Mixing Times in Quantum Walks on Two-Dimensional Grids

TL;DR

This paper investigates the mixing times of discrete-time quantum walks on 2D grids, deriving exact stationary distributions and analyzing how these relate to the efficiency of quantum search algorithms.

Contribution

It provides an exact expression for stationary distributions on odd-sided lattices and explores the connection between mixing times and search algorithm complexity.

Findings

Exact stationary distribution for odd-sided lattices derived.

Numerical analysis of mixing times with modified coin operators.

Discussion of the relation between mixing times and search algorithm running times.

Abstract

Mixing properties of discrete-time quantum walks on two-dimensional grids with torus-like boundary conditions are analyzed, focusing on their connection to the complexity of the corresponding abstract search algorithm. In particular, an exact expression for the stationary distribution of the coherent walk over odd-sided lattices is obtained after solving the eigenproblem for the evolution operator for this particular graph. The limiting distribution and mixing time of a quantum walk with a coin operator modified as in the abstract search algorithm are obtained numerically. On the basis of these results, the relation between the mixing time of the modified walk and the running time of the corresponding abstract search algorithm is discussed.