in an optimal infidelity scaling $\frac{|\beta|^2 \pi }{2} P_\oned^{-1/2}$ for $\Delta_{12}^+ = \Delta_{12}^-=\frac{1}{2} \sqrt{\Gamma^* \Gamma_\oned}$.

#### **Four atoms:** 

A similar behaviour can be obtained by choosing $\Delta_{12}^- = 0$ such that the infidelity is arbitrarily close to $0$ (see figure \[fig3\]a). Slight deviations from this ideal value do not change this behaviour drastically (see figure \[fig3\]b). However, when $\Delta_{12}^-$ is not negligible, it leads to two types of errors that decrease the fidelity (see figure \[fig3\]c): i) virtual population of non-DFS states, which leads to an error rate proportional to $|\Delta_{12}^-|^2/\Gamma_\oned$ as for two atoms; and ii) transitions to auxiliary states, in particular ${|A_{1,2}G_{3,4}+A_{3,4}G_{1,2}\rangle}/\sqrt{2}$. The latter can be made far off-resonance by applying a detuning on the second qubit such that $|\Delta_{12}^-| \ll |\Delta_{34}^+| \ll \Gamma_\oned$. With a large off-resonance ratio $r_\Delta = |\Delta_{34}^+ / \Delta_{12}^-| \gg 1$, one still achieves a small infidelity (see figure \[fig3\]d).

As shown in figures \[fig3\]a-d, the detuning of the third and fourth atom is important when $\Delta_{12}^-$ cannot be neglected. As expected, the infidelity decreases by increasing the off-resonance ratio $r_\Delta$ (see figure \[figSM1\]a). For large enough $r_{\Delta}$, the infidelity can be analytically approximated by$$1-F
    \stackrel{r_\Delta \rightarrow \infty}{\longrightarrow} \frac{\pi}{8} 
        \left( \frac{\Gamma^*}{|\Delta_{12}^+|} + 2 