Quantum Fields from Quantum Cellular Automata? A No-Go Theorem and a Path Forward

A quantum walk (QW) is a unitary analogue of a classical random walk, and a quantum cellular automaton (QCA) is a unitary analogue of a classical cellular automaton. QWs can be treated as the one-particle sector of a QCA. Since quantum walks on the body-centered cubic lattice give rise to relativistic wave equations in the long wavelength limit, it is natural to seek for QCAs that give rise to quantum field theories in a similar limit. We show that this can be done in one spatial dimension, with the QCA being naturally described in terms of creation and annihilation operators that create or destroy particle locally, evolve straightforwardly under the QCA unitary, and obey the usual anticommutation relations (ACR). However, generalizing this construction to two or more spatial dimensions fails: the requirements on the creation and annihilation operators are inconsistent with a local QCA. For a QCA to give rise to a quantum field theory in the long-wavelength limit, one must give up at least one of the desired properties. A likely choice is to give up the requirement that the creation and annihilation operators create and destroy localized states.