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.
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.
We present a rigorous mathematical proof of concept demonstrating the theoretical feasibility of quantum linear regression through variational Hamiltonian formulation. Our framework systematically transforms classical linear regression into a quantum optimization problem by encoding regression weights as expectation values of Pauli-Z operators and reformulating the loss function as a quantum Hamiltonian. We provide explicit mathematical constructions, matrix representations, and convergence analysis to establish the theoretical foundation for quantum machine learning applications. Unlike previous quantum linear regression approaches that focus on exponential speedups through quantum linear systems solvers, our variational approach is designed for near-term quantum devices and naturally incorporates regularization through quantum parameter constraints. The mathematical framework demonstrates computational equivalence between classical and quantum approaches while revealing inherent advantages in constrained optimization scenarios. This work provides the theoretical groundwork for future empirical applications to financial time series prediction, including potential implementation on Thai SET Index forecasting using data from Yahoo Finance and quantum computing platforms. Our key contributions include: (1) complete mathematical mapping from classical to quantum regression formulation, (2) explicit Hamiltonian constructions with matrix representations, (3) theoretical convergence analysis for quantum gradient descent, and (4) identification of quantumspecific regularization properties that distinguish this approach from classical methods.
We define a local Riemannian metric tensor in the manifold of Gaussian channels and the distance that it induces. We adopt an information-geometric approach and define a metric derived from the Bures-Fisher metric for quantum states. The resulting metric inherits several desirable properties from the Bures-Fisher metric and is operationally motivated by distinguishability considerations: It serves as an upper bound to the attainable quantum Fisher information for the channel parameters using Gaussian states, under generic constraints on the physically available resources. Our approach naturally includes the use of entangled Gaussian probe states. We prove that the metric enjoys some desirable properties like stability and covariance. As a by-product, we also obtain some general results in Gaussian channel estimation that are the continuous-variable analogs of previously known results in finite dimensions. We prove that optimal probe states are always pure and bounded in the number of ancillary modes, even in the presence of constraints on the reduced state input in the channel. This has experimental and computational implications. It limits the complexity of optimal experimental setups for channel estimation and reduces the computational requirements for the evaluation of the metric: Indeed, we construct a converging algorithm for its computation. We provide explicit formulas for computing the multiparametric quantum Fisher information for dissipative channels probed with arbitrary Gaussian states and provide the optimal observables for the estimation of the channel parameters (e.g., bath couplings, squeezing, and temperature).
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.
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.
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.
Quantum state tomography faces exponential scaling with system size, while recent neural network approaches achieve polynomial scaling at the cost of losing the geometric structure of quantum state space. We introduce geometric latent space tomography, combining classical neural encoders with parameterized quantum circuit decoders trained via a metric-preservation loss that enforces proportionality between latent Euclidean distances and quantum Bures geodesics. On two-qubit mixed states with purity 0.85--0.95 representing NISQ-era decoherence, we achieve high-fidelity reconstruction (mean fidelity F=0.942±0.03) with an interpretable 20-dimensional latent structure. Critically, latent geodesics exhibit strong linear correlation with Bures distances (Pearson r=0.88, R2=0.78), preserving 78\% of quantum metric structure. Geometric analysis reveals intrinsic manifold dimension 6.35 versus 20 ambient dimensions and measurable local curvature (κ=0.011±0.006), confirming non-trivial Riemannian geometry with O(d2) computational advantage over O(4n) density matrix operations. Unlike prior neural tomography, our geometry-aware latent space enables direct state discrimination, fidelity estimation from Euclidean distances, and interpretable error manifolds for quantum error mitigation without repeated full tomography, providing critical capabilities for NISQ devices with limited coherence times.