Quantum Speedups for Markov Chain Monte Carlo Methods with Application to Optimization
Guneykan Ozgul, Xiantao Li, Mehrdad Mahdavi, Chunhao Wang
TL;DR
The paper addresses the problem of efficiently sampling from $\pi(\mathbf{x}) = e^{-f(\mathbf{x})}/Z$ in high dimensions and using quantum techniques to accelerate optimization tasks that arise in empirical risk and Bayesian inference. It develops quantum variance-reduced samplers (QSVRG-HMC, QCV-HMC, QSVRG-LMC) for finite-sum and Log-Sobolev settings, achieving faster gradient/evaluation complexities and provable convergence in Wasserstein/KL metrics. In the zeroth-order regime, it presents quantum gradient estimation based on Jordan's algorithm and unbiased quantum mean estimation, with robustMLMC augmentation, enabling faster gradient and sampling complexity under smoothness and variance assumptions. The framework extends to zeroth-order sampling and to optimization of non-smooth or approximately convex objectives via smoothing, yielding nontrivial speedups in dimension and accuracy, with practical implications for large-scale ERM and Bayesian inference. Overall, the work advances quantum-accelerated sampling and optimization by combining variance-reduction, phase-oracle gradient estimation, and MLMC techniques to achieve provable improvements over classical methods across multiple problem regimes.
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 $π\propto e^{-f}$, 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.