  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 (although they are
highly entangled). We consider a variety of models in both one and two
dimensions, involving Clifford gates and Toffoli gates. For some of these
models, we explicitly construct conserved densities; thus these models are
"integrable." The other models do not seem to be integrable; yet, for some
system sizes and boundary conditions, their eigenstate entanglement is strongly
subthermal. Some of the models have exponentially many eigenstates in which one
or more sites are "disentangled" from the rest of the system, as a consequence
of reflection symmetry.
