Localization of Two-Dimensional Five-State Quantum Walks

TL;DR

This paper rigorously analyzes a two-dimensional five-state quantum walk, deriving the wave function and probability distribution, and investigates whether localization occurs, i.e., if the particle remains localized or disperses over time.

Contribution

It provides a detailed mathematical analysis of a 2D five-state quantum walk, including wave function calculation and localization criteria, advancing understanding of quantum walk dynamics.

Findings

Wave function explicitly derived for any initial state.

Probability distribution over the plane characterized.

Localization behavior analyzed with conditions for convergence or dispersion.

Abstract

We investigate a generalized Hadamard walk in two dimensions with five inner states. The particle governed by a five-state quantum walk (5QW) moves, in superposition, either leftward, rightward, upward, or downward according to the inner state. In addition to the four degrees of freedom, it is allowed to stay at the same position. We calculate rigorously the wave function of the particle starting from the origin in the plane for any initial state, and give the spatial distribution of probability of finding the particle. We also investigate the localization problem for the two-dimensional five-state quantum walk: Does the probability of finding a particle anywhere on the plane converge to zero even after infinite time steps except initial states?