Bulk--Boundary Correspondence for Chiral Symmetric Quantum Walks

TL;DR

This paper develops a framework for defining chiral symmetry in discrete-time quantum walks using timeframes, and establishes a bulk-boundary correspondence linking topological invariants to edge states.

Contribution

It introduces the concept of timeframes for quantum walks, enabling a consistent definition of chiral symmetry and topological invariants in these systems.

Findings

Defined chiral symmetry for quantum walks using timeframes

Identified a bulk ZxZ topological invariant for 1D quantum walks

Demonstrated bulk-boundary correspondence with edge states in example walks

Abstract

Discrete-time quantum walks (DTQW) have topological phases that are richer than those of time-independent lattice Hamiltonians. Even the basic symmetries, on which the standard classification of topological insulators hinges, have not yet been properly defined for quantum walks. We introduce the key tool of timeframes, i.e., we describe a DTQW by the ensemble of time-shifted unitary timestep operators belonging to the walk. This gives us a way to consistently define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using an "inversion symmetric" pulse sequence. For one-dimensional DTQW's with CS, we identify the bulk ZxZ topological invariant that controls the number of topologically protected 0 and pi energy edge states at the interfaces between different domains, and give simple formulas for these invariants. We illustrate this bulk--boundary correspondence for DTQW's on the example of the "4-step quantum walk", where tuning CS and particle-hole symmetry realizes edge states in various symmetry classes.