Discrete quantum walks are dynamical protocols for controlling a singlequantum particle. Despite of its simplicity, quantum walks display richtopological phenomena and provide one of the simplest systems to study andunderstand topological phases. In this article, we review the physics ofdiscrete quantum walks in one and two dimensions in light of topologicalphenomena and provide elementary explanations of topological phases and theirphysical consequence, namely the existence of boundary states. We demonstratethat quantum walks are versatile systems that simulate many topological phaseswhose classifications are known for static Hamiltonians. Furthermore,topological phenomena appearing in quantum walks go beyond what has been knownin static systems; there are phenomena unique to quantum walks, being anexample of periodically driven systems, that do not exist in static systems.Thus the quantum walks not only provide a powerful tool as a quantum simulatorfor static topological phases but also give unique opportunity to studytopological phenomena in driven systems.
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