Dirac equation with ultraviolet cutoff and quantum walk

TL;DR

This paper explains the unique properties of quantum walks using the Dirac equation with an ultraviolet cutoff, linking quantum walk behavior to relativistic quantum mechanics and causality constraints.

Contribution

It introduces a relativistic quantum walk model based on the Dirac equation with a cutoff, explaining the bounded support and inverted-bell limit distributions.

Findings

Limit distributions have bounded support due to relativistic causality.

The Dirac equation with cutoff models quantum walk pseudovelocities.

Lower-dimensional walks' distributions derive from the 3D Dirac model.

Abstract

The weak convergence theorems of the one- and two-dimensional simple quantum walks, SQW$^{(d)}, d=1,2$, show a striking contrast to the classical counterparts, the simple random walks, SRW$^{(d)}$. The limit distributions have novel structures such that they are inverted-bell shaped and the supports of them are bounded. In the present paper we claim that these properties of the SQW$^{(d)}$ can be explained by the theory of relativistic quantum mechanics. We show that the Dirac equation with a proper ultraviolet cutoff can provide a quantum walk model in three dimensions, where the walker has a four-component qubit. We clarify that the pseudovelocity $\V(t)$ of the quantum walker, which solves the Dirac equation, is identified with the relativistic velocity. Since the quantum walker should be a tardyon, not a tachyon, $|\V(t)| < c$, where $c$ is the speed of light, and this restriction (the causality) is the origin of the finiteness of supports of the limit distributions universally found in quantum walk models. By reducing the number of components of momentum in the Dirac equation, we obtain the limit distributions of pseudovelocities for the lower dimensional quantum walks. We show that the obtained limit distributions for the one- and two-dimensional systems have the common features with those of SQW$^{(1)}$ and SQW$^{(2)}$.