Following this fundamental idea of mapping, one of the most natural ways to map a N -level quantum system is to respect its original symmetry, which is described by the special unitary symmetry group 31 SU (N ). For N = 2, it is well-known that the quantum dynamics of two electronic states can be mapped as the spin precession around a magnetic field (represented by the Hamiltonian) described by the motion of the Bloch vector on the Bloch sphere, due to the SU (2) symmetry shared by both problems. Thoss and Stock have developed a semiclassical initial-value representation of the corresponding propagator using the SU (2) mapping. Recently, Runeson and Richardson used this spin mapping approach to map a two-level system on a single spin- 2 DOFs. The same mapping formalism was also used to develop nonadiabatic path-integral approaches. One can, in principle, generalize this idea by mapping a N -level system with the generators of the su(N ) Lie algebra. The unique advantage of this mapping procedure compared to the MMST formalism is that the commutation relations among operators as well as the size of the Hilbert space are exactly preserved in the SU (N ) representation. The corresponding quantum equations of motion (EOMs) for the SU (N ) mapping were first introduced by Hioe and Eberly, which can be viewed as the generalization of the spin precession to N -dimensions with SU (N ) symmetry. The SU (N ) mapping has also been used recently for density matrix mapping. Meyer, McCurdy, and Miller also used a similar idea to map two-state or three-state systems with spin- 2 and spin-1 operators, although the matrices of the spin-1 operators (which are not necessarily traceless) are different than the su(N ) generators (which are traceless). Note that the SU (N ) mapping formalism is also different than the recent spin-mapping formalism introduced by Cot-ton and Miller, which maps N states onto N spin-1 2 particles, hence having the symmetry of ⊗ N SU (2) that is different than SU (N ).