where r ∈ [0, 1] is a random number sampled from the standard uniform distribution at each time step for each trajectory. A brief derivation of the approach described in Eqs. 27a and 27b is provided in Appendix A. Note that while Eqs. 27a and 27b preserve the trace of the expectation value of the density matrix estimator among the ensemble of trajectories, the trace of an individual trajectory's density matrix estimator may not be conserved due to the nature of the sampling. Furthermore, unlike the L-MFE approach, which has a fully deterministic change in the coefficient magnitudes, the change in the magnitude of the ground-state coefficient in Eq. 27b is stochastic, which was necessary to deal with the potential complex population estimators for each individual trajectory (Z F a,ξ (t)Z B* b,ξ (t) in Eq. 23) that arise in the PLDM method. The propagated forward mapping variables can be recovered as follows