  Here we present a problem related to the local Hamiltonian problem
(identifying whether the ground state energy falls within one of two ranges)
which is restricted to being translationally invariant. We prove that for
problems with a fixed local dimension and O(\log(N))-body local terms, or local
dimension N and 2-body terms, there are instances of the problem which are
QMA-complete. We discuss the implications for the computational complexity of
finding ground states of these systems, and hence for any classical
approximation techniques that one could apply including DMRG, Matrix Product
States and MERA. One important example is a 1D lattice of bosons with
nearest-neighbor hopping at constant filling fraction i.e. a generalization of
the Bose-Hubbard model.
