A QMA-Complete Translationally Invariant Hamiltonian Problem and the Complexity of Finding Ground State Energies in Physical Systems

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.