Comparison between a quantum annealer and a classical approximation algorithm for computing the ground state of an Ising spin glass.

Finding the ground state of an Ising spin glass on general graphs belongs to the class of NP-hard problems, widely believed to have no efficient polynomial-time algorithms to solve them. An approach developed in computer science for dealing with such problems is to devise approximation algorithms; these are algorithms, whose run time scales polynomially with the input size, that provide solutions with provable guarantees on their quality in terms of the optimal unknown solution. Recently, several algorithms for the Ising spin-glass problem on a bounded degree graph that provide different approximation guarantees were introduced. D-Wave, a Canadian-based company, has constructed a physical realization of a quantum annealer and has enabled researchers and practitioners to access it via their cloud service. D-Wave is particularly suited for computing an approximation for the ground state of an Ising spin glass on its Chimera and Pegasus graphs-both with a bounded degree. To assess the quality of D-Wave's solution, it is natural to compare it to classical approximation algorithms specifically designed to solve the same problem. In this work, we compare the performance of a recently developed approximation algorithm to solve the Ising spin-glass problem on graphs of bounded degree against the performance of the D-Wave computer. We also compared the performance of D-Wave's computer in the Chimera architecture against the performance of a heuristic tailored specifically to handle the Chimera graph. We found that the D-Wave computer was able to find better approximations for all the random instances of the problem we studied-Gaussian weights, uniform weights, and discrete binary weights. Furthermore, the convergence times of D-Wave's computer were also significantly better. These results indicate the merit of D-Wave's computer under certain specific instances. More broadly, our method is relevant to a wider class of performance comparison studies, and we suggest that it is important to compare the performance of quantum computers not only against exact classical algorithms with exponential run-time scaling, but also against approximation algorithms with polynomial run-time scaling and a provable guarantee of performance.