Markov chain Monte Carlo (MCMC) is the widely-used classical method of random sampling from a probability distribution π by simulating a Markov chain which “mixes” to π 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.