# Quantum walks: Schur functions meet symmetry protected topological phases

## TL;DR

This paper establishes a novel connection between Schur functions and symmetry protected topological phases in quantum walks, enabling a linear algebra approach to classify these phases even without translation invariance.

## Contribution

It introduces a Schur function framework to classify topological phases of quantum walks, extending analysis beyond translation-invariant systems.

## Key findings

- Topological indices are encoded by matrix Schur functions.
- Classification applies to non-translation invariant quantum walks.
- Schur approach covers all symmetry types of the tenfold way.

## Abstract

This paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetry indices of finite-dimensional unitaries obtained by evaluating such matrix Schur functions at the symmetry protected points $\pm1$. The Schur representation fully covers the complete set of symmetry indices for 1D quantum walks with a group of symmetries realizing any of the symmetry types of the tenfold way. The main advantage of the Schur approach is its validity in the absence of translation invariance, which allows us to go beyond standard Fourier methods, leading to the complete classification of non-translation invariant phases for typical examples.