Abstract: Systems and methods formulate problems for solving by a quantum processor using hardware graph decomposition. A decomposition of a primal graph may be built in a first stage based on a hardware specific graph, and refined in a second stage by, for example, removing vertices from the decomposition. The hardware specific graph may be a graph that is specific to a piece of hardware, for instance a quantum processor comprising a plurality of qubits and couplers operable to communicatively couple pairs of qubits.