Estimating mixing properties of local Hamiltonian dynamics and continuous quantum random walks is PSPACE-hard

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.