= \ket{\Psi_{\mathrm{down}}}\bra{\Psi_{\mathrm{down}}},$$ where $\ket{\Psi_{\mathrm{down}}} = \ket{\downarrow}^{\otimes N}$. For small $N$, we use a deterministic Runge-Kutta solver to provide reference solutions.

In Figure \[fig:1-4spins\_Zbasis\], we show the numerical results using QKMC with $Z$-basis. The cases with one spin to four spins are considered, and the coupling intensity $\gamma_0$ is set to be $0.05$. For all the four cases, 1,000,000 trajectories are used. It can be seen that when $t$ is large, the numerical solution of QKMC becomes oscillatory and unreliable, due to the large sample variance. As the number of spins increases, the reliable part of the curve becomes shorter.

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Better results can be obtained using the QKMC with simple tensors. Figure \[fig:1-4spins\] shows the numerical results with the same settings. The number of trajectories is again 1,000,000. For this method, when only one spin is present, the algorithm is identical to the deterministic ODE solver. In all the four cases, two complete cycles are obtained without obvious oscillation.

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In the case of 4 spins, we also check the numerical error for QKMC with simple tensors. Using the result of the deterministic solver as the reference solution $\ket{\Psi^{\mathrm{ref}}(t)}$, we plot the evolution of 