How does Grover walk recognize the shape of crystal lattice?

TL;DR

This paper analyzes the limit distribution support of Grover walks on various crystal lattices, revealing that the outer boundary of the support is always an ellipse determined by the fundamental lattice.

Contribution

It computes the orbit regions for Grover walks on triangular, hexagonal, and kagome lattices, showing the support boundary is an ellipse depending on the lattice realization.

Findings

Outer support boundary is an ellipse for all studied lattices.

Shape of the ellipse depends solely on the fundamental lattice.

Provides explicit orbit regions for specific crystal lattices.

Abstract

We consider the support of the limit distribution of the Grover walk on crystal lattices with the linear scaling. The orbit of the Grover walk is denoted by the parametric plot of the pseudo-velocity of the Grover walk in the wave space. The region of the orbit is the support of the limit distribution. In this paper, we compute the regions of the orbits for the triangular, hexagonal and kagome lattices. We show every outer frame of the support is described by an ellipse. The shape of the ellipse depends only on the realization of the fundamental lattice of the crystal lattice in $\mathbb{R}^2$.