Mixing of Quantum Walks on Generalized Hypercubes

TL;DR

This paper investigates the robustness of quantum uniform mixing on hypercube-like graphs, revealing conditions under which mixing is preserved or lost, and characterizing the mixing properties of Hamming and bunkbed graphs.

Contribution

It provides a tight characterization of quantum uniform mixing on Hamming graphs and explores how augmentations affect mixing on hypercubes and related graphs.

Findings

Quantum mixing on hypercubes is robust under adding certain perfect matchings.

Hamming graphs are not uniform mixing if arity exceeds 5.

Bunkbed graphs with Fourier support constraints are not uniform mixing.

Abstract

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: (1) The graph obtained by augmenting the hypercube with an additive matching is instantaneous uniform mixing whenever the parity of the matching is even, but with a slower mixing time. This strictly includes Moore-Russell's result on the hypercube. (2) The class of Hamming graphs is not uniform mixing if and only if its arity is greater than 5. This is a tight characterization of quantum uniform mixing on Hamming graphs; previously, only the status of arity less than 5 was known. (3) The bunkbed graph B(A[f]), defined by a hypercube-circulant matrix A and a Boolean function f, is not uniform mixing if the Fourier transform of f has small support. This explains why the hypercube is uniform mixing and why the join of two hypercubes is not.