Classical annealing of the Sherrington-Kirkpatrick spin glass using Suzuki-Kubo mean-field Ising dynamics

We propose and demonstrate numerically a fast classical annealing scheme for the Sherrington-Kirkpatrick (SK) spin glass model, employing the Suzuki-Kubo mean-field Ising dynamics (supplemented by a modified Thouless-Anderson-Palmer reaction field). The resultant dynamics, starting from any arbitrary paramagnetic phase (with local magnetizations m_{i}=±1, for the ith spin, and the global magnetization m=0), takes the system quickly to an appropriate state with small local values of magnetization (m_{i}) commensurate with the (frustrated) interactions. As the temperature decreases with the annealing, the configuration practically remains (in an effective adiabatic way) close to a low-energy configuration as the magnitudes of m_{i}'s and the spin glass order parameter q grow to unity. While the configuration reached by the procedure is not the ground state, for an N-spin SK model (with N up to 10 000), the deviation in the energy per spin E_{N}^{0}-E^{0} found by the annealing procedure scales as N^{-2/3}, with E^{0}=-0.7629±0.0002, suggesting that in the thermodynamic limit the energy per spin of the low-energy configurations converges to the ground state of the SK model (analytical estimate being E^{0}=-0.7631667265⋯), fluctuation σ_{N} in E_{N}^{0} decreases as ∼N^{-3/4}, and the annealing time τ_{N}∼N, making this protocol highly efficient in estimating the ground state energy of the SK model.