Homotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$

TL;DR

This paper characterizes the entire homotopy type of the space of finite propagation unitary operators on  sequences over , revealing its higher homotopy groups and extending previous index-based results.

Contribution

It provides a complete description of the homotopy type of the space of finite propagation unitary operators, including higher homotopy groups, which was previously unknown.

Findings

Determined the homotopy groups of the space of finite propagation unitary operators.

Extended the index theory to describe the entire homotopy type.

Analyzed the space of periodic finite propagation unitary operators.

Abstract

The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that $\pi_0$ of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable $\mathbb{C}$-valued $\mathbb{Z}$-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators.