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” QT≡m/HR2, where HR 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 QT. The formulae tell that we can make quantum expectation value of particle coordinates x approximate to the minimum point x0 in QT→∞. For systems where we have almost degenerate eigenvalues, examination working with our QT, that x→x0 in QT→∞, is analytically shown also efficient. Similar results that x→x0 under QT→∞ are also obtained when we utilize random-walk quantum Monte Carlo method to represent tunneling phenomena according to conventional quantum annealing.
