A Quantum Approach to Classical Statistical Mechanics

Abstract

We present a new approach to study the thermodynamic properties of $d$-dimensional classical systems by reducing the problem to the computation of ground state properties of a $d$-dimensional quantum model. This classical-to-quantum mapping allows us to deal with standard optimization methods, such as simulated and quantum annealing, on an equal basis. Consequently, we extend the quantum annealing method to simulate classical systems at finite temperatures. Using the adiabatic theorem of quantum mechanics, we derive the rates to assure convergence to the optimal thermodynamic state. For simulated and quantum annealing, we obtain the asymptotic rates of $T(t) \approx (p N) /(k_B \log t)$ and $γ(t) \approx (Nt)^{-\bar{c}/N}$, for the temperature and magnetic field, respectively. Other annealing strategies, as well as their potential speed-up, are also discussed.