On the other hand, the mapping-based methods (γ-SQC and spin-LSC) show much more accurate relaxation dynamics from the upper polariton state, compared to the HEOM results, and the steady-state population for the upper polariton is also similar to the exact HEOM dynamics (Fis. 2g-2l). We also observe that the transitions into the dark states are better captured with these mapping-based methods, although the γ-SQC method (Fig. -Fig. ) slightly outperforms the spin-LSC method (Fig. -Fig. ) in predicting the longer time transitions from the dark states to the lower polari-