  A major topic of (classical) ergodic theory is to examine qualitatively how
the phase space of dynamical systems is penetrated by the orbits of their
dynamics. We consider interacting qubit systems with dynamics according to
4-local Hamiltonians and continuous quantum random walks. For these systems one
could use the von Neumann entropy of the time-average to characterize the
mixing properties of the corresponding orbits, i.e., what portion of the state
space and how uniformly it is filled out by the orbits. We show that the
problem of estimating this entropy is PSPACE-hard.
