computational subspace in the ideal case ($\Omega_{12}^+ = \Delta_{12}^- = 0$) are possible when $\Omega_{12}^- \neq 0$, that is when $$\mathcal{Q}_\mathrm{CS} H \mathcal{P}_\mathrm{CS}
    =\mathcal{Q}_\mathrm{CS} H_\mathrm{eff} \mathcal{P}_\mathrm{CS} \neq 0\,,$$ where we use that $\mathcal{P}_\mathrm{CS} \mathcal{P} = \mathcal{P}_\mathrm{CS}$. However, the transitions to these states can be made far off-resonant by setting $$\Omega_n = 0, \ \mathrm{and} \  |\Omega_{12}^-| \ll \Delta_n 
    = \Delta_\mathrm{D} \ll \Gamma_\oned, \ n \geq 3,$$ because the auxiliary states inside the DFS that the computational subspace couples to extend over more than two atoms [^2] and can therefore be detuned as $$\mathcal{Q}_\mathrm{CS} H \mathcal{Q}_\mathrm{CS} \sim \Delta_D \mathcal{Q}_\mathrm{CS}\,,
    \label{eq:DeltaD}$$ while keeping the desired transition driven by $\Omega_{12}^-$ as resonant. One has to make sure that the Stark-shift introduced by this off-resonant transition is small and possibly correct the detuning that it will induce by choosing appropriately the applied laser frequency $\omega_L$.

Controlled Pauli-Z
------------------

For universal quantum computation, a controlled two-qubit gate is required. In this case, the minimal system to encode the operation is the $N=4$ atom case, where two decoherence-free logical qubits can be obtained.

#### **Four atoms:** 

In order to build the controlled-$Z$ gate, we use one of the auxiliary states, ${|A_{1,2}G_{3,4}+A_{3,4}G_{1,2}\rangle}/\sqrt{2}$. Now, it is possible to 