On Computational Complexity of 3D Ising Spin Glass: Lessons from D-Wave Annealer

Abstract: Finding an exact ground state of a 3D Ising spin glass is proven to be an NP-hard problem. Given validity of the exponential time hypothesis, its computational complexity was proven to be no less than $2^{N^{2/3}}$, where $N$ is the total number of spins. Here we report results of extensive experimentation with D-Wave 3D annealer with $N\leq 5627$. We found exact ground states (in a probabilistic sense) for typical realizations of 3D spin glasses with the efficiency, which scales as $2^{N/\beta}$ with $\beta\approx 10^{3}$. Based on statistical analysis of low energy states, we argue that with an improvement of annealing protocols and device noise reduction, $\beta$ can be increased even further. This suggests that, for $N<\beta^3$, annealing devices provide a most efficient way to find the ground state.