Open Quantum Random Walks and the Mean Hitting Time Formula

TL;DR

This paper extends classical hitting time concepts to open quantum walks, deriving a quantum mean hitting time formula and exploring its implications for quantum dynamics on graphs.

Contribution

It introduces a quantum version of the Mean Hitting Time Formula within the open quantum walk framework, connecting linear maps to hitting times.

Findings

Established a quantum fundamental matrix for ergodic open quantum walks

Proved a quantum version of the Random Target Lemma

Derived a mean hitting time formula involving the minimal polynomial

Abstract

We make use of the Open Quantum Random Walk setting due to S. Attal, F. Petruccione, C. Sabot and I. Sinayskiy [J. Stat. Phys. (2012) 147:832-852] in order to discuss hitting times and a quantum version of the Mean Hitting Time Formula from classical probability theory. We study an open quantum notion of hitting probability on a finite collection of sites and with this we are able to describe the problem in terms of linear maps and its matrix representations. After setting an open quantum version of the fundamental matrix for ergodic Markov chains we are able to prove our main result and as consequence a version of the Random Target Lemma. We also study a mean hitting time formula in terms of the minimal polynomial associated to the matrix representation of the quantum walk. We discuss applications of the results to open quantum dynamics on graphs together with open questions.