Study of Optimization Problems by Quantum Annealing

TL;DR

This paper explores quantum annealing by introducing quantum fluctuations into optimization problems, demonstrating improved convergence and shorter relaxation times compared to classical methods through simulations of the Ising model and TSP.

Contribution

It presents a novel approach of quantum annealing applied to classical optimization problems, showing enhanced performance over traditional thermal annealing methods.

Findings

Quantum annealing increases the probability of reaching the ground state.

Quantum systems exhibit shorter relaxation times.

Quantum annealing outperforms classical thermal methods in convergence speed.

Abstract

We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. The idea is tested by the two models, the transverse Ising model and the traveling salesman problem (TSP). Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations. The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method. TSP can be described by the Ising spin, so that we also apply the same technique to TSP. We directory solve the time-dependent Schr\"odinger equation for small-size systems and perform the quantum Monte Carlo simulation for large-size systems. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters. We also find that the relaxation time is quite short for quantum systems by numerical simulations. We consider this is one of the reasons why the annealing in quantum systems have a better performance of finding the optimal state in comparison with classical systems.