Convergence of Quantum Annealing with Real-Time Schrödinger Dynamics

Convergence conditions for quantum annealing are derived for optimizationproblems represented by the Ising model of a general form. Quantum fluctuationsare introduced as a transverse field and/or transverse ferromagneticinteractions, and the time evolution follows the real-time Schrodingerequation. It is shown that the system stays arbitrarily close to theinstantaneous ground state, finally reaching the target optimal state, if thestrength of quantum fluctuations decreases sufficiently slowly, in particularinversely proportionally to the power of time in the asymptotic region. This isthe same condition as the other implementations of quantum annealing, quantumMonte Carlo and Green's function Monte Carlo simulations, in spite of theessential difference in the type of dynamics. The method of analysis is anapplication of the adiabatic theorem in conjunction with an estimate of a lowerbound of the energy gap based on the recently proposed idea of Somma et. al.for the analysis of classical simulated annealing using a classical-quantumcorrespondence.Comment: 6 pages, minor correction