# An index theorem for split-step quantum walks

## TL;DR

This paper establishes a new index theorem for split-step quantum walks, linking their Witten index to winding numbers of associated functions, providing an alternative derivation of recent index formulas.

## Contribution

It proves that the Witten index equals the difference in winding numbers, offering a novel mathematical connection and an alternative proof for existing index formulas in split-step quantum walks.

## Key findings

- Witten index coincides with the difference of winding numbers.
- Provides an alternative derivation of the index formula.
- Connects supersymmetric quantum walk indices with topological invariants.

## Abstract

Split-step quantum walks are models of supersymmetric quantum walk, and thus their Witten indices can be defined. We prove that the Witten index of a split-step quantum walk coincides with the difference between the winding numbers of functions corresponding to the right-limit of coins and the left-limit of coins. As a corollary, we give an alternative derivation of the index formula for split-step quantum walks, which is recently obtained by Suzuki and Tanaka.