We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph G is universal mixing if the instantaneous or average probability distribut...
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.
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.
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.
The fundamental problem of sampling from the limiting distribution of quantum walks on networks, known as mixing, finds widespread applications in several areas of quantum information and computation. Of particular interest in most of these applications is the minimum time beyond which the instantaneous probability distribution of the quantum walk remains close to this limiting distribution, known as the "quantum mixing time". However, this quantity is only known for a handful of specific networks. In this Letter, we prove an upper bound on the quantum mixing time for almost all networks, i.e., the fraction of networks for which our bound holds, goes to one in the asymptotic limit. To this end, using several results in random matrix theory, we find the quantum mixing time of Erdös-Renyi random networks: networks of n nodes where each edge exists with probability p independently. For example, for dense random networks, where p is a constant, we show that the quantum mixing time is O(n^{3/2+o(1)}). In addition to opening avenues for the analytical study of quantum dynamics on random networks, our work could find applications beyond quantum information processing. Owing to the universality of Wigner random matrices, our results on the spectral properties of random graphs hold for general classes of random matrices that are ubiquitous in several areas of physics. In particular, our results could lead to novel insights into the equilibration times of isolated quantum systems defined by random Hamiltonians, a foundational problem in quantum statistical mechanics.
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.