# Convergence theorems on multi-dimensional homogeneous quantum walks

## TL;DR

This paper develops a comprehensive framework for multi-dimensional homogeneous quantum walks, proving their limit distributions and convergence properties of associated 1-cocycles, advancing understanding of their asymptotic behavior.

## Contribution

It introduces a general framework for d-dimensional quantum walks and establishes new convergence theorems for their limit distributions and 1-cocycles.

## Key findings

- Existence of limit distribution for homogeneous quantum walks.
- Convergence of averages of 1-cocycles related to position observables.
- Support of initial vectors not required to be finite.

## Abstract

We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. Among them, existence of limit distribution of homogeneous walks is proved. In this theorem, the support of the initial unit vector is not necessarily finite. We also pay attention on 1-cocycles. For homogeneous walks, convergence of averages of 1-cocycles associated to the position observable is also proved.