  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.
