# Mean hitting times of quantum Markov chains in terms of generalized inverses

## TL;DR

This paper investigates quantum Markov chains on graphs, defining a quantum mean first visit time using generalized inverses, and compares these results with classical Markov chain theory.

## Contribution

It introduces a quantum analogue of mean hitting times expressed via generalized inverses, extending classical Markov chain concepts to quantum settings.

## Key findings

- Derived formulas for quantum mean hitting times using generalized inverses
- Illustrated similarities and differences between quantum and classical Markov chains
- Provided computational examples demonstrating the theory

## Abstract

We study quantum Markov chains on graphs, described by completely positive maps, following the model due to S. Gudder (J. Math. Phys. 49, 072105, 2008) and which includes the dynamics given by open quantum random walks as defined by S. Attal et al. (J. Stat. Phys. 147:832-852, 2012). After reviewing such structures we examine a quantum notion of mean time of first visit to a chosen vertex. However, instead of making direct use of the definition as it is usually done, we focus on expressions for such quantity in terms of generalized inverses associated with the walk and most particularly the so-called fundamental matrix. Such objects are in close analogy with the theory of Markov chains and the methods described here allow us to calculate examples that illustrate similarities and differences between the quantum and classical settings.