  Adiabatic Quantum Computing (AQC) is an attractive paradigm for solving hard
integer polynomial optimization problems. Available hardware restricts the
Hamiltonians to be of a structure that allows only pairwise interactions. This
requires that the original optimization problem to be first converted -- from
its polynomial form -- to a quadratic unconstrained binary optimization (QUBO)
problem, which we frame as a problem in algebraic geometry. Additionally, the
hardware graph where such a QUBO-Hamiltonian needs to be embedded -- assigning
variables of the problem to the qubits of the physical optimizer -- is not a
complete graph, but rather one with limited connectivity. This "problem graph
to hardware graph" embedding can also be framed as a problem of computing a
Groebner basis of a certain specially constructed polynomial ideal. We develop
a systematic computational approach to prepare a given polynomial optimization
problem for AQC in three steps. The first step reduces an input polynomial
optimization problem into a QUBO through the computation of the Groebner basis
of a toric ideal generated from the monomials of the input objective function.
The second step computes feasible embeddings. The third step computes the
spectral gap of the adiabatic Hamiltonian associated to a given embedding.
These steps are applicable well beyond the integer polynomial optimization
problem. Our paper provides the first general purpose computational procedure
that can be used directly as a $translator$ to solve polynomial integer
optimization. Alternatively, it can be used as a test-bed (with small size
problems) to help design efficient heuristic quantum compilers by studying
various choices of reductions and embeddings in a systematic and comprehensive
manner. An added benefit of our framework is in designing Ising architectures
through the study of $\mathcal Y-$minor universal graphs.
