Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk
Hideaki Obuse, János K. Asbóth, Yuki Nishimura +1 more
Quantum walks, whose dynamics is prescribed by alternating unitary coin and shift operators, possess topological phases akin to those of Floquet topological insulators, driven by a time-periodic field. While there is ample theoretical work on topological phases of quantum walks where the coin operators are spin rotations, in experiments a different coin, the Hadamard operator, is often used instead. This was the case in a recent photonic quantum walk experiment, where protected edge states were observed between two bulks whose topological invariants, as calculated by the standard theory, were the same. This hints at a hidden topological invariant in the Hadamard quantum walk. We establish a relation between the Hadamard and the spin rotation operator, which allows us to apply the recently developed theory of topological phases of quantum walks to the one-dimensional Hadamard quantum walk. The topological invariants we derive account for the edge state observed in the experiment; we thus reveal the hidden topological invariant of the one-dimensional Hadamard quantum walk.
arXiv:1505.03264•Physical Review B•Jul 23, 2015•79 citations