From classical to quantum walks with stochastic resetting on networks

TL;DR

This paper develops a unified framework for classical and quantum random walks with stochastic resetting on networks, analyzing how quantum effects influence long-term distributions and sampling properties.

Contribution

It introduces a formalism based on graph Laplacians for classical and quantum walks with resets, bridging the gap between classical and quantum stochastic processes on networks.

Findings

Quantum effects alter stationary distributions.

Resets impact classical and quantum walks differently.

Analytical results match numerical simulations on various networks.

Abstract

Random walks are fundamental models of stochastic processes with applications in various fields including physics, biology, and computer science. We study classical and quantum random walks under the influence of stochastic resetting on arbitrary networks. Based on the mathematical formalism of quantum stochastic walks, we provide a framework of classical and quantum walks whose evolution is determined by graph Laplacians. We study the influence of quantum effects on the stationary and long-time average probability distribution by interpolating between the classical and quantum regime. We compare our analytical results on stationary and long-time average probability distributions with numerical simulations on different networks, revealing differences in the way resets affect the sampling properties of classical and quantum walks.