  Efficient control of qubits plays a key role in quantum information
processing. In the current work, an alternative set of differential equations
are derived for an optimal quantum control of single or multiple qubits with or
without interaction. The new formulation enables a great reduction of the
computation load by eliminating redundant complexity involved in previous
formulations. A relaxation technique is designed for numerically detecting
optimal paths involving entanglement. Interesting continuous symmetries are
identified in the Lagrangian, which indicates the existence of physically
equivalent classes of paths and may be utilized to remove neutral directions in
the Jacobian of the evolution. In the 'ground state' solution among the set of
optimal paths, the time-reversal symmetry of the system shows up, which is
expected to be universal for the symmetry-related initial and final state.
