Quantum Annealing in Sherrington-Kirkpatrick Spin Glass in Presence of Time-Dependent Longitudinal Field

Abstract: Motivated by the recent development of quantum technology using quantum annealing technique and the recent works on the static properties of the Sherrington-Kirkpatrick (SK) spin glass model, we study quantum annealing of the spin glass model by tuning both transverse and longitudinal fields. We numerically solve the time-dependent Schr\"odinger equation of the total Hamiltonian when both the fields are made time-dependent and eventually vanish at the same time. We have computed the time-evolution of the probability of finding the system in one of two degenerate ground states of the classical spin glass. At the end of annealing, using the configuration averaged probability, we have shown a clear advantage while the longitudinal field is annealed rather than keeping it constant throughout the process of quantum annealing. We further investigate the order parameter distribution of a quantum SK spin glass in presence of a small longitudinal field and find, from our exact diaginalization results for small system sizes, evidence for quantum tunneling induced disappearance of the classical Almeida-Thouless phase boundary separating the replica symmetry broken (nonergodic) and replica symmetric (ergodic) spin glass phase (reported already in $2022$). We believe that this longitudinal field induced ergodicity in quantum SK model to be responsible for the observed enhancement of quantum annealing (reported earlier for smaller systems by us in $2014$).