Mixing in Continuous Quantum Walks on Graphs

Details Mixing in Continuous Quantum Walks on Graphs Mixing in Continuous Quantum Walks on Graphs

2002-09-18

arxiv.org/abs/quant-ph/0209106v5

Quantum Information and Computation 3 (2003), 611-618.

Physics

Quantum Physics

8 pages, no figures

Scientific paper

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs lack this mixing property as well.

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