Abstract: Methods for solving discrete quadratic models are described. The methods compute an energy of each state of each variable based on its interaction with other variables, exponential weights, and normalized probabilities proportional to the exponential weights. The energy of each variable is computed as a function of the magnitude of each variable and a current state of all other variables, exponential weights, the feasible region for each variable, and normalized probabilities, proportional to the exponential weights and respecting constraints. Methods executed via a hybrid computing system obtain two candidate values for each variable; constructs a Hamiltonian that uses a binary value to determine which candidate values each variable should take, then constructs a binary quadratic model based on the Hamiltonian. Samples from the binary quadratic model are obtained via a quantum processor.