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We give a quantum speedup for solving the canonical semidefinite programmingrelaxation for binary quadratic optimization. This class of relaxations forcombinatorial optimization has so far eluded quantum speedups. Our methodscombine ideas from quantum Gibbs sampling and matrix exponent updates. Ade-quantization of the algorithm also leads to a faster classical solver. Forgeneric instances, our quantum solver gives a nearly quadratic speedup overstate-of-the-art algorithms. Such instances include approximating the groundstate of spin glasses and MaxCut on Erd\"{o}s-R\'enyi graphs. We also providean efficient randomized rounding procedure that converts approximately optimalSDP solutions into approximations of the original quadratic optimizationproblem.

Faster quantum and classical SDP approximations for quadratic binary optimization