Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk

TL;DR

This paper reveals a hidden topological invariant in the one-dimensional Hadamard quantum walk by relating it to spin rotation operators, explaining observed edge states in experiments.

Contribution

It establishes a connection between Hadamard and spin rotation operators, enabling the application of topological phase theory to Hadamard quantum walks.

Findings

Identifies a hidden topological invariant in Hadamard quantum walks.

Explains the experimentally observed edge states.

Provides a theoretical framework linking Hadamard and spin rotation operators.

Abstract

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.