general framework for the algorithm. To apply the algorithm, we need a decomposition satisfying the conditions A) to C). In general, such decomposition should be studied case by case. One standard way is to decompose the Hamiltonian into non-interacting and interacting parts, where the non-interacting part will be regarded as $H_{\mathrm{easy}}$ and the interacting part as $H_{\mathrm{hard}}$. Since the interacting part usually induces many-body entanglement to the system and is therefore considered as “hard”.

QKMC with $Z$-basis {#sec:Z-basis}
-------------------

Let us now consider specific examples of the quantum kinetic Monte Carlo algorithm. In the first example, we take $${\mathcal{A}} = \Bigl\{\ket{\Psi} =  a \ket{\psi_1} \otimes \cdots \otimes \ket{\psi_N} \;\big\vert\; a \in {\mathbb{C}},\, \ket{\psi_i} \in \{ \ket{\uparrow}, \ket{\downarrow} \}, \, \forall i
  \Bigr\}$$ We call this the $Z$-basis, since each $\ket{\Psi} \in {\mathcal{A}}$ is an eigenvector of (one-body or many-body) Pauli matrices in the $z$-direction.

It is obvious that we can choose $H_{\mathrm{easy}}$ as $$H_{\mathrm{easy}}(t) = \sum_{k=1}^N m_3^{(k)} \sigma_3^{(k)}
  + \sum_{1 \le j < k \le N} \gamma_{jk}(t) \sigma_3^{(j)} \sigma_3^{(k)},$$ since for $U(t,s)$ defined as , $$U(t,s) \ket{\Psi} = \exp\left(
    -{\mathrm{i}}(t-s) \sum_{k=1}^N \kappa_k m_3^{(k)}
    -{\mathrm{i}}\sum_{1\le j<k \le N} \kappa_j \kappa_k
      \int_s^t \gamma_{jk}(\tau) \,\mathrm{d}\tau
  \right) \ket{\Psi},$$ where $\ket{\Psi}$ is an arbitrary state 