We study the estimation of repeatedly nested expectations (RNEs) with a constant horizon (number of nestings) using quantum computing. We propose a quantum algorithm that achieves $\varepsilon$-error with cost $\tilde O(\varepsilon^{-1})$, up to logarithmic factors. Standard lower bounds show this scaling is essentially optimal, yielding an almost quadratic speedup over the best classical algorithm. Our results extend prior quantum speedups for single nested expectations to repeated nesting, and therefore cover a broader range of applications, including optimal stopping. This extension requires a new derandomized variant of the classical randomized Multilevel Monte Carlo (rMLMC) algorithm. Careful de-randomization is key to overcoming a variable-time issue that typically increases quantized versions of classical randomized algorithms.

翻译：我们研究了利用量子计算对具有恒定层数（嵌套次数）的重复嵌套期望（RNEs）进行估计的问题。我们提出了一种量子算法，该算法能以$\tilde O(\varepsilon^{-1})$的成本（忽略对数因子）达到$\varepsilon$误差。标准下界表明该缩放比例本质上是最优的，相比最佳经典算法实现了近乎二次方的加速。我们的结果将先前针对单层嵌套期望的量子加速推广到了重复嵌套情形，从而覆盖了更广泛的应用，包括最优停止问题。这一推广需要一种新的、经过去随机化处理的经典随机化多级蒙特卡洛（rMLMC）算法变体。细致的去随机化是克服经典随机化算法量子化版本中通常存在的变时问题的关键。