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International Journal of Quantum Information

NON-UNIFORM MIXING OF QUANTUM WALK ON CYCLES

William Adamczak, Kevin Andrew, Leon Bergen, Dillon Ethier, Peter Hernberg, Jennifer Lin, Christino Tamon

Published Dec 1, 2007• 27 citations
Publisher

Abstract

A classical lazy random walk on cycles is known to mix with the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibits strong non-uniform mixing properties. First, we prove that the instantaneous distribution of a quantum walk on most even-length cycles is never uniform. More specifically, we prove that a quantum walk on a cycle Cnis not instantaneous uniform mixing, whenever n satisfies either: (a) n = 2u, for u ≥ 3; or (b) n = 2uq, for u ≥ 1 and q ≡ 3 (mod 4). Second, we prove that the average distribution of a quantum walk on any Abelian circulant graph is never uniform. As a corollary, the average distribution of a quantum walk on any standard circulant graph, such as the cycles, complete graphs, and even hypercubes, is never uniform. Nevertheless, we show that the average distribution of a quantum walk on the cycle Cnis O(1/n)-uniform.

DOI
10.1142/s0219749907003195

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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.
We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph GGG is {\em universal mixing} if the instantaneous or average probability distribution of the quantum walk on GGG ranges over all probability distributions on the vertices as the weights are varied over non-negative reals. The graph is {\em uniform} mixing if it visits the uniform distribution. Our results include the following: 1) All weighted complete multipartite graphs are instantaneous universal mixing. This is in contrast to the fact that no {\em unweighted} complete multipartite graphs are uniform mixing (except for the four-cycle K2,2K_{2,2}K2,2​). 2) For all n≥1n \ge 1n≥1, the weighted claw K1,nK_{1,n}K1,n​ is a minimally connected instantaneous universal mixing graph. In fact, as a corollary, the unweighted K1,nK_{1,n}K1,n​ is instantaneous uniform mixing. This adds a new family of uniform mixing graphs to a list that so far contains only the hypercubes. 3) Any weighted graph is average almost-uniform mixing unless its spectral type is sublinear in the size of the graph. This provides a nearly tight characterization for average uniform mixing on circulant graphs. 4) No weighted graphs are average universal mixing. This shows that weights do not help to achieve average universal mixing, unlike the instantaneous case. Our proofs exploit the spectra of the underlying weighted graphs and path collapsing arguments.
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Continuous-time quantum walks on graphs is a generalization of continuous-time Markov chains on discrete structures. Moore and Russell proved that the continuous-time quantum walk on the nnn-cube is instantaneous exactly uniform mixing but has no average mixing property. On complete (circulant) graphs KnK_{n}Kn​, the continuous-time quantum walk is neither instantaneous (except for n=2,3,4n=2,3,4n=2,3,4) nor average uniform mixing (except for n=2n=2n=2). We explore two natural {\em group-theoretic} generalizations of the nnn-cube as a GGG-circulant and as a bunkbed $G \rtimes \Int_{2}$, where GGG is a finite group. Analyses of these classes suggest that the nnn-cube might be special in having instantaneous uniform mixing and that non-uniform average mixing is pervasive, i.e., no memoryless property for the average limiting distribution; an implication of these graphs having zero spectral gap. But on the bunkbeds, we note a memoryless property with respect to the two partitions. We also analyze average mixing on complete paths, where the spectral gaps are nonzero.