Quantum vs classical Markov chains; Exactly solvable examples

TL;DR

This paper introduces a quantum analogue of classical reversible Markov chains, providing exactly solvable examples based on hypergeometric orthogonal polynomials to compare quantum and classical dynamics.

Contribution

It presents a novel coinless quantisation method for reversible Markov chains and offers explicit solvable models for quantum-classical comparison.

Findings

Quantum Hamiltonian derived from classical transition matrix

Explicit solutions for five Markov chain examples

Framework for analyzing quantum vs classical Markov dynamics

Abstract

A coinless quantisation procedure of general reversible Markov chains on graphs is presented. A quantum Hamiltonian H is obtained by a similarity transformation of the fundamental transition probability matrix K in terms of the square root of the reversible distribution. The evolution of the classical and quantum Markov chains is described by the solutions of the eigenvalue problem of the quantum Hamiltonian H. About twenty plus exactly solvable Markov chains based on the hypergeometric orthogonal polynomials of Askey scheme, derived by Odake-Sasaki, would provide a good window for scrutinising the quantum/classical contrast of Markov chains. Among them five explicit examples, related to the Krawtchouk, Hahn, q-Hahn, Charlier and Meixner, are demonstrated to illustrate the actual calculations.