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.

翻译：条件风险价值（CVaR）是金融领域领先的尾部风险度量指标，在监管框架和投资组合优化中均占据核心地位。CVaR及其梯度的经典估计依赖于蒙特卡洛模拟，实现ε精度需要O(1/ε²)的样本复杂度。在本研究中，我们基于振幅估计设计并分析了一种用于CVaR最小化的量子次梯度预言机。通过三方命题，我们证明即使需要估计风险价值（VaR）阈值本身，CVaR次梯度也能以O(1/ε)的量子查询次数进行估计。我们进一步量化了VaR阶段估计误差向CVaR梯度的传播，并推导了使用该预言机的随机投影次梯度下降法的收敛速率。我们的分析表明，相比经典蒙特卡洛方法，查询复杂度实现了近二次方的改进。使用模拟量子电路的数值实验验证了理论速率，并展示了对于阈值估计噪声的稳健性。这是首次对面向尾部风险最小化的量子次梯度方法进行严格的复杂度分析。