Fig. presents the population dynamics σ z (t) = C 1σz (t) for a 1-D Spin-Boson model. For the spin mapping simulation, a time-step of dt = 0.01 a.u. is used, and the converged results are obtained with 10 4 trajectories for the focused initial conditions (upper panels) and 10 5 trajectories for the sampled initial conditions (lower panels). The population dynamics of the spin mapping approach are compared to the numerically exact calculations (red dots), LSC-IVR (blue) and Ehrenfest dynamics (green). The SM approach is in a very good agreement with the exact calculations for the model in panel (a), including all of the longer time recurrence of the electronic Rabi oscillations. It is important to note that the sampled and focused initial conditions yield almost exactly the same dynamics for this model. The LSC-IVR approach and the Ehrenfest dynamics, on the other hand, capture the initial electronic oscillations but fail to reproduce the longer time recurrence. This less accurate longer time dynamics from LSC-IVR or Ehrenfest was thought causing by the zero point energy (ZPE) leakage problem associated with the classical Wigner dynamics of the nuclear DOF, 100,101 which is typical for linearized path-integral approaches based on the classical Wigner dynamics. The ZPE leakage origi-nates from the fact that classical dynamics does not preserve the ZPE incorporated in the nuclear initial Wigner distribution, 100,101 causing an incorrect energy flow from the nuclear DOF to the electronic DOFs, equalizing the longer time populations and giving σ z (t) =0. In our previous work on non-adiabatic ring polymer molecular dynamics, we have shown that quantizing the nuclear DOF with a ring polymer can effectively incorporate nuclear quantum distribution and alleviate ZPE leaking problem, even when using the MMST mapping formalism. Here, our numerical results suggest that by using the SU (N ) mapping formalism which exactly preserves the size of the electronic Hilbert space, this problem can be largely alleviated, compare to the traditional MMST mapping formalism which can get outside of the singly excited oscillator (SEO) mapping subspace, even though the classical Wigner type of dynamics is used for the nuclear DOF.