We study the out-of-equilibrium dynamics of the quantum cellular
automaton Rule 54 using a time-channel approach. We exhibit a family of
(non-equilibrium) product states for which we are able to describe
exactly the full relaxation dynamics. We use this to prove that finite
subsystems relax to a one-parameter family of Gibbs states. We also
consider inhomogeneous quenches. Specifically, we show that when the two
halves of the system are prepared in two different solvable states,
finite subsystems at finite distance from the centre eventually relax to
the non-equilibrium steady state (NESS) predicted by generalised
hydrodynamics. To the best of our knowledge, this is the first exact
description of the relaxation to a NESS in an interacting system and,
therefore, the first independent confirmation of generalised
hydrodynamics for an inhomogeneous quench.