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Quantum Information and Computation

Hybrid Circuit–Spintronic Quantum Framework for Financial Risk Analysis with QCVaR Estimation Using Variational Quantum Algorithms and Maximum-Likelihood Amplitude Estimation

S. S. Gayathri, P. Muthulakshmi, R. Palanivel

Published Mar 1, 2026• 0 citations
Publisher

Abstract

Abstract Problem Statement Accurate estimation of extreme financial risks, such as Conditional Value-at-Risk (CVaR) at high confidence levels ( α ≥ 0.95), poses significant computational challenges for classical Monte Carlo methods, which require O (1/ ϵ 2 ) samples and struggle to scale under rare-event scenarios. Methodology To address this, we propose a hybrid variational quantum–spintronic framework integrating Variational Quantum State Preparation (VQA), a threshold comparator oracle, and Maximum-Likelihood Amplitude Estimation (MLAE) to enable quantum-accelerated CVaR estimation suitable for NISQ devices. Results Using only 6 qubits and K = 6 amplification levels, our approach achieves CVaR ^ 0.95 = 0.281 ± 0.012 {\widehat {{\rm{CVaR}}}_{0.95}} = 0.281 \pm 0.012 (error < 1.1% compared to classical benchmarks), with tail probability p ^ = 0.0478 ± 0.0009 \hat p = 0.0478 \pm 0.0009 and state fidelity 0.967 on the ibm_brisbane backend. MLAE reduces circuit depth by 3.3 × relative to canonical QAE, and a conceptual spintronic implementation achieves ∼2.0 fJ/gate, corresponding to an 80% energy reduction versus CMOS. Contributions This work introduces the first NISQ-compatible, energy-efficient quantum–spintronic pipeline for regulatory-compliant financial risk analytics, providing a quadratic sampling speedup, accurate tail-risk estimation, and a pathway toward sustainable quantum finance.

DOI
10.2478/qic-2026-0002

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We present a rigorously derived Quantum Risk Management Framework (QRMF) integrating quantum amplitude estimation for tail risk metrics, entanglement-based systemic risk analytics, and variational quantum risk optimization. Our comprehensive approach encompasses three novel modules: (i) Quantum Risk Amplitude Estimation (QRAE) achieving quadratic sample complexity improvements over classical Monte Carlo for Value-at-Risk (VaR) and Conditional VaR (CVaR) computation through binary search with amplitude estimation subroutines; (ii) Quantum Systemic Risk Analysis (QSRA) encoding financial exposure networks into entangled bipartite states with purity-based early warning observables and QAOA partitioning for cascade mitigation; and (iii) Variational Quantum Risk Optimizer (VQRO) implementing constraint-aware portfolio optimization using risk-adjusted Hamiltonians with feasible-state ansätze and penalty-based formulations. We establish formal complexity theorems under explicit QRAM/QROM cost models, prove convergence guarantees for variational components under convex surrogate conditions, and demonstrate theoretical correspondence between entanglement purity responses and classical contagion path-sum sensitivities in linearized network models. The hybrid framework combines gate-model quantum circuits for amplitude estimation and entanglement encoding with variational quantum eigensolvers for optimization, validated through comprehensive simulations on synthetic financial networks and real market data. Extensive ablation studies on circuit depth versus accuracy trade-offs, noise resilience analysis, and resource scaling complement theoretical results, with all quantum amplitude estimation and entanglement simulations performed using Qiskit on classical hardware while baseline classical Monte Carlo and network centrality methods provide benchmarking baselines. This distinction ensures accurate interpretation of quantum advantages across the integrated risk management pipeline.
The analysis of credit risk is crucial for the efficient operation of financial institutions. Quantum Amplitude Estimation (QAE) offers the potential for a quadratic speedup over classical methods used to estimate metrics such as Value at Risk (VaR) and Conditional Value at Risk (CVaR). However, numerous limitations remain in efficiently scaling the implementation of quantum circuits that solve these estimation problems. One of the main challenges is the use of costly and restrictive arithmetic that must be implemented within the quantum circuit. In this paper, we propose using Quantum Singular Value Transformation (QSVT) to significantly reduce the cost of implementing the state preparation operator, which underlies QAE for credit risk analysis. We also present an end-to-end code implementation and the results of a simulation study to validate the proposed approach and demonstrate its benefits.
Conditional Value-at-Risk (CVaR) is a leading tail-risk measure in finance, central to both regulatory and portfolio optimization frameworks. Classical estimation of CVaR and its gradients relies on Monte Carlo simulation, incurring O(1/ε2)O(1/ε^2)O(1/ε2) sample complexity to achieve εεε-accuracy. In this work, we design and analyze a quantum subgradient oracle for CVaR minimization based on amplitude estimation. Via a tripartite proposition, we show that CVaR subgradients can be estimated with O(1/ε)O(1/ε)O(1/ε) quantum queries, even when the Value-at-Risk (VaR) threshold itself must be estimated. We further quantify the propagation of estimation error from the VaR stage to CVaR gradients and derive convergence rates of stochastic projected subgradient descent using this oracle. Our analysis establishes a near-quadratic improvement in query complexity over classical Monte Carlo. Numerical experiments with simulated quantum circuits confirm the theoretical rates and illustrate robustness to threshold estimation noise. This constitutes the first rigorous complexity analysis of quantum subgradient methods for tail-risk minimization.
Conditional Value-at-Risk (CVaR) is a central tail-risk measure in stochastic structural mechanics, yet its accurate evaluation under high-dimensional, spatially correlated material uncertainty remains computationally prohibitive for classical Monte Carlo methods. Leveraging bounded-expectation reformulations of CVaR compatible with quantum amplitude estimation, we develop a quantum-enhanced inference framework that casts CVaR evaluation as a statistically consistent, confidence-constrained maximum-likelihood amplitude estimation problem. The proposed method extends iterative quantum amplitude estimation (IQAE) by embedding explicit maximum-likelihood inference within a rigorously controlled interval-tracking architecture. To ensure global correctness under finite-shot noise and the non-injective oscillatory response induced by Grover amplification, we introduce a stabilized inference scheme incorporating multi-hypothesis feasibility tracking, periodic low-depth disambiguation, and a bounded restart mechanism governed by an explicit failure-probability budget. This formulation preserves the quadratic oracle-complexity advantage of amplitude estimation while providing finite-sample confidence guarantees and reduced estimator variance. The framework is demonstrated on benchmark problems with spatially correlated lognormal Young's modulus fields generated using a Nystrom low-rank Gaussian kernel model. Numerical results show that the proposed estimator achieves substantially lower oracle complexity than classical Monte Carlo CVaR estimation at comparable confidence levels, while maintaining rigorous statistical reliability. This work establishes a practically robust and theoretically grounded quantum-enhanced methodology for tail-risk quantification in stochastic continuum mechanics.