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arXiv:2510.04736

Quantum Subgradient Estimation for Conditional Value-at-Risk Optimization

Vasilis Skarlatos, Nikos Konofaos

Published Oct 6, 2025• 1 citation
arXivHTMLPDF

Abstract

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 $O(1/ε^2)$ sample complexity to achieve $ε$-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 $O(1/ε)$ 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.

DOI
10.48550/arxiv.2510.04736

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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.
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.
Risk measures are important key figures to measure the adequacy of the reserves of a company. The most common risk measures in practice are Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). Recently, quantum-based algorithms are introduced to calculate them. These procedures are based on the so-called quantum amplitude estimation algorithm, which leads to a quadratic speed-up compared to classical Monte-Carlo based methods. Based on these ideas, we construct quantum-based algorithms to calculate alternatives for VaR and CVaR, namely the Expectile Value-at-Risk (EVaR) and the Range Value-at-Risk (RVaR). These algorithms are also based on quantum amplitude estimation. In two case studies, we compare their performance with the quantum-based algorithms for VaR and CVaR. We find that all of the algorithms perform sufficiently well on a quantum simulator. Further, the calculations of EVaR and VaR are robust against noise on a real quantum device. This is not the case for CVaR and RVaR.
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.
Value-at-risk (VaR) is a broadly used measure for evaluating financial tail risk, but its estimation depends on assessing threshold exceeding probabilities over complex loss distributions. Probability estimation tasks using quantum algorithms works based on amplitude estimation that has a potential to offer quadratic speedup; however, their implementation on noisy intermediate-scale quantum devices is obstructed by the qubit and circuit depth requirements of arithmetic-heavy comparator oracles. This work proposes a qubit-efficient Boolean quantum oracle for threshold-based risk screening motivated by VaR analysis that has the ability to directly flag loss scenarios that violate a predefined threshold condition without utilizing high depth quantum adders or quantum subtractors. The proposed oracle employs a risk ancilla together with a reusable work qubit and utilizes a shallow circuit depth, making it a feasible candidate for near-term quantum hardware. In this work we analyze the resource scaling of the proposed oracle and benchmark it against arithmetic-based approaches under noisy simulations. This work further shows how the proposed oracle can be integrated within a hybrid quantum–classical variational optimization framework to restrain high-risk configurations.
Abstract Value-at-risk (VaR) is a broadly used measure for evaluating financial tail risk, but its estimation depends on assessing threshold exceeding probabilities over complex loss distributions. Probability estimation tasks using quantum algorithms works based on amplitude estimation that has a potential to offer quadratic speedup; however, their implementation on noisy intermediate-scale quantum devices is obstructed by the qubit and circuit depth requirements of arithmetic-heavy comparator oracles. This work proposes a qubit-efficient Boolean quantum oracle for threshold-based risk screening motivated by VaR analysis that has the ability to directly flag loss scenarios that violate a predefined threshold condition without utilizing high depth quantum adders or quantum subtractors. The proposed oracle employs a risk ancilla together with a reusable work qubit and utilizes a shallow circuit depth, making it a feasible candidate for near-term quantum hardware. In this work we analyze the resource scaling of the proposed oracle and benchmark it against arithmetic-based approaches under noisy simulations. This work further shows how the proposed oracle can be integrated within a hybrid quantum–classical variational optimization framework to restrain high-risk configurations.
Monte Carlo simulations underpin computational quantitative finance, where they are used for derivative pricing, risk quantification and scenario analysis whenever closed-form solutions are unavailable. Their main limitation is the slow O(N^-1/2) convergence of classical sampling. Quantum amplitude estimation promises a quadratic improvement in query complexity over classical Monte Carlo, reducing the sample requirement from O(eps^-2) to O(eps^-1) under idealised conditions. This thesis measures the extent to which the quadratic improvement in query complexity survives a concrete risk task under gate noise when compared to classical Monte Carlo. A four-obligor correlated portfolio, whose loss distribution can be computed exactly, is encoded into quantum circuits, and its value at risk and expected shortfall are estimated by iterative amplitude estimation under exactly simulated depolarising noise on every two-qubit gate. Six configurations of 11 to 15 qubits vary the portfolio, the factor grid and the arithmetic construction. At matched loss-model evaluation budgets and without noise, amplitude estimation reduces the RMSE of the tail probability against classical Monte Carlo by a factor of 13.6 at the largest budget tested. The error budget is dominated by the circuit encoding, not the estimator. As a result, the value at risk at the 95% level picks the wrong loss level in none of the noiseless repetitions, while the encoding understates expected shortfall by 0.177 loss units. Under noise, the signal decays with every amplification round. The number of rounds that still help falls from 64 at an error rate of 3x10^-6 to none at 3x10^-2. This work concludes that amplification benefits only while the product of gate count and error rate stays small against one. Because every obligor added enlarges the circuit, the error rate at which amplification remains useful falls as the portfolio grows, and advantage starts depending on circuit compression instead of estimator design.