Continuous-Time Classical and Quantum Random Walk on Direct Product of Cayley Graphs

TL;DR

This paper introduces a method to compute probabilities of continuous-time classical and quantum random walks on direct product graphs, simplifying analysis on complex structures like Cayley graphs, and explores their long-term behaviors.

Contribution

It presents a new recipe for calculating walk probabilities on direct product graphs, extending analysis to complex Cayley graph structures for classical and quantum walks.

Findings

Probability on product graphs is obtained by multiplying probabilities on sub-graphs.

Classical walks reach a uniform stationary distribution as time approaches infinity.

Quantum walks do not always reach a stationary distribution.

Abstract

In this paper we define direct product of graphs and give a recipe for obtained probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on direct product of graph obtain by multiplication of probability on the corresponding to sub-graphs, where this method is useful to determine probability of walk on complicated graphs. Using this method, we calculate the probability of continuous-time classical and quantum random walks on many of finite direct product cayley graphs (complete cycle, complete $K_n$, charter and $n$-cube). Also, we inquire that the classical state the stationary uniform distribution is reached as $t\longrightarrow \infty$ but for quantum state is not always satisfy.