The mean of a random variable can be understood as a linear functional on the space of probability distributions. Quantum computing is known to provide a quadratic speedup over classical Monte Carlo methods for mean estimation. In this paper, we investigate whether a similar quadratic speedup is achievable for estimating non-linear functionals of probability distributions. We propose a quantum-inside-quantum Monte Carlo algorithm that achieves such a speedup for a broad class of non-linear estimation problems, including nested conditional expectations and stochastic optimization. Our algorithm improves upon the direct application of the quantum multilevel Monte Carlo algorithm introduced by An et al. (2021). The existing lower bound indicates that our algorithm is optimal up polylogarithmic factors. A key innovation of our approach is a new sequence of multilevel Monte Carlo approximations specifically designed for quantum computing, which is central to the algorithm's improved performance.

翻译：随机变量的均值可理解为概率分布空间上的线性泛函。已知量子计算在均值估计问题上相比经典蒙特卡洛方法可实现二次加速。本文研究对于概率分布的非线性泛函估计，是否可实现类似的二次加速。我们提出一种量子嵌套量子蒙特卡洛算法，该算法在包括嵌套条件期望与随机优化在内的广泛非线性估计问题上实现了此类加速。我们的算法改进了An等人（2021）提出的量子多级蒙特卡洛算法的直接应用方式。现有下界表明我们的算法在多项式对数因子范围内是最优的。本方法的核心创新在于专门为量子计算设计的新型多级蒙特卡洛逼近序列，这是算法性能提升的关键所在。