  Quantum annealing is getting increasing attention in combinatorial
optimization. The quantum processing unit by D-Wave is constructed to
approximately solve Ising models on so-called Chimera graphs. Ising models are
equivalent to quadratic unconstrained binary optimization (QUBO) problems and
maximum cut problems on the associated graphs. We have tailored branch-and-cut
as well as semidefinite programming algorithms for solving Ising models for
Chimera graphs to provable optimality and use the strength of these approaches
for comparing our solution values to those obtained on the current quantum
annealing machine D-Wave 2000Q. This allows for the assessment of the quality
of solutions produced by the D-Wave hardware. It has been a matter of
discussion in the literature how well the D-Wave hardware performs at its
native task, and our experiments shed some more light on this issue.
