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Quantum walks on lattices can give rise to relativistic wave equations in thelong-wavelength limit, but going beyond the single-particle case has provenchallenging, especially in more than one spatial dimension. We constructquantum cellular automata for distinguishable particles based on two differentquantum walks, and show that by restricting to the antisymmetric and symmetricsubspaces, respectively, a multiparticle theory for free fermions and bosons inthree spatial dimensions can be produced. This construction evades a no-gotheorem that prohibits the usual fermionization constructions in more than onespatial dimension. In the long-wavelength limit, these recover Dirac fieldtheory and Maxwell field theory, i.e., free QED.

Fermionic and bosonic quantum field theories from quantum cellular automata in three spatial dimensions