We study the fundamental problem of efficiently computing the stationary distribution of general classes of structured Markov processes. In strong contrast with previous work, we consider this problem within the context of quantum computational environments from a mathematical perspective and devise the first quantum algorithms for computing the stationary distribution of structured Markov processes. We derive a mathematical analysis of the computational properties of our quantum algorithms together with related theoretical results, establishing that our quantum algorithms provide the potential for significant computational improvements over that of the best-known classical algorithms in various settings of both theoretical and practical importance. Although motivated by structured Markov processes, our quantum algorithms have the potential for being exploited to address a much larger class of numerical computation problems.

翻译：研究结构化马尔可夫过程一般类的平稳分布高效计算这一基础问题。与以往工作形成鲜明对比的是，我们从数学视角将这一问题置于量子计算环境中加以考量，并设计了首个用于计算结构化马尔可夫过程平稳分布的量子算法。通过对量子算法计算特性及相关理论结果的数学分析，我们证实：在多个具有理论和实践重要性的场景中，所提出的量子算法相比现有最优经典算法具有显著的计算性能提升潜力。尽管本文以结构化马尔可夫过程为出发点，但所提出的量子算法有望推广应用于更广泛的数值计算问题。