drive only the transition between this state and ${|10\rangle}^\mathrm{L}$ without affecting the other states within the DFS by the choice $\Omega_n = 0$, $\Delta_3=\Delta_4 = 0$ and $\Delta_1 = -\Delta_2 \neq 0$. A $\pi$-pulse on the state ${|10\rangle}^\mathrm{L}$ leads to a relative phase of $-1$ on this state, i.e. $${|10\rangle}^\mathrm{L}_{1,3} \longrightarrow -{|10\rangle}^\mathrm{L}_{1,3},$$ for $\frac{1}{\sqrt{2}} \Delta_{12}^- T = 2 \pi$ without affecting the other states of the computational subspace. Hence, we have defined a a controlled controlled-$(-Z)$ gate which is equivalent up to single qubit unitaries to a CNOT-gate [@nielsen_book00a].

#### **For $N>4$ atoms:** 

One can restrict the dynamics to the subspace of four atoms in a similar way as for the single-qubit rotations. With the choice of $$\Omega_n = 0, \ \mathrm{and} \ |\Delta_{12}^-| \ll \Delta_n =\Delta_\mathrm{D} \ll \Gamma_\oned , \ n\geq 5,$$ transitions to states over more than four atoms are far off-resonant. As before, this adds an error rate proportional to $\frac{|\Delta_{12}^- |^2}{\Delta_\mathrm{D}}$ with a proportionality factor depending on the coupling strength after the projection onto the DFS.

Error analysis: spontaneous emission and imperfect addressing. {#sec4}
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So far we have considered only the interaction within the ideal Quantum Zeno Dynamics, where the only possible sources of error 