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

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.

11 pages, 7 figures

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• Dynamics of discrete-time quantum walk with time-correlated unitary noise
• Limit theorems and localization of three state quantum walks on a line defined by generalized Grover coins
• Topological delocalization in the completely disordered two-dimensional quantum walk
• Controlled Alternate Quantum Walks based Quantum Hash Function
• Conditional limit measure of one-dimensional quantum walk with absorbing sink
• Quantum Simulation of Neutrino Oscillation and Dirac Particle Dynamics in Curved Space-time
• Three-state quantum walk on the Cayley Graph of the Dihedral Group