Quasi-optimal quantum Markov chain spectral gap estimation

TL;DR

This paper introduces a quantum algorithm for estimating the spectral gap of Markov chains that is nearly optimal and offers significant advantages over classical methods, with potential applications in speeding up Markov chain Monte Carlo sampling.

Contribution

It presents a quasi-optimal quantum algorithm for spectral gap estimation and develops new block-encoding techniques for Markov chain transition matrices.

Findings

Achieves near-quadratic speedup over classical algorithms

Develops explicit block-encoding methods for certain Markov chains

Improves quantum spectral gap estimation techniques

Abstract

This paper proposes a quantum algorithm for Markov chain spectral gap estimation that is quasi-optimal (i.e., optimal up to a polylogarithmic factor) in the number of vertices for all parameters, and additionally quasi-optimal in the reciprocal of the spectral gap itself, if the permitted relative error is above some critical value. In particular, these results constitute an almost quadratic advantage over the best-possible classical algorithm. Our algorithm also improves on the quantum state of the art, and we contend that this is not just theoretically interesting but also potentially practically impactful in real-world applications: knowing a Markov chain's spectral gap can speed-up sampling in Markov chain Monte Carlo. Our approach uses the quantum singular value transformation, and as a result we also develop some theory around block-encoding Markov chain transition matrices, which is potentially of independent interest. In particular, we introduce explicit block-encoding methods for the transition matrices of two algebraically-defined classes of Markov chains.