Optimal Quantum Speedups for Repeatedly Nested Expectation Estimation

TL;DR

This paper introduces a quantum algorithm for estimating repeatedly nested expectations with a near-optimal cost of , extending quantum speedups to more complex expectation estimation problems.

Contribution

It presents a new quantum algorithm for nested expectation estimation that achieves near-optimal complexity and extends prior results to multiple nestings, including applications like optimal stopping.

Findings

Quantum algorithm achieves  error with  complexity.

The algorithm extends quantum speedups to repeated nested expectations, broadening application scope.

Lower bounds confirm the near-optimality of the proposed quantum complexity.

Abstract

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.