\frac{|\Delta_{12}^+|}{\Gamma_\oned} \right).$$ This leads to a minimal infidelity $\propto P_\oned^{-1/2}$ (see figure \[figSM1\]b) for $\Delta_{12}^+ = \Delta_{12}^- = \sqrt{\Gamma^* \Gamma_\oned/2}$.

![Dependence of infidelity on off-resonance ratio $r_\Delta=|\Delta_{34}^+ / \Delta_{12}^-|$ for $\pi/2$ phase shift on first qubit with $\Delta_{12}^- = \Delta_{12}^+$. (a) Infidelity varied over the coupling strength $\Delta_{12}^+$ for $P_\oned=10^4$. The results correspond to $r_\Delta=5$ (black), $r_\Delta=10$ (red) and $r_\Delta=15$ (blue). (b) Minimal infidelity depending on the Purcell Factor $P_\oned$. The numerical results (points) corresponding to the values as in (a) fit well with the approximation (line) of $(1-F)_\mathrm{min} \propto  P_\oned^{-1/2}+ C(r_\Delta)$, where the $C(r_\Delta)$ is a number which does not depend on $P_\oned$.[]{data-label="figSM1"}](figure4v2.pdf){width="98.00000%"}

Pauli-X Gate
------------

For rotations around the $x$-axis, we set $\Delta_{1} = \Delta_2= 0$, and $\Delta_n = \Delta_\mathrm{D} \gg \Omega_{12}^-$ for all $n\geq 3$ to avoid transitions to auxiliary states. In contrast to the phase shift gate, even in the ideal case, $\Omega_{12}^+ = 0$, errors will occur because $\Omega_{12}^-$ couples to state outside the DFS, as shown schematically in figure \[fig2\](a) for the two atom case. Moreover, we also include a short discussion on deviations due to imperfect control on $\Delta_{1(2)} \neq 0$ and $\Omega_{12}^+$.

#### **Two atoms:** 

Using $\Omega_{12}^+  = 0$, the error rate 