Topological phenomena in quantum walks; elementary introduction to the physics of topological phases

TL;DR

Quantum walks serve as simple, versatile platforms for exploring and simulating topological phases, revealing both known and novel phenomena, including boundary states and effects unique to driven systems.

Contribution

This paper provides an elementary introduction to topological phases in quantum walks, highlighting their ability to simulate static topological phases and uncovering phenomena exclusive to driven systems.

Findings

Quantum walks exhibit rich topological phenomena including boundary states.

They can simulate known topological phases classified for static Hamiltonians.

Unique topological effects arise in quantum walks due to their driven nature.

Abstract

Discrete quantum walks are dynamical protocols for controlling a single quantum particle. Despite of its simplicity, quantum walks display rich topological phenomena and provide one of the simplest systems to study and understand topological phases. In this article, we review the physics of discrete quantum walks in one and two dimensions in light of topological phenomena and provide elementary explanations of topological phases and their physical consequence, namely the existence of boundary states. We demonstrate that quantum walks are versatile systems that simulate many topological phases whose classifications are known for static Hamiltonians. Furthermore, topological phenomena appearing in quantum walks go beyond what has been known in static systems; there are phenomena unique to quantum walks, being an example of periodically driven systems, that do not exist in static systems. Thus the quantum walks not only provide a powerful tool as a quantum simulator for static topological phases but also give unique opportunity to study topological phenomena in driven systems.