Continuous-Time Quantum Markov Chains And Discretizations Of p-Adic Schr\"odinger Equations: Comparisons And Simulations

TL;DR

This paper explores the relationship between p-adic Schr"odinger equations, quantum Markov chains, and classical heat equations, using mathematical analysis and simulations to compare their behaviors and distributions.

Contribution

It introduces a broad class of p-adic Schr"odinger equations and compares associated quantum and classical Markov chains through analysis and numerical simulations.

Findings

Quantum Markov chains have greater limiting distributions than classical counterparts.

Numerical simulations confirm the theoretical comparison of distributions.

The study broadens understanding of p-adic quantum and classical stochastic processes.

Abstract

The continuous-time quantum walks (CTQWs) are a fundamental tool in the development of quantum algorithms. Recently, it was shown that discretizations of p-adic Schr\"odinger equations give rise to continuous-time quantum Markov chains (CTQMCs); this type of Markov chain includes the CTQWs constructed using adjacency matrices of graphs as a particular case. In this paper, we study a large class of p-adic Schr\"odinger equations and the associated CTQMCs by comparing them with p-adic heat equations and the associated continuous-time Markov chains (CTMCs). The comparison is done by a mathematical study of the mentioned equations, which requires, for instance, solving the initial value problems attached to the mentioned equations, and through numerical simulations. We conducted multiple simulations, including numerical approximations of the limiting distribution. Our simulations show that the limiting distribution of quantum Markov chains is greater than the stationary probability of their classical counterparts, for a large class of CTQMCs.