While there are several possible choices for constructing c 1,ξ (t + dt) and c 0,ξ (t + dt) that agree with Eqs. 37a-41b, it is important to consider how these choices affect the Hamiltonian dynamics (evolved by e L Ĥ dt ) that occur in conjunction with the Lindblad dynamics. Certain choices of c 1,ξ (t+dt) and c 0,ξ (t+dt) may cause the Hamiltonian dynamics to diverge from the results of the exact Liouvillian dynamics, for example, when they do not conserve the trace of the density matrix within a trajectory. To avoid this, we add on an additional constraint to the coefficient evolution that