  We propose a novel hybrid quantum-classical approach to calculate Graver
bases, which have the potential to solve a variety of hard linear and
non-linear integer programs, as they form a test set (optimality certificate)
with very appealing properties. The calculation of Graver bases is
exponentially hard (in general) on classical computers, so they not used for
solving practical problems on commercial solvers. With a quantum annealer,
however, it may be a viable approach to use them. We test two hybrid
quantum-classical algorithms (on D-Wave)--one for computing Graver basis and a
second for optimizing non-linear integer programs that utilize Graver bases--to
understand the strengths and limitations of the practical quantum annealers
available today. Our experiments suggest that with a modest increase in coupler
precision--along with near-term improvements in the number of qubits and
connectivity (density of hardware graph) that are expected--the ability to
outperform classical best-in-class algorithms is within reach, with respect to
non-linear integer optimization.
