Spin systems and minimal switching decompositions

The control of spin chains represents a very interesting problem from the point of view of quantum computation. The problem consists in defining a procedure to obtain any possible evolution operator of the spin chain by means of in external magnetic field. The set of possible evolution operators of the system corresponds to the unitary group SU(2^N) (where N is the number of atoms in the chain) and the interactions involved can he set to correspond to elements in the corresponding Lie algebra. As a consequence, the whole problem can be formulated in Lie algebraic terms and the design issues be reduced to a suitable decomposition of the group elements. The goal of this paper is to introduce a Cartan decomposition for the unitary group based on a minimal switching decomposition of the special orthogonal group. We analyze its implications from the point of view of time optimality in the construction of a program as a sequence of quantum gates.