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

We present an extensive numerical study of the Sherrington-Kirkpatrick model in transverse field. Recent numerical studies of quantum spin-glasses have focused on exact diagonalization of the full Hamiltonian for small systems ($\approx$ 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 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 state. For the energy gap, we derive a novel formula that enables it to be calculated using just the ground-state wavefunction, 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.