  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
$\Delta$, 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 $\Delta$ 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.
