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：我们提出了一种新颖的混合量子计算策略，用于每一步生成多个提议的并行MCMC算法。该策略通过利用Gumbel-max技巧将广义接受-拒绝步骤转化为离散优化问题，从而使并行MCMC中的限速步骤适用于量子并行化。结合并行MCMC文献的最新见解，这种方法使我们能够将目标密度评估嵌入到Grover量子搜索算法的一个著名扩展中。设$P$表示单次MCMC迭代中的提议数量，该组合策略将所需的目标评估次数从$\mathcal{O}(P)$减少到$\mathcal{O}(P^{1/2})$。在下文中，我们回顾了量子计算、量子搜索和Gumbel-max技巧的基本原理，以便尽可能广泛的读者能够理解其组合方式。