Effects of low-lying excitations on ground-state energy and energy gap of the Sherrington-Kirkpatrick model in a transverse field

We present an extensive numerical study of the Sherrington-Kirkpatrick model in a transverse field. Recent numerical studies of quantum spin glasses have focused on exact diagonalization of the full Hamiltonian for small systems ($\ensuremath{\approx}20$ spins). However, such exact numerical treatments are difficult to apply on larger systems. We propose making an approximation by using only a subspace of the full Hilbert space spanned by low-lying excitations consisting of one-spin-flipped and two-spin-flipped states. The approximation procedure is carried out within the theoretical framework of the Hartree-Fock approximation and configuration interaction. Although not exact, our approach allows us to study larger system sizes comparable to that achievable by state-of-the-art quantum Monte Carlo simulations. We calculate two quantities of interest due to recent advances in quantum annealing, the ground-state energy and the energy gap between the ground and first excited states. For the energy gap, we derive a formula that enables it to be calculated using just the ground-state wave function, thereby circumventing the need to diagonalize the Hamiltonian. We calculate the scalings of the energy gap and the leading correction to the extensive part of the ground-state energy with system size, which are difficult to obtain with current methods.