Faster quantum mixing for slowly evolving sequences of Markov chains

TL;DR

This paper introduces a quantum algorithm that significantly speeds up the mixing process of slowly evolving Markov chains, especially for large state spaces, with advantages in memory efficiency and output format.

Contribution

The authors develop a quantum memory-efficient mixing algorithm with a runtime of O(√(δ^{-1})·N^{1/4}), improving over previous methods, and demonstrate its optimality under certain conditions.

Findings

Achieves faster mixing times for slowly evolving Markov chains

Provides quantum encodings of distributions for output

Shows the algorithm's optimality under specific assumptions

Abstract

Markov chain methods are remarkably successful in computational physics, machine learning, and combinatorial optimization. The cost of such methods often reduces to the mixing time, i.e., the time required to reach the steady state of the Markov chain, which scales as $\delta^{-1}$, the inverse of the spectral gap. It has long been conjectured that quantum computers offer nearly generic quadratic improvements for mixing problems. However, except in special cases, quantum algorithms achieve a run-time of $\mathcal{O}(\sqrt{\delta^{-1}} \sqrt{N})$, which introduces a costly dependence on the Markov chain size $N,$ not present in the classical case. Here, we re-address the problem of mixing of Markov chains when these form a slowly evolving sequence. This setting is akin to the simulated annealing setting and is commonly encountered in physics, material sciences and machine learning. We provide a quantum memory-efficient algorithm with a run-time of $\mathcal{O}(\sqrt{\delta^{-1}} \sqrt[4]{N})$, neglecting logarithmic terms, which is an important improvement for large state spaces. Moreover, our algorithms output quantum encodings of distributions, which has advantages over classical outputs. Finally, we discuss the run-time bounds of mixing algorithms and show that, under certain assumptions, our algorithms are optimal.