Quantum walks with a one-dimensional coin

TL;DR

This paper classifies quantum walks with a one-dimensional coin on various groups, showing trivial evolution on infinite Abelian groups and characterizing walks on the infinite dihedral group, including well-known examples like Weyl, Dirac, and Hadamard walks.

Contribution

It provides a complete classification of scalar quantum walks on infinite Abelian groups and characterizes walks on the infinite dihedral group, linking them to spinorial walks on Z.

Findings

Scalar QWs on infinite Abelian groups are trivial.

Class of QWs on the infinite dihedral group is characterized.

Includes well-known walks like Weyl, Dirac, and Hadamard.

Abstract

Quantum walks (QWs) describe particles evolving coherently on a lattice. The internal degree of freedom corresponds to a Hilbert space, called coin system. We consider QWs on Cayley graphs of some group $G$. In the literature, investigations concerning infinite $G$ have been focused on graphs corresponding to $G=\mathbb{Z}^d$ with coin system of dimension 2, whereas for one-dimensional coin (so called scalar QWs) only the case of finite $G$ has been studied. Here we prove that the evolution of a scalar QW with $G$ infinite Abelian is trivial, providing a thorough classification of this kind of walks. Then we consider the infinite dihedral group $D_\infty$, that is the unique non-Abelian group $G$ containing a subgroup $H\cong\mathbb{Z}$ with two cosets. We characterize the class of QWs on the Cayley graphs of $D_\infty$ and, via a coarse-graining technique, we show that it coincides with the class of spinorial walks on $\mathbb{Z}$ which satisfies parity symmetry. This class of QWs includes the Weyl and the Dirac QWs. Remarkably, there exist also spinorial walks that are not coarse-graining of a scalar QW, such as the Hadamard walk.