Quantum speedup for mixing a Markov chain can be achieved based on the construction of slowly-varying $r$ Markov chains where the initial chain can be easily prepared and the spectral gaps have uniform lower bound. The overall complexity is proportional to $r$. We present a multi-level approach to construct such a sequence of $r$ Markov chains by varying a resolution parameter $h.$ We show that the density function of a low-resolution Markov chain can be used to warm start the Markov chain with high resolution. We prove that in terms of the chain length the new algorithm has $O(1)$ complexity rather than $O(r).$
Quantum speedup for mixing a Markov chain can be achieved based on the construction of slowly-varying r Markov chains where the initial chain can be easily prepared and the spectral gaps have uniform lower bound. The overall complexity is proportional to r. We present a multi-level approach to construct such a sequence of r Markov chains by varying a resolution parameter h. We show that the density function of a low-resolution Markov chain can be used to warm start the Markov chain with high resolution. We prove that in terms of the chain length the new algorithm has O(1) complexity rather than O(r).
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), where δ 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), 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 N-state Markov chains, akin to the setting of simulated annealing methods. The complexity of preparation we achieve is O(δ−14N), 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.
Most approximation algorithms for #P-complete problems (e.g., evaluating the partition function of a monomer-dimer or ferromagnetic Ising system) work by reduction to the problem of approximate sampling from a distribution $\ensuremath{\pi}$ over a large set S. This problem is solved using the Markov chain Monte Carlo method: a sparse, reversible Markov chain P on S with stationary distribution $\ensuremath{\pi}$ is run to near equilibrium. The running time of this random walk algorithm, the so-called mixing time of P, is $O({\ensuremath{\delta}}^{\ensuremath{-}1}\text{ }\text{log}\text{ }1∕{\ensuremath{\pi}}_{\ensuremath{\ast}})$ as shown by Aldous, where $\ensuremath{\delta}$ is the spectral gap of P and ${\ensuremath{\pi}}_{\ensuremath{\ast}}$ is the minimum value of $\ensuremath{\pi}$. A natural question is whether a speedup of this classical method to $O(\sqrt{{\ensuremath{\delta}}^{\ensuremath{-}1}}\text{ }\text{log}\text{ }1∕{\ensuremath{\pi}}_{\ensuremath{\ast}})$ is possible using quantum walks. We provide evidence for this possibility using quantum walks that decohere under repeated randomized measurements. We show that (i) decoherent quantum walks always mix, just like their classical counterparts, (ii) the mixing time is a robust quantity, essentially invariant under any smooth form of decoherence, and (iii) the mixing time of the decoherent quantum walk on a periodic lattice Znd is O(ndlogd), which is indeed $O(\sqrt{{\ensuremath{\delta}}^{\ensuremath{-}1}}\text{ }\text{log}\text{ }1∕{\ensuremath{\pi}}_{\ensuremath{\ast}})$ and is asymptotically no worse than the diameter of Znd (the obvious lower bound) up to at most a logarithmic factor.
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.
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.
The Markov-chain Monte Carlo method is at the heart of efficient approximation schemes for a wide range of problems in combinatorial enumeration and statistical physics. It is therefore very natural and important to determine whether quantum computers can speed up classical mixing processes based on Markov chains. To this end, we present a quantum algorithm, making it possible to prepare a quantum sample---i.e., a coherent version of the stationary distribution of a reversible Markov chain. Our algorithm has a significantly better running time than that of a previous algorithm based on adiabatic-state generation. We also show that our methods provide a greater speedup over a recently proposed method for obtaining the ground states of (classical) Hamiltonians.