Sufficient condition for the coherent control of n-qubit systems

We study quantum systems with even numbers N of levels that are completely state controlled by unitary transformations generated by Lie algebras isomorphic to sp(N) of dimension N(N+1)/2 as discussed by Albertini and D'Allesandro [IEEE Trans. Autom. Control 48, 1399 (2003)]. These Lie algebras are smaller than the corresponding su(N) with dimension N-2-1. We show that this reduction constrains the field-free Hamiltonian to have symmetric energy levels. An example of such a system is an n-qubit system with state-independent interaction terms. Using Clifford's geometric algebra to represent the quantum wave function of a finite system, we present an explicit example of a two-qubit system that can be controlled by the elements of the Lie algebra sp(4) [isomorphic to spin(5) and so(5)] with dimension 10 rather than su(4) with dimension 15, but only if its field-free energy levels are symmetrically distributed about an average. These results enable one to envision more efficient algorithms for the design of fields for quantum-state engineering in certain quantum-computing applications, and provide more insight into the fundamental structure of quantum control.