  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.
