Quantum walks (QWs) describe particles evolving coherently on a graph. The internal degree of freedom corresponds to a Hilbert space, called a 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 a coin system of dimension 2, whereas for a 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.

• Received 24 March 2016

DOI:https://doi.org/10.1103/PhysRevA.93.062334