  We study the out-of-equilibrium dynamics of the quantum cellular automaton
known as "Rule 54". For a class of low-entangled initial states, we provide an
analytic description of the effect of the global evolution on finite subsystems
in terms of simple quantum channels, which gives access to the full
thermalization dynamics at the microscopic level. As an example, we provide
analytic formulae for the evolution of local observables and R\'enyi entropies.
We show that, in contrast to other known examples of exactly solvable quantum
circuits, Rule 54 does not behave as a simple Markovian bath on its own parts,
and displays typical non-equilibrium features of interacting integrable
many-body quantum systems such as finite relaxation rate and
interaction-induced dressing effects. Our study provides a rare example where
the full thermalization dynamics can be solved exactly at the microscopic
level.
