Exceptional configurations of quantum walks with Grover's coin

TL;DR

This paper investigates how Grover's coin affects quantum walk search on a 2D grid, revealing a broad class of configurations where the search probability remains low, extending previous knowledge of exceptional cases.

Contribution

It identifies a wide class of exceptional configurations for Grover's coin in quantum walks, expanding understanding of search limitations.

Findings

Grover's coin leads to many exceptional configurations.

Probability of finding marked locations does not increase over time in these cases.

Extends known exceptional configuration classes beyond previous diagonal cases.

Abstract

We study search by quantum walk on a two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [AKR05]. We show what the most natural coin transformation - Grover's diffusion transformation - has a wide class of exceptional configurations of marked locations, for which the probability of finding any of the marked locations does not grow over time. This extends the class of known exceptional configurations; until now the only known such configuration was the "diagonal construction" by Ambainis and Rivosh [AR08]