Quantum Algorithms for Gibbs Expectation of Non-log-concave and Heavy-tailed Distributions

TL;DR

This paper introduces a comprehensive unbiased quantum sampling framework for Gibbs expectations applicable to non-log-concave and heavy-tailed distributions, achieving quantum speedups over classical methods.

Contribution

It develops an unbiased quantum-accelerated multilevel Monte Carlo method with broader applicability and improved complexity for complex distributions.

Findings

Quantum complexity $ ilde{O}(rac{1}{})$ for error 

Classical MLMC complexity is 

Unbiased quantum sampling for heavy-tailed distributions

Abstract

We establish a systematic framework of unbiased quantum sampling and estimation protocols for the classical Gibbs expectation. This framework generalizes existing approaches to the partition function estimation and has broader applications in various fields. We consider sampling and estimation for a wide class of non-log-concave distributions, particularly heavy-tailed ones, under relaxed assumptions beyond strong convexity, such as dissipativity. We develop an unbiased extension of quantum-accelerated multilevel Monte Carlo (QA-MLMC) to eliminate all biases from discretization and time truncation, together with introducing a change-of-measure approach and the Girsanov theorem via Radon-Nikodym derivatives. As a result, our approach achieves quantum complexity $\widetilde{\mathcal{O}}(\epsilon^{-1})$ within error $\epsilon$, whereas the classical MLMC requires $\widetilde{\mathcal{O}}(\epsilon^{-2})$ and existing quantum algorithms yield biased estimators under stronger assumptions. Furthermore, our unified framework enables unbiased quantum sampling and estimation for certain heavy-tailed distributions after transformation. We provide several concrete applications of our approach in statistics, machine learning, and finance, towards more practical scenarios of the quantum acceleration of stochastic processes.