Computational complexity of three-dimensional Ising spin glass: Lessons from D-Wave annealer

Abstract

Finding an exact ground state of a three-dimensional (3D) Ising spin glass is proven to be an NP-hard problem (i.e., at least as hard as any problem in the nondeterministic polynomial-time (NP) class). 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 $N\le 5627$. We found exact ground states (in a probabilistic sense) for typical realizations of 3D spin glasses with the efficiency, which scales as $2^{N/{\hat{I}}^2}$ with ${\hat{I}}^2\approx 10^3$. Based on statistical analysis of low-energy states, we argue that with an improvement of annealing protocols and device noise reduction, ${\hat{I}}^2$ can be increased even further. This suggests that, for $N<β^3$, annealing devices provide most efficient way to find an exact ground state.

7 pages, 7 figures