Alleviating the quantum Big-M problem

A major obstacle for quantum optimizers is the reformulation of constraints as a quadratic unconstrained binary optimization (QUBO). Current QUBO translators exaggerate the weight M of the penalty terms. Classically known as the “Big-M” problem, the issue becomes even more daunting for quantum solvers, since it affects the physical energy scale. We take a systematic, encompassing look at the quantum big-M problem, revealing NP-hardness in finding the optimal M and establishing bounds on the Hamiltonian spectral gap Δ as a function of the weight M, inversely related to the expected run-time of quantum solvers. We propose a practical translation algorithm, based on SDP relaxation, that outperforms previous methods in numerical benchmarks. Our algorithm gives values of Δ orders of magnitude greater, e.g. for portfolio optimization instances. Solving such instances with an adiabatic algorithm on 6-qubits of an IonQ device, we observe significant advantages in time to solution and average solution quality. Our findings are relevant to quantum and quantum-inspired solvers alike.