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Markov chain Monte Carlo (MCMC) is the widely-used classical method of random
sampling from a probability distribution pi by simulating a Markov chain which
``mixes'' to pi at equilibrium. Despite the success quantum walks
have been shown to have in speeding up random walk algorithms for search
problems (``hitting'') and simulated annealing, it remains to prove a
general speedup theorem for MCMC sampling algorithms. We review the progress
toward this end, in particular using decoherent quantum walks.