\,\mathrm{d}\mu_{\omega_M}
  \,\mathrm{d}t_M.
\end{split}$$ with the unitary evolution operator $U(t, s)$ given by $$\label{eq:U_definition}
U(t,s) := {\mathcal{T}}
  \exp \left( -{\mathrm{i}}\int_s^t H_{\mathrm{easy}}(\tau) \,\mathrm{d}\tau \right),$$ where ${\mathcal{T}}$ is the time-ordering operator.

The expansion inspires us to use Monte Carlo method to evaluate $\ket{\Psi(t)}$. In such a method, each sample would be the integrand on the right hand side of . To determine the integrand, we need to specify the following:

1.  A non-negative integer $M$;

2.  A sequence of random times: $0 \leq t_1 \leq t_2 \leq \cdots \leq t_M \leq t$;

3.  A sequence of random samples in $\Omega$: $\omega_1, \cdots, \omega_M$.

This links the Monte Carlo method to a marked point process with mark space $\Omega$ (see the textbook [@Jacobsen2006] for an introduction of the marked point process and also point process in general). We denote $\Xi = ((t_m), (\omega_m))_{m\geqslant 1}$ one realization of the marked point process, in which $\omega_m \in \Omega$ is marked at time $t_m$. The marked point process is generated by an intensity function $\lambda(t,\omega)$, *i.e.* $$\mathbb{P}(\text{A mark in } \Sigma \text{ appears in } [t,t+h)) =
  \int_{\Sigma} \lambda(t,\omega) h \,\mathrm{d}\mu_{\omega} + o(h),
  \qquad \forall t > 0, \quad \forall \Sigma \subset \Omega.$$ Any intensity function that 