Ergodicity in discrete-time quantum walks

TL;DR

This paper analyzes ergodicity in discrete-time quantum walks, establishing spectral conditions for equidistribution in one dimension and criteria for ergodicity in higher dimensions, with numerous illustrative examples.

Contribution

It provides a complete spectral characterization of ergodicity in one dimension and introduces a new spectral property, 'No Repeating Graphs', for higher dimensions.

Findings

Complete equivalence between spectrum and equidistribution in 1D.

Spectral criterion 'No Repeating Graphs' for ergodicity in higher dimensions.

Multiple examples illustrating ergodicity criteria.

Abstract

We undertake a detailed analysis of ergodicity for homogeneous discrete-time quantum walks on the integer lattice. The most significant result of our paper holds in dimension one, and gives a complete equivalence between the absolutely continuous spectrum of the unitary operator encoding the walk, and the equidistribution of its dynamics in position space, which appears for the first time in the context of large-volume quantum ergodicity. In higher dimensions, we give a criterion for full and partial ergodicity in terms of a finer property of the spectrum which we dub ``No Repeating Graphs'', and we distinguish how strongly the equidistribution is taking place (weak convergence vs total variation). Many examples are included to illustrate the criterion and to distinguish between the types of ergodicity.