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

TL;DR

This paper investigates the computational complexity of finding ground states in 3D Ising spin glasses and reports experimental results using a D-Wave annealer, showing promising scaling behavior and potential for improved quantum annealing methods.

Contribution

It provides empirical evidence on the efficiency of quantum annealing for 3D spin glasses and suggests that with improvements, annealers could efficiently find exact ground states for larger systems.

Findings

Exact ground states found probabilistically for N ≤ 5627

Efficiency scales as 2^{N/β} with β≈1000

Potential for improved annealing protocols to increase β

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 $2^{N/ \beta}$ 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 most efficient way to find an exact ground state.