Unitarity in one dimensional nonlinear quantum cellular automata

Abstract

Unitarity of the global evolution is an extremely stringent condition on finite state models in discrete spacetime. Quantum cellular automata, in particular, are tightly constrained. In previous work we proved a simple No-go Theorem which precludes nontrivial homogeneous evolution for linear quantum cellular automata. Here we carefully define general quantum cellular automata in order to investigate the possibility that there be nontrivial homogeneous unitary evolution when the local rule is nonlinear. Since the unitary global transition amplitudes are constructed from the product of local transition amplitudes, infinite lattices require different treatment than periodic ones. We prove Unitarity Theorems for both cases, expressing the equivalence in $1+1$ dimensions of global unitarity and certain sets of constraints on the local rule, and then show that these constraints can be solved to give a variety of multiparameter families of nonlinear quantum cellular automata. The Unitarity Theorems, together with a Surjectivity Theorem for the infinite case, also imply that unitarity is decidable for one dimensional cellular automata.