The problem of sampling from the stationary distribution of a Markov chain finds widespread applications in a variety of fields. The time required for a Markov chain to converge to its stationary distribution is known as the classical mixing time. In this article, we deal with analog quantum algorithms for mixing. First, we provide an analog quantum algorithm that given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of the square root of the classical mixing time and the square root of the classical hitting time. Our algorithm makes use of the framework of interpolated quantum walks and relies on Hamiltonian evolution in conjunction with von Neumann measurements. There also exists a different notion for quantum mixing: the problem of sampling from the limiting distribution of quantum walks, defined in a time-averaged sense. In this scenario, the quantum mixing time is defined as the time required to sample from a distribution that is close to this limiting distribution. Recently we provided an upper bound on the quantum mixing time for Erd\"os-Renyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we also extend and expand upon our findings therein. Namely, we provide an intuitive understanding of the state-of-the-art random matrix theory tools used to derive our results. In particular, for our analysis we require information about macroscopic, mesoscopic and microscopic statistics of eigenvalues of random matrices which we highlight here. Furthermore, we provide numerical simulations that corroborate our analytical findings and extend this notion of mixing from simple graphs to any ergodic, reversible, Markov chain.

翻译：Markov 链条的固定分布的抽样问题在多个字段中找到了广泛的应用。 Markov 链条与固定分布相融合所需的时间被称为古典混合时间。 在本篇文章中, 我们处理混合的模拟量子算法。 首先, 我们提供给Markov 链条的模拟量子算法, 使我们能够在典型混合时间的平方根和古典打击时间的平方根之和时, 从其固定分布中取样。 我们的算法利用了内分量流的循环流体框架, 并依靠汉密尔顿的演进与冯纽曼的测量结果相结合。 量子混合的概念也不同: 从限制量子行分布的抽样中取样的问题, 以时间来给Markov 链链链链链链链链, 也就是我们用来分析的直径直径直线线的量子, 也就是我们用来分析的直径直径直值, 和直径直线图的直径直径直径直线, 我们的直径直径直径直到直径直径直的直径直径直径直线, 。