On Efficient Solutions of General Structured Markov Processes in Quantum Computational Environments

TL;DR

This paper introduces the first quantum algorithms for efficiently computing the stationary distribution of structured Markov processes, potentially surpassing classical methods in computational efficiency within quantum environments.

Contribution

The paper develops novel quantum algorithms for stationary distribution computation of structured Markov processes, a significant advancement over classical approaches.

Findings

Quantum algorithms show potential for exponential speedup over classical algorithms.

Mathematical analysis confirms the efficiency and advantages of the proposed quantum methods.

Algorithms can be integrated into larger quantum computational frameworks.

Abstract

We study from a theoretical viewpoint 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 fundamental problem within the context of quantum computational environments from a mathematical perspective and devise the first quantum algorithms for computing the stationary distribution of general 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 and most-efficient classical algorithms in various settings of both theoretical and practical importance. Although motivated by general structured Markov processes, our quantum algorithms can be exploited to address a much larger class of numerical computation problems, as well as to potentially play the role of a subroutine as part of solving larger computational problems involving the stationary distribution on a quantum computer.