In the optimal control of n-level quantum systems there is a class of  models which on one hand occur very often and, on the other hand,  are  amenable of explicit solutions. These are the so-called K-P systems. They include, among others, systems whose energy diagram is a bi-partite graph such as Lambda-systems, double Lambda systems and one quantum bit. For these models, it is possible to find the time optimal control under an energy constraint explicitly. The consideration of the minimum time transfer between two states is a natural requirement in applications such as quantum computing especially in view of the fact that fast transfer is a way to mitigate the influence of the environment. The K-P structure refers to an underlying  Cartan-type  K-P decomposition of the Lie algebra su(n) such that only the operators corresponding to the P part of the decomposition appear in the Schrodinger equation of the system.

We describe  the case of the quantum bit in detail and use it to illustrate the general theory. In particular, we  explicitly derive the minimum time trajectories between any two states for this system.  This analysis also displays some general features of the optimal synthesis such as:  the critical locus, i.e., the locus of points where optimal trajectories lose optimality, the geometry of the set of reachable states at each time and the diameter of the system, i.e., the worst case minimum time. Furthermore, such an analysis leads to the general consideration of the role of symmetries in optimal control problems. The explicit nature of the solution of the optimal control problem provided lends itself to generalizations to other systems of interest in applications. We shall in particular illustrate the case of N qubits controlled in parallel in minimum time.