Quantum Tunneling Parameter in Global Optimization

Abstract

Quantum tunneling that helps particles escape from local minima has been applied in “quantum annealing” method to global optimization of nonlinear functions. To control size of kinetic energy of quantum particles, we form a “quantum tunneling parameter” Q_T≡m/H_R², where H_R corresponds to a physical constant h, Planck's constant divided by 2π, that determines the lowest eigenvalue of quantum particles with mass m. Assumptions on profiles of the function V(x) around its minimum point x₀, harmonic oscillator type and square well type, make us possible to write down analytical formulae of the kinetic energy K in terms of Q_T. The formulae tell that we can make quantum expectation value of particle coordinates x approximate to the minimum point x₀ in Q_T→∞. For systems where we have almost degenerate eigenvalues, examination working with our Q_T, that x→x₀ in Q_T→∞, is analytically shown also efficient. Similar results that x→x₀ under Q_T→∞ are also obtained when we utilize random-walk quantum Monte Carlo method to represent tunneling phenomena according to conventional quantum annealing.