Simple derivation of the Weyl and Dirac quantum cellular automata

Abstract

We consider quantum cellular automata on a body-centred 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, Physical Review A 90, 062106 (2014)]. Our derivation relies on the notion of Gram matrix and emphasises the link between the transition matrices that characterise the automata and the body-centred 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.

v2: corrected typos, rephrasing of misleading sentences, one short proof added. To be published in PRA

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• Discrete Lorentz covariance for Quantum Walks and Quantum Cellular Automata
• Physics Without Physics: The Power of Information-theoretical Principles
• Special Relativity in a Discrete Quantum Universe
• Quantum walks with a one-dimensional coin

Cited by in corpus (7)

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• Discrete spacetime, quantum walks and relativistic wave equations
• Quantum cellular automata and quantum field theory in two spatial dimensions
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• Fermionic and bosonic quantum field theories from quantum cellular automata in three spatial dimensions
• An overview of Quantum Cellular Automata