Localization in Quantum Walks on a Honeycomb Network

TL;DR

This paper investigates localization phenomena in discrete-time quantum walks on honeycomb networks, deriving conditions for localization, analyzing initial state effects, and confirming findings with numerical simulations.

Contribution

It provides a rigorous mathematical framework for localization in quantum walks on honeycomb structures, including new conditions and analysis for various coin operators.

Findings

Localization probability depends on initial coin state

Sufficient conditions for localization are identified

Theoretical results match numerical simulations

Abstract

We systematically study the localization effect in discrete-time quantum walks on a honeycomb network and establish the mathematical framework. We focus on the Grover walk first and rigorously derive the limit form of the walker's state, showing it has a certain probability to be localized at the starting position. The relationship between localization and the initial coin state is concisely represented by a linear map. We also define and calculate the average probability of localization by generating random initial states. Further, coin operators varying with positions are considered and the sufficient condition for localization is discussed. We also similarly analyze another four-state Grover walk. Theoretical predictions are all in accord with numerical simulation results. Finally, our results are compared with previous works to demonstrate the unusual trapping effect of quantum walks on a honeycomb network, as well as the advantages of our method.