Quasi-optimal quantum Markov chain spectral gap estimation

TL;DR

The paper tackles the problem of estimating the spectral/singular gap of Markov chains, a task classically hard to speed up. It advances a quantum approach based on quantum singular value transformation (QSVT) that leverages block-encoded Markov-chain discriminants, achieving quasi-optimal scaling in the number of vertices $N$ and, under a controlled relative error, quasi-optimal scaling in $1/\gamma$; this yields an almost quadratic quantum improvement over the best classical methods. A key contribution is the design of new singular-value filters, including a Dolph-Chebyshev-based kernel, and the construction of explicit unscaled block-encodings for certain algebraically-defined Markov chains, alongside a detailed analysis of the algorithmic complexity. The results have potential practical impact for speeding up Markov chain Monte Carlo through faster mixing-time proxies, and they open avenues for extending block-encoding techniques to a broader class of chains and properties.

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.

Figures (2)

• Figure 1: Relationship between some important ergodic Markov chain classes.
• Figure 2: A comparison of the three polynomial filters. In (a), the step function approximation is limited in how fast it can rise as it must "turn flat again"; conversely there is no such limitation in (b), where the monomial (approximated by a polynomial with degree about square root of the monomial degree) grows rapidly at $x=1$. In the case of (c), the function grows slowly enough that there is a region of $x$ with a useful lower-bound (the attractive feature of (a)), whilst still requiring relatively low degree (the attractive feature of (b)).

Theorems & Definitions (62)

• Definition 1
• Definition 2
• Lemma 3
• proof
• Definition 4
• Remark 5
• Definition 6
• Remark 7
• Lemma 8
• proof