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Quantum walks on lattices can give rise to one-particle relativistic waveequations in the long-wavelength limit. In going to multiple particles, quantumcellular automata (QCA) are natural generalizations of quantum walks. In onespatial dimension, the quantum walk can be "promoted" to a QCA that, in thelong-wavelength limit, gives rise to the Dirac quantum field theory (QFT) fornoninteracting fermions. This QCA/QFT correspondence has both theoretical andpractical applications, but there are obstacles to similar constructions in twoor more spatial dimensions. Here we show that a method of constructionemploying distinguishable particles confined to the completely antisymmetricsubspace yields a QCA in two spatial dimensions that gives rise to the 2D DiracQFT. Generalizing to 3D will entail some additional complications, but noconceptual barriers. We examine how this construction evades the "no go"results in earlier work.

Quantum cellular automata and quantum field theory in two spatial dimensions