Comparison of unitary evolution operator decompositions in quantum systems

The purpose of controlling a quantum system is to steer the system from an initial state to a target state. Such a control problem can be changed into the problem of generating the desired unitary evolution operators by using an appropriate decomposition. By means of the desired state´s exponential factor product theorem in the right-invariant system, the desired unitary evolution operators of a quantum system can be obtained. This paper investigates the functions and features of the Magnus decomposition, the Wei-Norman decomposition, the Lie Group decomposition and the Cartan decomposition contrastively and respectively. Properties and realization performance of those decompositions are analyzed, too. Each type of decomposition applied to different situations is also given.