Quantum Information Processing

, Volume 11, Issue 5, pp 1107–1148 | Cite as

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

  • Takuya Kitagawa


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.


Quantum walk Topological phases Periodically driven systems Floquet states Non-equilibrium phenomena Topological phenomena in driven systems Dynamically induced phase Quantum simulator Quantum Hall effect Topological insulator Zero energy state 


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Physics DepartmentHarvard UniversityCambridgeUSA

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