where e L L is the decay dynamics propagation outlined above (in Eq. 28 and Eq. 27) and e L Ĥ is the PLDM propagation based on the system Hamiltonian. In the scheme of Eq. 29, we have used the symmetrized Trotter decomposition to reduce the error of using a finite dt. This stochastic approach is briefly compared to Lindblad dynamics in Appendix B.