  The control laws based on quantum Lyapunov control method are designed to
prepare operators for two level open quantum systems in this paper. A novel
Lyapunov function is proposed according to a matrix logarithm function. The
higher accuracy and faster convergence of the novel Lyapunov function proposed
are analyzed by comparing with the operator distance . We adopt three control
fields, and design two types of control law forms. The first form is to use one
control law to offset the influence of the dissipation, while the other one is
to use all three control laws to offset the dissipation. Moreover, the
robustness of the system Hamiltonian with uncertainty is further investigated
to comprehend the performances of control laws. NOT gates are prepared by the
designed control laws for open quantum systems as well as closed quantum
systems in numerical experiments to verify the superiority of, in which the
performance indexes of distance and fidelity are compared and analyzed.
