Fermion Doubling in Quantum Cellular Automata

TL;DR

This paper analyzes Fermion Doubling issues in Quantum Cellular Automata (QCA) for simulating relativistic quantum particles, proposing a flavor-staggering fix that preserves chiral symmetry and enables lattice models of chiral fermions.

Contribution

It extends Fermion Doubling analysis to discrete-time QCAs and introduces a flavor-staggering method that avoids FD without breaking chiral symmetry.

Findings

Fermion Doubling exists in QCAs with discrete time and space.

A flavor-staggering fix can eliminate FD while preserving chiral symmetry.

The method enables lattice models of chiral fermions interacting via weak force.

Abstract

A Quantum Cellular Automaton (QCA) is essentially an operator driving the evolution of particles on a lattice, through local unitaries. Because $\Delta_t=\Delta_x = \epsilon$, QCAs constitute a privileged framework to cast the digital quantum simulation of relativistic quantum particles and their interactions with gauge fields, e.g., $(3+1)$D Quantum Electrodynamics (QED). But before they can be adopted, simulation schemes for high-energy physics need prove themselves against specific numerical issues, of which the most infamous is Fermion Doubling (FD). FD is well understood in particular in the real-time, discrete-space \emph{but} continuous-time settings of Hamiltonian Lattice Gauge Theories (LGTs), as the appearance of spurious solutions for all $\Delta_x=\epsilon\neq 0$. We rigorously extend this analysis to the real-time, discrete-space \emph{and} discrete-time schemes that QCAs are. We demonstrate the existence of FD issues in QCAs for $\Delta_t =\Delta_x = \epsilon \neq 0$. By applying a covering map on the Brillouin zone, we provide a flavor-staggering-only way of fixing FD that does not break the chiral symmetry of the massless scheme. We explain how this method coexists with the Nielsen-Ninomiya no-go theorem, and give an example of neutrino-like QCA showing that our model allows to put chiral fermions interacting via the weak interaction on a spacetime lattice, without running into any FD problem.