  The simulation of quantum systems is a task for which quantum computers are
believed to give an exponential speedup as compared to classical ones. While
ground states of one-dimensional systems can be efficiently approximated using
Matrix Product States (MPS), their time evolution can encode quantum
computations, so that simulating the latter should be hard classically.
However, one might believe that for systems with high enough symmetry, and thus
insufficient parameters to encode a quantum computation, efficient classical
simulation is possible. We discuss supporting evidence to the contrary: We
provide a rigorous proof of the observation that a time independent local
Hamiltonian can yield a linear increase of the entropy when acting on a product
state in a translational invariant framework. This criterion has to be met by
any classical simulation method, which in particular implies that every global
approximation of the evolution requires exponential resources for any MPS based
method.
