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

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

2009-04-14

arxiv.org/abs/0904.2057v1

Commun. Theor. Phys. 51 (2009) 1003-1009

Physics

Quantum Physics

21, page. Accepted for publication on CTP

Scientific paper

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.

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