Bounds for mixing time of quantum walks on finite graphs

TL;DR

This paper establishes bounds on the mixing time of discrete-time quantum walks on finite graphs, showing no speed-up over classical walks in specific cases and analyzing decoherence effects on convergence.

Contribution

It provides new inequalities for quantum walk mixing times, compares quantum and classical mixing, and introduces criteria for convergence of non-unitary quantum walks.

Findings

Quantum walks on certain graphs do not outperform classical walks in mixing speed.

New bounds are established for the mixing time of quantum walks.

Decoherence in quantum walks affects their convergence to stationary distributions.

Abstract

Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov, Ambainis, Kempe and Vazirani (2001) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speed-up in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e., walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.