  We study the nonequilibrium dynamics of the Floquet quantum East model (a
Trotterized version of the kinetically constrained quantum East spin chain) at
its "deterministic point", where evolution is defined in terms of CNOT
permutation gates. We solve exactly the thermalization dynamics for a broad
class of initial product states by means of "space evolution". We prove: (i)
the entanglement of a block of spins grows at most at one-half the maximal
speed allowed by locality (i.e., half the speed of dual-unitary circuits); (ii)
if the block of spins is initially prepared in a classical configuration, speed
of entanglement is a quarter of the maximum; (iii) thermalization to the
infinite temperature state is reached exactly in a time that scales with the
size of the block.
