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Quantum speedup of non-linear Monte Carlo problems

Jose Blanchet, Yassine Hamoudi, Mario Szegedy, Guanyang Wang

TL;DR

The paper tackles estimating nonlinear functionals of probability distributions via nested expectations, where standard Monte Carlo methods incur high cost. It introduces Q-NestExpect, a quantum-inside-quantum MLMC algorithm that redefines the multilevel decomposition to leverage quantum acceleration both inside and across levels. Under standard assumptions, it achieves a near-optimal $\tilde{O}(1/\epsilon)$ cost for $\epsilon$-accurate estimation, improving over direct QA-MLMC approaches. The approach broadens the applicability of quantum speedups to nonlinear stochastic problems, with practical implications for Bayesian design, EVPPI, and financial derivatives, while acknowledging current hardware limitations and outlining directions for future work.

Abstract

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.

Quantum speedup of non-linear Monte Carlo problems

TL;DR

The paper tackles estimating nonlinear functionals of probability distributions via nested expectations, where standard Monte Carlo methods incur high cost. It introduces Q-NestExpect, a quantum-inside-quantum MLMC algorithm that redefines the multilevel decomposition to leverage quantum acceleration both inside and across levels. Under standard assumptions, it achieves a near-optimal cost for -accurate estimation, improving over direct QA-MLMC approaches. The approach broadens the applicability of quantum speedups to nonlinear stochastic problems, with practical implications for Bayesian design, EVPPI, and financial derivatives, while acknowledging current hardware limitations and outlining directions for future work.

Abstract

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.

Paper Structure

This paper contains 24 sections, 12 theorems, 37 equations, 1 figure, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Nested expectation
  3. Our contribution
  4. Multilevel Monte Carlo
  5. Classical MLMC
  6. Quantum-accelerated MLMC
  7. Main Algorithm
  8. Roadmap
  9. Our algorithm
  10. Proof Sketch:
  11. Applications
  12. Limitations and Future directions
  13. Proofs for the Main Algorithm
  14. Analysis of the Bl algorithms
  15. MSE:
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $\mu$ denote a quantity that we want to estimate. Suppose for each integer $l\geq 0$, we have an algorithm $\mathcal{A}_l$ that outputs a random variable $\Delta_l$ with variance $V_l$ and computational cost $C_l$. Define $s_l := \sum_{k=0}^l \mathbb{E} \bc{\Delta_k}$ and assume for some $(\alp here $\Delta_l^{(1)}, \Delta_l^{(2)}, \ldots$ are i.i.d. copies of $\Delta_l$ for each $l$.

Figures (1)

  • Figure : Classical MLMC for nested expectation: $\mathcal{A}_l$ ($l\geq 1)$

Theorems & Definitions (19)

  • Theorem 2.1: Theorem 1 in giles2015multilevel
  • Theorem 2.2: Theorem 2 in an2021quantum
  • Proposition 2.3: Lower bound for the general QA-MLMC
  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Lemma 3.4: Median trick
  • Theorem 3.5: Theorem 1.1 in kothari2023mean
  • Lemma 3.6: Informal
  • Lemma 3.7: Informal
  • ...and 9 more