Quantum walks with a one-dimensional coin

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.

8 pages, 4 figures

References in corpus (5)

• Exponential algorithmic speedup by quantum walk
• Quantum Walks and discrete Gauge Theories
• Discrete Lorentz covariance for Quantum Walks and Quantum Cellular Automata
• One-Dimensional Coinless Quantum Walks
• Moments of Coinless Quantum Walks on Lattices

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• Noisy quantum walks of two indistinguishable interacting particles
• Solutions of a two-particle interacting quantum walk
• Discrete spacetime, quantum walks and relativistic wave equations
• Isotropic quantum walks on lattices and the Weyl equation
• Detection of discrete spacetime by matter interferometry
• Discrete-time quantum walks as fermions of lattice gauge theory
• Quantum field theory from a quantum cellular automaton in one spatial dimension and a no-go theorem in higher dimensions
• Virtually Abelian Quantum Walks
• Coarse-grained quantum cellular automata
• Simple derivation of the Weyl and Dirac quantum cellular automata
• Chirality from quantum walks without quantum coin
• Scalar fermionic cellular automata on finite Cayley graphs
• Quantum field theory from first principles
• Simulating Non Commutative Geometry with Quantum Walks