We propose a novel hybrid quantum computing strategy for parallel MCMC algorithms that generate multiple proposals at each step. This strategy makes the rate-limiting step within parallel MCMC amenable to quantum parallelization by using the Gumbel-max trick to turn the generalized accept-reject step into a discrete optimization problem. When combined with new insights from the parallel MCMC literature, such an approach 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 the rudiments of quantum computing, quantum search and the Gumbel-max trick in order to elucidate their combination for as wide a readership as possible.

翻译：我们提出了一种新的混合量子计算策略，用于并行 MCMC 算法，该算法在每一步产生多个提议。该策略通过使用 Gumbel-max 技巧将广义接受-拒绝步骤转化为离散优化问题，使并行 MCMC 中的瓶颈步骤易于量子并行化。与并行 MCMC 文献中的新见解相结合，这种方法允许我们将目标密度评估嵌入格罗弗量子搜索算法的一个广为人知的扩展中。让 $P$ 表示单个 MCMC 迭代中的提议数量，组合策略将所需的目标评估数从 $\mathcal {O} (P)$ 降低到 $\mathcal {O} (P ^ {1/2})$。在下文中，我们回顾量子计算、量子搜索和 Gumbel-max 技巧的基本知识，以便让尽可能广泛的读者理解其组合。