Quantum Speedups for Markov Chain Monte Carlo Methods with Application to Optimization

2025-04-04

AI Summary

This paper proposes the first quantum-accelerated algorithmic framework with rigorous theoretical guarantees for two fundamental computational problems: (1) Markov chain Monte Carlo (MCMC) sampling from exponential-family distributions (π ∝ e⁻ᶠ), and (2) empirical risk minimization (ERM) over nonsmooth approximately convex functions. Methodologically, it integrates quantum stochastic gradient estimation, quantum Gibbs sampling, and phase estimation to design novel quantum variants of Hamiltonian and Langevin Monte Carlo under a stochastic potential oracle and a two-point joint query model. Key contributions include: (1) the first polynomial quantum speedup in dimension *d*, accuracy *ε*, and Lipschitz constant *L*; (2) a two-point synchronous quantum stochastic gradient estimator that substantially reduces gradient/function query complexity; and (3) provable quantum acceleration for nonsmooth ERM, achieving an Ω(√*n*) improvement in convergence rate over classical optimal algorithms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Security and privacy of machine learning and AI applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We propose quantum algorithms that provide provable speedups for Markov Chain Monte Carlo (MCMC) methods commonly used for sampling from probability distributions of the form $pi propto e^{-f}$, where $f$ is a potential function. Our first approach considers Gibbs sampling for finite-sum potentials in the stochastic setting, employing an oracle that provides gradients of individual functions. In the second setting, we consider access only to a stochastic evaluation oracle, allowing simultaneous queries at two points of the potential function under the same stochastic parameter. By introducing novel techniques for stochastic gradient estimation, our algorithms improve the gradient and evaluation complexities of classical samplers, such as Hamiltonian Monte Carlo (HMC) and Langevin Monte Carlo (LMC) in terms of dimension, precision, and other problem-dependent parameters. Furthermore, we achieve quantum speedups in optimization, particularly for minimizing non-smooth and approximately convex functions that commonly appear in empirical risk minimization problems.

Problem

• Quantum speedups for MCMC sampling methods
• Improved gradient and evaluation complexities
• Quantum optimization for non-smooth convex functions

Innovation

• Quantum Gibbs sampling with gradient oracles
• Stochastic gradient estimation via dual-point queries
• Dimension-precision optimized quantum MCMC acceleration