as a weight multiplied by an element in the finite set $${\mathcal{S}} = \left\{ \ket{\psi_1} \otimes \cdots \otimes \ket{\psi_N}
  \Big\vert \ket{\psi_i} \in \{ \ket{\uparrow}, \ket{\downarrow} \}, \forall i
\right\}.$$ From this point of view, the marked point process can be interpreted as a jump process with state space ${\mathcal{S}}$. In detail, supposing the current state in the jump process is $\ket{\Psi}$, we can interpret the mark $\omega$ at time $t$ as the jump $\ket{\Psi} \rightarrow \ket{\Phi}$, where $\ket{\Phi}$ is the only state in ${\mathcal{S}}$ such that $\bra{\Phi} A(t,\omega) \ket{\Psi} \neq 0$. At each jump, the weight is changed to fit the result of applying ${\widetilde{A}}(t, \omega)$. Such an interpretation matches the currently proposed method with the surface hopping method proposed in [@CaiLu2017], where each element in ${\mathcal{S}}$ is considered as a “surface”, and the weight is considered as evolving on the surfaces.

QKMC with simple tensors
------------------------

In this case, we choose the subset ${\mathcal{A}}$ to be all simple tensors in the tensor product space ${\mathcal{H}} = ({\mathbb{C}}^2)^{\otimes N}$. $${\mathcal{A}} = \Bigl\{\ket{\Psi} =  \ket{\psi_1} \otimes \cdots \otimes \ket{\psi_N} \;\big\vert\; \ket{\psi_i} \in {\mathbb{C}}^2, \, \forall i
  \Bigr\}$$ It is clear that this set is larger than the one of $Z$-basis.

In 