Quantum-Merlin-Arthur-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 Hamiltonians with a fixed local dimension and $O\mathbf{\left(}\log \left(N\right)\mathbf{\right)}$-body local terms, or local dimension $N$ and two-body terms, there are instances where finding the ground-state energy is quantum-Merlin-Arthur-complete and simulating the dynamics is BQP-complete (BQP denotes “bounded error, quantum polynomial time”). 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 density-matrix renormalization group, matrix product states, and multiscale entanglement renormalization ansatz. One important example is a one-dimensional lattice of bosons with nearest-neighbor hopping at constant filling fraction—i.e., a generalization of the Bose-Hubbard model.