Fermion Doubling in Quantum Cellular Automata

TL;DR

The paper analyzes Fermion Doubling (FD) in real-time, discrete-spacetime Quantum Cellular Automata (QCAs) used to simulate (1+1)D and (3+1)D QED. It shows that standard QCA/QW discretizations exhibit FD via spurious Brillouin-zone regions and propagator poles, and introduces a flavor-staggering-only fix that corresponds to a covering-map on the Brillouin zone, yielding FD-free, chiral-symmetric flavored QCAs. The approach uses a geometric, topological interpretation (covering maps) to relate the spectral-folding problem to sublattice flavors, producing 2-sheet and 8-sheet flavorings for 1+1D and 3+1D cases, respectively. The work also discusses compatibility with Nielsen–Ninomiya no-go and demonstrates how, with flavor, the weak interaction can be represented on a lattice while preserving chirality, thus offering a pathway for realistic lattice QFT simulations on QCAs.Overall, the paper provides a rigorous framework for diagnosing and resolving FD in QCAs, linking spectral properties to lattice geometry and symmetry considerations, with explicit constructions for QED QCAs and neutrino-like chiral fermions on a spacetime lattice.

Abstract

A Quantum Cellular Automaton (QCA) is essentially an operator driving the evolution of particles on a lattice, through local unitaries. Because $Δ_t=Δ_x = ε$, 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 $Δ_x=ε\neq 0$. 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 $Δ_t =Δ_x = ε\neq 0$. 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.

Figures (8)

• Figure 1: Fermion-doubling neighborhoods for the symmetric-finite-differences scheme (yellow, green and blue), and (see Sec. \ref{['subsubsec:QWFD']}) for the $(1+1)$D Dirac discrete-time quantum walk (blue only). The neighborhoods having the same color are in fact identical due to the toric geometry.
• Figure 2: Fermion doubling for the $(3+1)$D Dirac discrete-time quantum walk. We draw the neighbourhoods on the hyper-surfaces where $E=0$ (left) and $E= \pm \pi/\epsilon$ (right).The neighborhood having the same colour are identical due to the hyper-toric geometry.
• Figure 3: Schematic representation of the dispersion-relation expressions of the $(1+1)$D Dirac QW (left) and its flavored version (right). On the left, due to the periodicity of $\mathscr{D}^{1}(E,p)$, the FD neighborhoods will pollute the continuum limit. On the right is the two-sheeted (BZ vision), or flavored (direct-space vision) solution to this problem.
• Figure 4: Sublattices of the $(1+1)$D flavored Dirac QCA.
• Figure 5: Scheùatic representation of the dispersion-relation expressions of the $(3+1)$D Dirac QW (left) and its flavored version (right). On the left, the FD neighborhoods are projected on the hyperplanes $E$---$p_x$, $E$---$p_y$, and $E$---$p_z$. On the right, the eight-sheeted solution.

Theorems & Definitions (1)

• Definition 1