustness is assessed under depolarizing noise p∈{0,0.001,0.01,0.05}, with ranking preservation measured directly through Spearman ρ and Kendall τ between the noisy and noiseless decision scores rather than read off AUPRC, alongside the per-paradigm computational cost. At four qubits the classical baselines lead (AUPRC 0.60 to 0.74, CatBoost best), above the VQC (0.494) and the QSVM (0.240); the controlled QSVM-versus-RBF–SVM comparison puts the cost of the quantum kernel at about 0.45 AUPRC. Under noise the QSVM keeps its score ranking (ρ=0.998 at p=0.001, 0.906 at p=0.01) and an operational decision threshold (recall 0.87 to 0.89, stable calibration), while the VQC AUPRC peaks non-monotonically at p=0.01 (0.494 rising to 0.654, then easing to 0.569 at p=0.05) even as its ranking decays monotonically (ρ from 0.72 to near zero), so average precision on its own misrepresents how noise affects it. The quantum models do not surpass the tuned classical reference at four qubits; the contribution is methodological: under noise, AUPRC has to be read together with a genuine rank statistic, because the two can move in opposite directions.
Determining the fundamental limits of nonlinear functionals of quantum measurement statistics is a crucial yet generally intractable nonconvex optimization problem. We introduce a generic support-function-based outer-approximation framework for solving concave-minimization (or convex-maximization) problems over the quantum state space. By mapping the problem onto a reduced Z space, we characterize the exact quantum boundary through supporting half space derived from the largest eigenvalues of effective observables. This yields an effective method that produces tight bounds for general measurements in finite-dimensional quantum systems with preassigned numerical precision. As an initial application, we recover the exact variance-based uncertainty relations of [PRL \textbf{119}, 170404 (2017)] and efficiently compute optimal entropic uncertainty relations (EURs). Our results reveal that standard analytical and majorization-based EUR bounds are fundamentally loose for generic measurements, and we show that the resulting exact bounds directly enhance quantum steering detection under asymmetric settings. We further apply the framework to determine the maximal athermality resource certifiable from a restricted measurement scenario. Our method thus provides a universal computational tool for exploring the boundaries of quantum state space and the limits of quantum resources.