  Quantum walks on lattices can give rise to relativistic wave equations in the
long-wavelength limit, but going beyond the single-particle case has proven
challenging, especially in more than one spatial dimension. We construct
quantum cellular automata for distinguishable particles based on two different
quantum walks, and show that by restricting to the antisymmetric and symmetric
subspaces, respectively, a multiparticle theory for free fermions and bosons in
three spatial dimensions can be produced. This construction evades a no-go
theorem that prohibits the usual fermionization constructions in more than one
spatial dimension. In the long-wavelength limit, these recover Dirac field
theory and Maxwell field theory, i.e., free QED.
