  We extend the family of problems that may be implemented on an adiabatic
quantum optimizer (AQO). When a quadratic optimization problem has at least one
set of discrete controls and the constraints are linear, we call this a
quadratic constrained mixed discrete optimization (QCMDO) problem. QCMDO
problems are NP-hard, and no efficient classical algorithm for their solution
is known. Included in the class of QCMDO problems are combinatorial
optimization problems constrained by a linear partial differential equation
(PDE) or system of linear PDEs. An essential complication commonly encountered
in solving this type of problem is that the linear constraint may introduce
many intermediate continuous variables into the optimization while the
computational cost grows exponentially with problem size. We resolve this
difficulty by developing a constructive mapping from QCMDO to quadratic
unconstrained binary optimization (QUBO) such that the size of the QUBO problem
depends only on the number of discrete control variables. With a suitable
embedding, taking into account the physical constraints of the realizable
coupling graph, the resulting QUBO problem can be implemented on an existing
AQO. The mapping itself is efficient, scaling cubically with the number of
continuous variables in the general case and linearly in the PDE case if an
efficient preconditioner is available.
