Mixing Times in Quantum Walks on the Hypercube

TL;DR

This paper analyzes the mixing times of discrete-time quantum walks on the hypercube, showing they reach a specific distribution faster than classical walks and that decoherence can optimize uniform mixing.

Contribution

It provides an explicit expression for the stationary distribution of quantum walks on the hypercube and investigates how decoherence influences mixing times.

Findings

Quantum walks mix to a non-uniform distribution in O(n) time.

Decoherence at a critical rate minimizes mixing time to uniform distribution.

Controlled decoherence can optimize and preserve uniformity in quantum walks.

Abstract

The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time O(n log n). We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally non-uniform) distribution pi(x) in time O(n) and the stationary distribution is determined by the initial state of the walk. An explicit expression for pi(x) is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered. We find that the probability distribution mixes to the uniform distribution as expected. However, the mixing time has a minimum at a critical decoherence rate $p \approx 0.1$. A similar effect was previously reported for the QW on the N-cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps to obtain--and preserve--a uniform distribution over the $2^n$ sites of the hypercube in the shortest possible time.