The optimal control problem on SO (4) and its applications to quantum control

In this note, we consider the problem of steering control via an input electro-magnetic field for a system of two interacting spin 1/2 particles. This model is of interest in applications because it is used to perform logic operations in quantum computing that involve two quantum bits. The describing model is a bilinear system whose state varies on the Lie group of special unitary matrices of dimension 4, SU (4). By using decompositions of the latter Lie group, the problem can be decomposed into a number of subproblems for a system whose state varies on the (smaller) Lie group of 4 x 4 proper orthogonal matrices, SO (4). We tackle the time optimal control problem for this system and show that the extremals can be computed explicitly and they are the superposition of a constant field and a sinusoidal one.