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We consider quantum cellular automata on a body-centred cubic lattice andprovide a simple derivation of the only two homogenous, local, isotropic, andunitary two-dimensional automata [G. M. D'Ariano and P. Perinotti, PhysicalReview A 90, 062106 (2014)]. Our derivation relies on the notion of Gram matrixand emphasises the link between the transition matrices that characterise theautomata and the body-centred cubic lattice: The transition matricesessentially are the matrix representation of the vertices of the lattice'sprimitive cell. As expected, the dynamics of these two automata reduce to theWeyl equation in the limit of small wave vectors and continuous time. We alsobriefly examine the four-dimensional case where we find two one-parameterfamilies of automata that reduce to the Dirac equation in a suitable limit.

Simple derivation of the Weyl and Dirac quantum cellular automata