Residual energies after slow quantum annealing

The features of residual energy after quantum annealing are investigated. The quantum annealing method exploits quantum fluctuations to find out the ground state of the classical disordered Hamiltonian. If the quantum fluctuation decreases sufficiently slowly and linearly with time, residual energy after quantum annealing decreases with the inverse square of annealing time. We show this feature of residual energy by numerical calculations for small systems and derive it on the basis of the quantum adiabatic theorem.