Classical Annealing of Sherrington-Kirkpatrick Spin Glass Using Suzuki-Kubo Mean-field Ising Dynamics

Abstract

We propose and demonstrate numerically a fast classical annealing scheme for the Sherrington-Kirkpatrick (SK) spin glass model, employing the Suzuki-Kubo meanfield Ising dynamics (supplemented by a modified Thouless-Anderson-Palmer reaction field). The resultant dynamics, starting from any arbitrary paramagnetic phase (with local magnetizations $m_i=\pm 1$ for the $i^{th}$ 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 10000) the deviation in the energy per spin $E^0_N - E^0$ found by the annealing procedure scales as $N^{-2/3}$, with $E^0 = -0.7629\pm 0.0002$, 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 $E^0 =-0.7631667265 \dots$), fluctuation $\sigma_N $ in $E^0_N$ decreases as $\sim N^{-3/4}$ and the annealing time $\tau_N \sim N$, making this protocol highly efficient in estimating the ground state of the SK model.