The importance of the spectral gap in estimating ground-state energies

The Local Hamiltonian problem, which is concerned with estimating the ground-state energy of a local Hamiltonian, is a central object in the field of Hamiltonian complexity and is complete for the class QMA. A major challenge in the field is to understand the complexity of the problem in more physically natural parameter regimes. Despite its importance in quantum many-body physics, the role played by the spectral gap in the complexity of the Local Hamiltonian is less well-understood. In this work, we make progress on this question by considering the precise regime, in which one estimates the ground-state energy to within inverse-exponential precision. In this setting, there is a surprising result that the complexity of Local Hamiltonian is magnified from QMA to PSPACE. We clarify the reason behind this boost in complexity. Specifically, we show that the full complexity of the high precision case only comes about when the spectral gap is exponentially small. As a consequence, we uncover important implications for the representability and circuit complexity of ground states of local Hamiltonians, the theory of uniqueness of quantum witnesses, and techniques for the amplification of quantum witnesses in the presence of postselection.