  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.
