which is indeed equivalent to Eq. 50 due to the constraint on the total population in Eq. 57. Despite the similar expression of Ĥ( R) s compared to the seminal MMST mapping Hamiltonian, 8-10 the SU (N ) mapping formalism should be viewed as a different mapping procedure compared to the MMST mapping formalism. As opposed to the Stock-Thoss mapping procedure, the starting point of the SU (N ) mapping formalism is completely different. The SU (N ) mapping formalism uses the generators of the su(N ) Lie algebra which exactly preserves the commutation relations among operators as well as the original electronic Hilbert space. As a result, there is no need for additional Hilbert space projection nor truncation that ruins the simple commutation relations of mapping operators (see Appendix F for detailed discussions) or necessity of projecting back to the subspace as required by MMST formalism. The exact quantum Liouvillian from the current SU (N ) mapping formalism (see Sec. V) is also different than the exact Liouvillian of the MMST formalism. A detailed discussion of the connection and difference between these two mapping approaches is provided in Appendix F.