Discrete-time quantum walks as fermions of lattice gauge theory

It is shown that discrete-time quantum walks can be used to digitize, i.e., to time discretize fermionic models of continuous-time lattice gauge theory. The resulting discrete-time dynamics is thus not only manifestly unitary, but also ultralocal, i.e., the particle's speed is upper bounded, as in standard relativistic quantum field theories. The lattice chiral symmetry of staggered fermions, which corresponds to a translational invariance, is lost after the requirement of ultralocality of the evolution; this fact is an instance of Meyer's 1996 no-go results stating that no nontrivial scalar quantum cellular automaton can be translationally invariant [D. A. Meyer, J. Stat. Phys. 85, 551 (1996); Phys. Lett. A 223, 337 (1996)]. All results are presented in a single-particle framework and for a (1+1)-dimensional space-time.

Figure 1

Comparison between the continuous-time (a) left-right and (b) staggered dynamics. The thick, solid blue (dashed red) arrows indicate how the ${\psi }_p^R$'s or ${\varphi }_{2p+1}$'s (${\psi }^L$'s or ${\varphi }_{2p}$'s) feed the ${\psi }_p^L$'s or ${\varphi }_{2p}$'s (${\psi }^R$'s or ${\varphi }_{2p+1}$'s). The thin, light-blue arrows indicate a possible way of performing the staggering starting from the left-right discretization.