not be a vector space, i.e., the parametrization is nonlinear;

2.  For any $\ket{\Psi} \in {\mathcal{A}}$, the action of $H_{\text{easy}}$ remains in ${\mathcal{A}}$: $$\mathrm{e}^{-{\mathrm{i}}t H_{\text{easy}}} \ket{\Psi} \in {\mathcal{A}}, \qquad \forall\, t$$ and is easy to obtain (either exactly or with a controllable error);

3.  It is possible to stochastically represent the action of $H_{\text{hard}}$, in the sense that there exists a stochastic operator $A(\omega)$ with $\omega$ corresponding to some random space $\Omega$, such that for any $\ket{\Psi} \in {\mathcal{A}}$, $$\label{eq:hard_op}
        {\mathbb{E}}_{\omega} A(\omega) \ket{\Psi} = H_{\text{hard}} \ket{\Psi}, \qquad \text{and} \qquad A(\omega) \ket{\Psi} \in {\mathcal{A}}, \quad \forall \omega.$$

In practice, it is often easier to first determine the set ${\mathcal{A}}$, and then look for the decomposition of the Hamiltonian and also the stochastic representation of the action of $H_{\text{hard}}$. Thus, in the following, when discussing two examples of algorithms developed under this framework, we will refer to them by the choice of the set ${\mathcal{A}}$.

With such a decomposition, the Schrödinger equation can be written as the following integral form by the Duhamel’s principle: $$\label{eq:int_form}
\begin{split}
\ket{\Psi(t)} &= \mathrm{e}^{-{\mathrm{i}}t H_{\mathrm{easy}}} \ket{\Psi(0)} -
  {\mathrm{i}}\int_0^t \mathrm{e}^{-{\mathrm{i}}(t-t_1) H_{\mathrm{easy}}}
    H_{\mathrm{hard}} \ket{\Psi(t_1)} \,\mathrm{d}t_1 \\
&= \mathrm{e}^{-{\mathrm{i}}t H_{\mathrm{easy}}} \ket{\Psi(0)} -
  {\mathrm{i}}\int_0^t \int_{\Omega} \mathrm{e}^{-{\mathrm{i}}(t-t_1) H_{\mathrm{easy}}}
    A(\omega_1) \ket{\Psi(t_1)} \,\mathrm{d}\mu_{\omega_1} \,\mathrm{d}t_1,
