Recurrences in three-state quantum walks on a plane

TL;DR

This paper investigates how dimensionality affects the behavior of three-state quantum walks on a plane, revealing that certain coin operators lead to recurrence without localization, with propagation resembling one-dimensional dynamics.

Contribution

It demonstrates that only a specific subclass of coin operators induces recurrence in three-state quantum walks on a plane, and shows the absence of localization.

Findings

Three-state Grover walk on a plane does not exhibit trapping or recurrence.

Only a special subclass of coin operators can cause recurrence.

Propagation in recurrent cases is quasi one-dimensional.

Abstract

We analyze the role of dimensionality in the time evolution of discrete time quantum walks through the example of the three-state walk on a two-dimensional, triangular lattice. We show that the three-state Grover walk does not lead to trapping (localization) or recurrence to the origin, in sharp contrast to the Grover walk on the two dimensional square lattice. We determine the power law scaling of the probability at the origin with the method of stationary phase. We prove that only a special subclass of coin operators can lead to recurrence and there are no coins leading to localization. The propagation for the recurrent subclass of coins is quasi one-dimensional.