Cartan Decomposition of SU(2^n), Constructive Controllability of Spin systems and Universal Quantum Computing

In this paper we provide an explicit parameterization of arbitrary unitary transformation acting on n qubits, in terms of one and two qubit quantum gates. The construction is based on successive Cartan decompositions of the semi-simple Lie group, SU(2^n). The decomposition highlights the geometric aspects of building an arbitrary unitary transformation out of quantum gates and makes explicit the choice of pulse sequences for the implementation of arbitrary unitary transformation on $n coupled spins. Finally we make observations on the optimality of the design procedure.