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

摘要

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