based methods, one simulates a SDE $X_t$ with infinitesimal generator ${\mathcal{A}}_t^{*}$ (the adjoint operator of ${\mathcal{A}}_t$) and with initial condition $X_0$ drawn from $\mu_0$ and thus the distribution of $X_t$ is exactly $\mu_t$. Similarly, as the quantum analog, [Lindblad]{} equation has the form $\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}t} {{\boldsymbol{\rho}}}= {\mathcal{L}}({{\boldsymbol{\rho}}})$ where ${{\boldsymbol{\rho}}}$ denotes the density matrix of a quantum system and ${\mathcal{L}}$ is the generator for a completely positive dynamical semigroup [@Lindblad76; @Gorini76]. [[One may solve it by sampling a SDE $X_t$ such that ${\mathbb{E}}\bigl[X_t X^{\dagger}_t\bigr]$ is exactly the solution of [Lindblad]{} equation (see Lemma \[lemma::lb\_unravel\] below for more details).]{}]{} [[There are various choices of the SDEs, [*e.g.*]{}, quantum state diffusion (QSD) [@Gisin92] and linear quantum state diffusion (LQSD) [@Brun00].]{}]{} While it is also possible to use other stochastic processes such as jumping process [@Dalibard92; @Brun00], we will limit the scope of our consideration to Monte Carlo methods based on diffusion processes.

The Monte Carlo method for both [Fokker-Planck]{} equation and [Lindblad]{} equation can be described under the same framework:

Given a ${{\mathbb{C}}^{{n}}}$-valued SDE $X_t$, one would like to approximate ${\mathbb{E}}\left[f(X_t)\right] \equiv \int f\ {\,\mathrm{d}}\mu_t$ for a collection of prescribed functions $f \in {\mathsf{F}}$, where $\mu_t$ is the distribution of $X_t$.

In the case 