# Stationary amplitudes of quantum walks on the higher-dimensional integer lattice

## TL;DR

This paper derives the stationary amplitudes for quantum walks on higher-dimensional integer lattices, extending the understanding from one-dimensional cases and providing explicit solutions for the eigenfunctions with eigenvalue 1.

## Contribution

It introduces a method to determine stationary amplitudes for quantum walks on d-dimensional lattices with finite support, including the Grover walk, filling a gap in higher-dimensional quantum walk analysis.

## Key findings

- Explicit stationary amplitudes for quantum walks on d-dimensional lattices.
- Stationary measures for Grover walks derived from these amplitudes.
- Eigenfunction solutions with eigenvalue 1 identified for stationary states.

## Abstract

Stationary measures of quantum walks on the one-dimensional integer lattice are well studied. However, the stationary measure for the higher dimensional case has not been clarified. In this paper, we give the stationary amplitude for quantum walks on the d-dimensional integer lattice with a finite support by solving the corresponding eigenvalue problem. As a corollary, we can obtain the stationary measures of the Grover walks. In fact, the amplitude for the stationary measure is an eigenfunction with eigenvalue 1.