  We determine the computational power of preparing Projected Entangled Pair
States (PEPS), as well as the complexity of classically simulating them, and
generally the complexity of contracting tensor networks. While creating PEPS
allows to solve PP problems, the latter two tasks are both proven to be
#P-complete. We further show how PEPS can be used to approximate ground states
of gapped Hamiltonians, and that creating them is easier than creating
arbitrary PEPS. The main tool for our proofs is a duality between PEPS and
postselection which allows to use existing results from quantum compexity.
