An exponential time upper bound for Quantum Merlin-Arthur games with unentangled provers

Abstract: We prove a deterministic exponential time upper bound for Quantum Merlin-Arthur games with k unentangled provers. This is the first non-trivial upper bound of QMA(k) better than NEXP and can be considered an exponential improvement, unless EXP=NEXP. The key ideas of our proof are to use perturbation theory to reduce the QMA(2)-complete Separable Sparse Hamiltonian problem to a variant of the Separable Local Hamiltonian problem with an exponentially small promise gap, and then to decide this instance using epsilon-net methods. Our results imply an exponential time algorithm for the Pure State N-Representability problem in quantum chemistry, which is in QMA(2), but is not known to be in QMA. We also discuss the implications of our results on the Best Separable State problem.