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arXiv:1503.01334 · Quantum

Faster quantum mixing for slowly evolving sequences of Markov chains

Davide Orsucci, Hans J. Briegel, Vedran Dunjko

Published Nov 9, 2018• 20 citations
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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−1N) , 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−1N4) , 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.

DOI
10.22331/q-2018-11-09-105

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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−1N) , 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−1N4) , 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.
In this work we consider the problem of preparation of the stationary distribution of irreducible, time-reversible Markov chains, which is a fundamental task in algorithmic Markov chain theory. For the classical setting, this task has a complexity lower bound of Ω(δ−1)\Omega(\delta^{-1})Ω(δ−1), where δ\deltaδ is the spectral gap of the Markov chain, and other dependencies contribute only logarithmically. In the quantum case, the conjectured complexity is O(δ−1)O(\sqrt{\delta^{-1}})O(δ−1​), with other dependencies contributing only logarithmically. However, this bound has only been achieved for a few special classes of Markov chains. In this work, we provide a method for the sequential preparation of stationary distributions for sequences of time-reversible NNN-state Markov chains, akin to the setting of simulated annealing methods. The complexity of preparation we achieve is O(δ−1N4)O(\sqrt{\delta^{-1}} \sqrt[4]{N})O(δ−1​4N​), neglecting logarithmic factors. While this result falls short of the conjectured optimal time, it provides a quadratic improvement over na\{i}ve approaches. Moreover, for the case when the output distributions are required to be encoded in pure quantum states we identify the settings where our algorithm is strictly optimal. The settings of slowly evolving sequences of Markov chains naturally appear in reinforcement learning, and consequently our results can be readily applied in quantum machine learning as well.
Copyright © 2020 by SIAM Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution. In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.
Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution. In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.
The preparation of the stationary distribution of irreducible, time-reversible Markov chains (MCs) is a fundamental building block in many heuristic approaches to algorithmically hard problems. It has been conjectured that quantum analogs of classical mixing processes may offer a generic quadratic speed-up in realizing such stationary distributions. Such a speed-up would also imply a speed-up of a broad family of heuristic algorithms. However, a true quadratic speed up has thus far only been demonstrated for special classes of MCs. These results often presuppose a regular structure of the underlying graph of the MC, and also a regularity in the transition probabilities. In this work, we demonstrate a true quadratic speed-up for a class of MCs where the restriction is only on the form of the stationary distribution, rather than directly on the MC structure itself. In particular, we show efficient mixing can be achieved when it is known beforehand that the distribution is monotonically decreasing relative to a known order on the state space. Following this, we show that our approach extends to a wider class of distributions, where only a fraction of the shape of the distribution is known to be monotonic. Our approach is built on the Szegedy-type quantization of transition operators.
Quantum-enhanced Markov chain Monte Carlo, an algorithm in which configurations are proposed through a measured quantum quench and accepted or rejected by a classical algorithm, has been proposed as a possible method for robust quantum speedup on imperfect quantum devices. While this procedure is resilient to noise and control imperfections, the potential for quantum advantage is unclear. By upper-bounding the gap of the Markov chain, we identify competing factors that limit the algorithm's performance. One needs the quantum dynamics to efficiently delocalize the system over a range of classical states, but it is also detrimental to introduce too much entropy through the quench. Specifically, we show that in the long-time limit, the gap of the Markov chain is bounded by the inverse participation ratio of the classical states in the eigenstate basis, showing there is no advantage when quenching to an ergodic system. For the paradigmatic Sherrington-Kirkpatrick and three-spin model, we identify the regime of optimal spectral gap scaling and link it to the system's eigenstate properties.
Quantum computing offers an alternative paradigm for addressing combinatorial optimization problems compared to classical computing. Despite recent hardware improvements, the execution of empirical quantum optimization experiments at scales known to be hard for state-of-the-art classical solvers is not yet in reach. In this work, we offer a different way to approach combinatorial optimization with near-term quantum computing. Motivated by the promising results observed in using quantum-enhanced Markov chain Monte Carlo (QeMCMC) for approximating complicated probability distributions, we combine ideas of sampling from the device with QeMCMC together with warm-starting and parallel tempering, in the context of combinatorial optimization. We demonstrate empirically that our algorithm recovers the global optima for instances of the Maximum Independent Set problem (MIS) up to 117 decision variables using 117 qubits on IBM quantum hardware. We show early evidence of a scaling advantage of our algorithm compared to similar classical methods for the chosen instances of MIS. MIS is practically relevant across domains like financial services and molecular biology, and, in some cases, already difficult to solve to optimality classically with only a few hundred decision variables.