Faster quantum mixing for slowly evolving sequences of Markov chains

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 δ −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 O( √ δ −1 √ 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 O( √ δ −1 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.Davide Orsucci: davide.orsucci@uibk.ac.at Hans J. Briegel: hans.briegel@uibk.ac.at Vedran Dunjko: v.dunjko@liacs.leidenuniv.nl ods [4]. In MC-based approaches the underlying objective is to produce samples from the steady state, i.e., the stationary distribution of a given MC. The MC is constructed so that this distribution encodes the solution of the problem at hand. The solution can then be reached by "mixing", i.e., by applying the MC transitions many times. For some problems, mixing processes constitute the fastest known classical solving algorithms, and play a vital role, e.g., in the Metropolis-Hastings methods [5], periodic Gibbs sampling [6], and Glauber dynamics [7].The fundamental parameter governing the time complexity of MC-based algorithms is thus the mixing time, that is, the number of steps required to attain stationarity. In most applications the MC is ergodic, i.e., has a unique stationary distribution, and timereversible, i.e., satisfies detailed balance [8,9]. The mixing time is tightly related to the spectral gap δ of the MC 1 and is bounded by Ω(δ −1 ) [10].Oftentimes direct mixing can be computationally prohibitive and thus heuristic methods, such as simulated annealing [11,12], are employed. Here one constructs a sequence of Markov chains which, for instance, encode the Gibbs (thermal) distributions at gradually decreasing values of the temperature, where the target distribution is specified by the final MC, i.e., the final temperature. Intuitively, this process increases efficiency by avoiding local minima, although the performance is typically not guaranteed. In simulated annealing, the neighbouring chains in the sequence are similar, in other words, the sequence is slowly evolving.The emergence of quantum computation offers a new possibil...

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