The cavity loss dynamics is simulated using the stochastic Lindblad approach with PLDM, as detailed in Ref. 47. For a cavity mode with loss rate Γ, the Lindblad loss dynamics is equivalent to updating the forward and backward coefficients outlined in Eq. A3 and Eqs. A5-A6. For any t ∈ [0, t 1 -∆t], the overall propagation of the reduced density matrix for the quantum subsystem during a time step ∆t can thus be summarized as