量子拟蒙特卡洛：预渐近量子优势的一个窗口
Quantum Quasi-Monte Carlo: a window for pre-asymptotic quantum advantage
- Centre for Quantum Technologies, NUS（新加坡国立大学量子技术中心）
- Standard Chartered Singapore, SCB-Singapore（渣打银行新加坡）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究提出量子拟蒙特卡洛算法，结合低差异网与量子振幅估计，探索预渐近量子优势窗口，可在有限函数查询下以更少计算达到目标精度，适用于实际计算场景。
AI中文摘要：
蒙特卡洛方法的数值积分是金融衍生品定价、风险管理等众多科学与工业应用中的核心计算任务。经典蒙特卡洛算法计算开销大：达到精度ε通常需要O(1/ε²)量级的函数评估次数。基于量子振幅估计的量子加速蒙特卡洛方法原则上可使这一依赖关系获得二次提升，但拟蒙特卡洛方法尚未在量子语境中被探索。本研究提出一种量子拟蒙特卡洛算法，将低差异网与量子振幅估计相结合，该方法以叠加态相干地制备拟随机点集。该方法未在渐近层面优于经典拟蒙特卡洛，因为总误差分为有限网决定的离散化误差和量子估计误差；转而探索预渐近优势窗口：对于经典需2^q个低差异点的目标精度，可在叠加态中制备尺寸为2^Q（Q>q）的更高分辨率网，用少得多的函数查询达到相同精度。该窗口可通过调节电路分辨率和振幅估计参数控制，使该方法适用于查询数有限而非渐近大的实际场景。
英文摘要：
Numerical integration with Monte Carlo methods is a central computational task in many scientific and industrial applications, including financial derivative pricing and risk management. Classical Monte Carlo algorithms are computationally demanding: achieving an accuracy $ε$ typically requires a number of function evaluations scaling as $O(1/ε^2)$. Quantum-accelerated Monte Carlo methods based on quantum amplitude estimation can in principle quadratically improve this dependence. However, \textit{quasi}-Monte Carlo methods have not been explored in the quantum context. In this work, we introduce a quantum quasi-Monte Carlo algorithm that combines low-discrepancy nets with quantum amplitude estimation. The proposed method prepares the quasi-random point set coherently in superposition. The method does not yield an asymptotic improvement over classical quasi-Monte Carlo, since the total error separates into a discretization error, determined by the finite net, and a quantum estimation error. Instead, we explore a pre-asymptotic advantage window: for a target accuracy that would classically require $2^q$ low discrepancy points, one can prepare a higher-resolution net of size $2^Q$, with $Q>q$, in superposition and reach the same accuracy using significantly fewer function queries. This window can be controlled by tuning the circuit resolution and amplitude-estimation parameters, making the approach relevant for practical regimes where the number of queries is finite rather than asymptotically large.
