Localization and limit laws of a three-state alternate quantum walk on a two-dimensional lattice

TL;DR

This paper introduces a three-state alternate quantum walk on a two-dimensional lattice, demonstrating localization and deriving limit laws, which differ from two-state models and have implications for physical systems.

Contribution

It presents a novel three-state quantum walk model with a parameterized coin, showing localization and establishing new limit theorems not observed in simpler models.

Findings

Localization observed in the three-state walk

Two limit theorems for return probability and position distribution

Differences from two-state and four-state quantum walks

Abstract

A two-dimensional discrete-time quantum walk (DTQW) can be realized by alternating a two-state DTQW in one spatial dimension followed by an evolution in the other dimension. This was shown to reproduce a probability distribution for a certain configuration of a four-state DTQW on a two-dimensional lattice. In this work we present a three-state alternate DTQW with a parameterized coin-flip operator and show that it can produce localization that is also observed for a certain other configuration of the four-state DTQW and non-reproducible using the two-state alternate DTQW. We will present two limit theorems for the three-state alternate DTQW. One of the limit theorems describes a long-time limit of a return probability, and the other presents a convergence in distribution for the position of the walker on a rescaled space by time. We will also outline the relevance of these walks in physical systems.