Methods are provided for implementing schemes for embedding a particular optimization problem into a particular hardware solution employing unique graph embedding techniques. The disclosed methods implement an adiabatic quantum optimization in a quantum computing device or a quantum processor. Heuristics for graph minor embedding are employed to map a problem graph structure of a particular binary unconstrained optimization problem onto a physical graph structure (topology) of the quantum computing device or quantum processor to provide an optimized hardware implementation. Known constraints that are presented with current schemes in their application to particular hardware solutions are avoided, including limited qubit connectivity and the presence of faulty qubits.