errors (from second order perturbation theory) occur because the computational states do not couple to the radiant ones. However, it is instructive to consider the errors that appear for situations where $\Delta_{12}^-\neq 0$, e.g, because of imperfect addressing, as this yields a useful understanding about how to deal with situations where the second order correction cannot be avoided.

#### **Two atoms:**

For the simplest situation the additional errors due to imperfect addressing, i.e., $\Delta_{12}^-\neq 0$, enter at a rate proportional to $|\Delta_{12}^-|^2/\Gamma_\oned$ through the same error channel as the spontaneous emission into all other modes with rate $\Gamma^*$, that is, via the quantum jump operator ${|0\rangle}^L {\langle 1|}$. Then, the infidelity, i.e., $1-F$, for a $\pi/2$ phase shift of the first logical qubit on the normalized state $\alpha {|0\rangle}^\mathrm{L} + \beta {|1\rangle}^\mathrm{L}$ can be approximated by $$1-F
    \approx \frac{|\beta|^2}{4} \frac{\pi}{\Delta_{12}^+} 
        \left( \Gamma^* + 4 \frac{| \Delta_{12}^-|^2}{\Gamma_{1\mathrm{D}}} \right).
    \label{eq:FidAppPh}$$ One observes, that in the ideal case, $\Delta_{12}^- = 0$, the infidelity can be arbitrarily close to $0$ for large $\Delta_{12}^+$. If $\Delta_{12}^-$ is not negligible, the transition strength $\Delta_{12}^+$ cannot be chosen arbitrarily large to decrease the infidelity. For example, in the worst case scenario where $\Delta_{12}^-  = \Delta_{12}^+$ this results 