  Any discrete quantum process is represented by a sequence of quantum
channels. We consider ergodic quantum processes obtained by a map that takes
the points along the trajectory of a discrete ergodic dynamical system to the
space of quantum channels. Under natural irreducibility conditions, we obtain a
theorem showing that the state under such a process converges exponentially
fast to an ergodic sequence depending on the process, but independent of the
initial state. As an application, we describe the thermodynamic limit of
ergodic matrix product states and prove that the 2-point correlations of local
observables in such states decay exponentially with their distance in the bulk.
