On analog quantum algorithms for the mixing of Markov chains
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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. We also make novel inroads into 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. This notion of quantum mixing has been explored for a handful of specific graphs and the derived upper bound for this quantity has been faster than its classical counterpart for some graphs while being slower for others. In this article, using several results in random matrix theory, we prove an upper bound on the quantum mixing time of Erdös-Renyi random graphs: graphs of n nodes where each edge exists with probability p independently. For example for dense random graphs, where p is a constant, we show that the quantum mixing time is O(n^3/2 + o(1)). Consequently, this allows us to obtain an upper bound on the quantum mixing time for almost all graphs, i.e. the fraction of graphs for which this bound holds, goes to one in the asymptotic limit.
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