Facilitated quantum cellular automata as simple models with non-thermal eigenstates and dynamics

We introduce and describe a class of simple facilitated quantum spin models in which the dynamics is due to the repeated application of unitary gates. The gates are applied periodically in time, so their combined action constitutes a Floquet unitary. The dynamics of the models we discuss can be classically simulated, and their eigenstates classically constructed (despite being highly entangled). We consider a variety of models in one and two dimensions: we are primarily concerned with models involving Clifford gates, but we also briefly discuss a model involving Clifford and Toffoli gates. Two of these models are integrable free-particle models; we construct their conserved densities. For all five models, however, the eigenstate entanglement (in all cases) and operator dynamics (in all but one case) differ from those of generic chaotic systems. In addition, two of these models have exponentially many eigenstates in which one or more sites 'disentangle' from the rest of the system, as a consequence of reflection symmetry.