On mixing in continuous-time quantum walks on some circulant graphs

Authors:
Amir Ahmadi;Ryan Belk;Christino Tamon;Carolyn Wendler
Affiliations:
Department of Chemical Engineering, Clarkson University, Potsdam, NY;Department of Mathematical Sciences, Binghamton University, State University of New York, Binghamton, NY;Department of Mathematics and Computer Science and Center for Quantum Device Technology, Clarkson University, Potsdam, NY;Department of Mathematics, Kenyon College, Gambier, OH
Venue:
Quantum Information & Computation
Year:
2003

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Abstract

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time 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 exactly 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 of the symmetric group lack this mixing property as well.