# Limit properties of global interaction stochastic quantum walks on directed graphs

## TL;DR

This paper investigates the limiting behaviors of global interaction stochastic quantum walks on directed graphs, revealing how graph connectivity and interaction type influence convergence and relaxation properties.

## Contribution

It provides new theoretical insights into how different graph structures and interaction models affect the limiting properties of quantum walks.

## Key findings

- Global interaction evolution is convergent on undirected graphs.
- Local interaction evolution is relaxing on strongly connected directed graphs.
- Connectivity significantly influences the limiting behavior of quantum walks.

## Abstract

The main results of our work is determining the differences between limiting properties in various models of quantum stochastic walks. In particular, we prove that in the case of strongly connected and a class of weakly connected directed graphs, local environment interaction evolution is relaxing, and in the case of undirected graphs, global environment interaction evolution is convergent. For other classes of directed graphs we show, that the character of connectivity large influence on the limiting properties. We also study the limiting properties for the non-moralizing global interaction case. We demonstrate that the digraph observance is recovered in this case.