Fermion Doubling in Quantum Cellular Automata

Abstract: A Quantum Cellular Automaton (QCA) is essentially an operator driving the evolution of particles on a lattice, through local unitaries. Because $\Delta_x=\Delta_t=\varepsilon$, 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 discrete-space but continuous-time settings of real-time/Hamiltonian Lattice Gauge Theories (LGTs), as the appearance of spurious solutions for all $\Delta_x=\varepsilon\neq 0$. We rigorously extend this analysis to the real-time discrete-space and discrete-time schemes that QCAs are. We demonstrate the existence of FD issues in QCAs. By applying a covering map on the Brillouin zone, we provide a flavoring-without-staggering way of fixing FD that does not break chiral symmetry. We explain how this method coexists with the Nielsen-Ninomiya no-go theorem, and illustrate this with a neutrino-like QCA.