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Optimal Quantum Speedups for Repeatedly Nested Expectation Estimation

Yihang Sun, Guanyang Wang, Jose Blanchet

TL;DR

This work delivers a near-optimal quantum algorithm for estimating repeatedly nested expectations with a constant horizon, extending quantum speedups from single nesting to the full RNE setting relevant for optimal stopping. The authors introduce a derandomized MLMC framework to replace the variable runtime of classical rMLMC, enabling stable quantum acceleration via Quantum-Accelerated Monte Carlo (QAMC). By carefully balancing deterministic level scheduling and quantum mean estimation, they prove RMSE guarantees with total cost scaling as \\tilde{O}(\\varepsilon^{-1}) (up to logarithmic factors), which is optimal up to lower-order terms. The results provide a broad, hardware-agnostic pathway to quantum speedups in risk estimation, probabilistic programming, and optimal stopping, with explicit notational and algorithmic structures that generalize previous quantum speeds for a single nesting to multiple nestings.

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.

Optimal Quantum Speedups for Repeatedly Nested Expectation Estimation

TL;DR

This work delivers a near-optimal quantum algorithm for estimating repeatedly nested expectations with a constant horizon, extending quantum speedups from single nesting to the full RNE setting relevant for optimal stopping. The authors introduce a derandomized MLMC framework to replace the variable runtime of classical rMLMC, enabling stable quantum acceleration via Quantum-Accelerated Monte Carlo (QAMC). By carefully balancing deterministic level scheduling and quantum mean estimation, they prove RMSE guarantees with total cost scaling as \\tilde{O}(\\varepsilon^{-1}) (up to logarithmic factors), which is optimal up to lower-order terms. The results provide a broad, hardware-agnostic pathway to quantum speedups in risk estimation, probabilistic programming, and optimal stopping, with explicit notational and algorithmic structures that generalize previous quantum speeds for a single nesting to multiple nestings.

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 -error with cost , 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.
Paper Structure (16 sections, 14 theorems, 64 equations, 6 algorithms)

This paper contains 16 sections, 14 theorems, 64 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Setup
  4. Main Results and Organization
  5. Notations
  6. Classical Algorithms
  7. Truncated rMLMC
  8. Derandomized MLMC
  9. Quantum Algorithms
  10. Quantum-Accelerated Monte Carlo
  11. The Direct Quantization
  12. Deterministic Level Scheduling
  13. Proof of Auxiliary Lemmas
  14. Proof of \ref{['prop:algo-2']}
  15. Proof of \ref{['prop:algo-3']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Under LBL assumptions (def:lbl) and any $\delta \in (0, 1/2)$, for each $0\le d\le D$, there exists $r_d\in (0, 1/2)$ such that given $\pi$-a.e. history $y_{<d}$ and $\mathbb{P}_d=\operatorname{Geo}(r_d)$, Algorithm 1 satisfies

Theorems & Definitions (20)

  • Theorem 1.1: Theorem 2.4 in rne
  • Corollary 1.2: Corollary 2.5 in rne
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Theorem 2.4: von Bahr-Esseen
  • proof : Proof Sketch of \ref{['prop:algo-2']}.
  • ...and 10 more