Quantum Merlin Arthur with Exponentially Small Gap

We will study the complexity of QMA proof systems with inverse exponentially small promise gap. We will show that this class, QMAexp, can be exactly characterized by PSPACE, the class of problems solvable with a polynomial amount of memory. As applications we show that a “precise” version of the Local Hamiltonian problem is PSPACE-complete, and give a provable setting in which the ability to prepare PEPS states is not as powerful as ability to prepare the ground state of general local Hamiltonians.