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.
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.
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.
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.
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 n-cube is instantaneous exactly uniform mixing but has no average mixing property. On complete (circulant) graphs Kn, the continuous-time quantum walk is neither instantaneous (except for n=2,3,4) nor average uniform mixing (except for n=2). We explore two natural {\em group-theoretic} generalizations of the n-cube as a G-circulant and as a bunkbed $G \rtimes \Int_{2}$, where G is a finite group. Analyses of these classes suggest that the n-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.
We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph G is {\em universal mixing} if the instantaneous or average probability distribution of the quantum walk on G 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 K2,2). 2) For all n≥1, the weighted claw K1,n is a minimally connected instantaneous universal mixing graph. In fact, as a corollary, the unweighted 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.
A classical lazy random walk on cycles is known to mix to the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibit strong non-uniform mixing properties. Our results include the following: - The instantaneous distribution of a quantum walk on most even-length cycles is never uniform. - 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.
We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform mixing on the hypercube is robust under the addition of perfect matchings but not much else. Our specific results include: • The graph obtained by augmenting the hypercube with an additive matching x ↦ x ⊕ η is instantaneous uniform mixing whenever |η| is even, but with a slower mixing time. This strictly includes the result of Moore and Russell1on the hypercube. • The class of Hamming graphs H(n,q) is not uniform mixing if and only if q ≥ 5. This is a tight characterization of quantum uniform mixing on Hamming graphs; previously, only the status of H(n,q) with q < 5 was known. • The bunkbed graph [Formula: see text] whose adjacency matrix is I ⊗ Qn+ X ⊗ Af, where Afis a [Formula: see text]-circulant matrix defined by a Boolean function f, is not uniform mixing if the Fourier transform of f has support of size smaller than 2n-1. This explains why the hypercube is uniform mixing and why the join of two hypercubes is not. Our work exploits the rich spectral structure of the generalized hypercubes and relies heavily on Fourier analysis of group-circulants.