Universal Mixing of Quantum Walk on Graphs

TL;DR

This paper investigates the conditions under which quantum walks on graphs can produce all possible probability distributions, revealing new classes of graphs with universal mixing properties and characterizing limitations for average mixing.

Contribution

It introduces the concept of universal mixing for weighted graphs, identifies new classes of graphs with this property, and characterizes the spectral conditions affecting mixing behaviors.

Findings

All weighted complete multipartite graphs are instantaneous universal mixing.

Weighted claw or star graphs are minimally connected instantaneous universal mixers.

No weighted graphs are average universal mixing.

Abstract

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 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 uniform mixing if it visits the uniform distribution. Our results include the following: (a) All weighted complete multipartite graphs are instantaneous universal mixing. This is in contrast to the fact that no unweighted complete multipartite graphs are uniform mixing (except for the four-cycle). (b) The weighted claw or star graph is a minimally connected instantaneous universal mixing graph. In fact, as a corollary, the unweighted claw is instantaneous uniform mixing. This adds a new family of uniform mixing graphs to a list that so far contains only the hypercubes. (c) 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. (d) 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.