Discrete time Dirac quantum walk in 3+1 dimensions

TL;DR

This paper develops a discrete-time quantum walk model in 3+1 dimensions that converges to the Dirac equation, providing exact Fourier implementation and analyzing particle-antiparticle dynamics.

Contribution

It introduces an exact Fourier-based implementation of Dirac quantum walks in multiple dimensions and analyzes their particle-antiparticle behavior.

Findings

Exact Fourier implementation of Dirac quantum walks in 1, 2, and 3 dimensions

Approximate dispersive equation describes particle-antiparticle kinematics

Analytical and numerical validation of position expectation jittering

Abstract

In this paper we consider quantum walks whose evolution converges to the Dirac equation one in the limit of small wave-vectors. We show exact Fast Fourier implementation of the Dirac quantum walks in one, two and three space dimensions. The behaviour of particle states, defined as states smoothly peaked in some wave-vector eigenstate of the walk, is described by an approximated dispersive differential equation that for small wave-vectors gives the usual Dirac particle and antiparticle kinematics. The accuracy of the approximation is provided in terms of a lower bound on the fidelity between the exactly evolved state and the approximated one. The jittering of the position operator expectation value for states having both a particle and an antiparticle component is analytically derived and observed in the numerical implementations.