Optimal control of single spin-1/2 quantum systems

The purpose of this study is to explore the optimal control problems for a class of single spin-1/2 quantum ensembles. The system in question evolves on a manifold in R-3 and is modelled as a bilinear control form whose states are represented as coherence vectors. An associated matrix Lie group system with state space SO(3) is introduced in order to facilitate solving the given problem. The controllability as well as the reachable set of the system is first analysed in detail. Then, the maximum principle is applied to the optimal control for system evolving on the Lie group of special orthogonal matrices of dimension 3, with cost that is quadratic in the control input. As an illustrative example, the authors apply their result to perform a reversible logic quantum operation NOT on single spin-1/2 system. Explicit expressions for the optimal control are given which are linked to the initial state of the system.