Constructive controllability for systems with drift motivated by and applied to quantum control

The purpose of this paper is fourfold: i) First, a brief description of how finite dimensional models arise in quantum control will be given. ii) Second, many notions in quantum engineering such as universality of logic gates, hard pulse approximation in NMR spectroscopy etc., will be related to controllability of systems on the unitary groups. iii) Recent results, of the authors, on constructive controllability, in the class of piecewise constant controls with constraints placed on them, for systems with drift on the unitary groups will be sketched. iv) Finally, a new decomposition of SU (2) matrices will be provided and used to illustrate how problems caused by periodicity arguments, to address the presence of drift, may be circumvented.