Cartan decomposition of SU(2ⁿ) and control of spin systems

In this paper, we give an explicit parameterization of any arbitrary unitary transformation on n qubits, in terms of one qubit and two qubit operations. Building on the previous work demonstrating the universality of two qubit quantum gates, we present here an explicit construction. The construction is based on the Cartan decomposition of the semi-simple Lie group SU(2ⁿ), and uses the geometric structure of the Riemannian symmetric space SU(2ⁿ)/SU(2^n-1) ⊗ SU(2^n-1) ⊗ U(1). This decomposition highlights the geometric aspects of the problem of building an arbitrary unitary transformation out of quantum gates and makes explicit the structure of pulse sequences for its approximate implementation in a network of n coupled 1/2 spins. Further work needs to be done to relate the parameters in our decomposition to other standard parameterization of SU(2ⁿ), in order to find explicit pulse sequences for synthesizing unitary transformations expressed in the standard parameterization of SU(2ⁿ).