It is unclear to what extent quantum algorithms can outperform classical algorithms for problems of combinatorial optimization. In this work, by resorting to computational learning theory and cryptographic notions, we give a fully constructive proof that quantum computers feature a super-polynomial advantage over classical computers in approximating combinatorial optimization problems. Specifically, by building on seminal work by Kearns and Valiant, we provide special instances that are hard for classical computers to approximate up to polynomial factors. Simultaneously, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The quantum advantage in this work is ultimately borrowed from Shor’s quantum algorithm for factoring. We introduce an explicit and comprehensive end-to-end construction for the advantage bearing instances. For these instances, quantum computers have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms.
