A quantum parallel Markov chain Monte Carlo

We propose a novel quantum computing strategy for parallel MCMC algorithms that generate multiple proposals at each step. This strategy makes parallel MCMC amenable to quantum parallelization by using the Gumbel-max trick to turn the generalized accept-reject step into a discrete optimization problem. This allows us to embed target density evaluations within a well-known extension of Grover's quantum search algorithm. Letting $P$ denote the number of proposals in a single MCMC iteration, the combined strategy reduces the number of target evaluations required from $\mathcal{O}(P)$ to $\mathcal{O}(P^{1/2})$. In the following, we review both the rudiments of quantum computing and the Gumbel-max trick in order to elucidate their combination for as wide a readership as possible.