Methods, systems and apparatus for constructing and programming quantum annealing hardware. In one aspect, a method includes deriving data characterizing an energy spectrum of a Hamiltonian Htotal that characterizes quantum states of a quantum processor, wherein the quantum processor is controllable such that the Hamiltonian Htotal evolves from an initial Hamiltonian Hi to a problem Hamiltonian Hp comprising an energy spectrum that encodes a solution to an optimization problem, the deriving being based on Hi and Hp at a time that Htotal has the energy spectrum; estimating an average phonon energy of a bath in which the quantum processor is located; and determining, based on the derived data and the calculated average phonon energy, an additional Hamiltonian HQG that when combined with Htotal limits evolution of the quantum state of the quantum processor into the ground state of Hp when Htotal evolves to Hp without diagonalizing Hp.