Abstract
We consider quantum cellular automata on a body-centered cubic lattice and provide a simple derivation of the only two homogenous, local, isotropic, and unitary two-dimensional automata [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)]. Our derivation relies on the notion of Gram matrix and emphasizes the link between the transition matrices that characterize the automata and the body-centered cubic lattice: The transition matrices essentially are the matrix representation of the vertices of the lattice's primitive cell. As expected, the dynamics of these two automata reduce to the Weyl equation in the limit of small wave vectors and continuous time. We also briefly examine the four-dimensional case, where we find two one-parameter families of automata that reduce to the Dirac equation in a suitable limit.
- Received 22 January 2017
DOI:https://doi.org/10.1103/PhysRevA.95.062344
©2017 American Physical Society