Optimal control of two-level quantum systems

We study the manipulation of two-level quantum systems. This research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two-level quantum system, a quantum bit. We consider the problem of driving the evolution operator to a desired state, while minimizing the energy-type cost. Mathematically, this problem translates into an optimal control problem for systems varying on the Lie group of special unitary matrices, with cost that is quadratic in the control. We develop a comprehensive theory of optimal control for two-level quantum systems. In particular, we prove the `normality´ of the given problem and the `regularity´ of the optimal control functions. The impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed