The Dirac equation as a quantum walk: higher dimensions, observational convergence

TL;DR

This paper demonstrates that a quantum walk model can accurately simulate the Dirac equation across higher dimensions, with proven convergence to the continuous solution, supporting its use in quantum simulation of relativistic particles.

Contribution

It provides a simple derivation of the quantum walk model for the Dirac equation in all dimensions and proves its convergence to the continuous solution with quantifiable error.

Findings

Quantum walk model converges to Dirac equation solutions with O(ε^2) error.

The derivation applies to all dimensions and hyperbolic symmetric systems.

Supports quantum simulation of relativistic particles.

Abstract

The Dirac equation can be modelled as a quantum walk, with the quantum walk being: discrete in time and space (i.e. a unitary evolution of the wave-function of a particle on a lattice); homogeneous (i.e. translation-invariant and time-independent), and causal (i.e. information propagates at a bounded speed, in a strict sense). This quantum walk model was proposed independently by Succi and Benzi, Bialynicki-Birula and Meyer: we rederive it in a simple way in all dimensions and for hyperbolic symmetric systems in general. We then prove that for any time t, the model converges to the continuous solution of the Dirac equation at time t, i.e. the probability of observing a discrepancy between the model and the solution is an O({\epsilon}^2), with {\epsilon} the discretization step. At the practical level, this result is of interest for the quantum simulation of relativistic particles. At the theoretical level, it reinforces the status of this quantum walk model as a simple, discrete toy model of relativistic particles. Keywords: Friedrichs symmetric hyperbolic systems, Quantum Walk, Quantum Lattice Gas Automata, Quantum Computation, Trotter-Kato, Baker-Campbell-Thomson, Operator splitting, Lax theorem