space and the additional states that must be either decoupled or used as auxiliary states. In particular, for $N=4$ the auxiliary states are ${|A_{1,2}G_{3,4}+A_{3,4}G_{1,2}\rangle}/\sqrt{3}$ and ${|A_{1,3}A_{2,4}+A_{1,4}A_{2,3}\rangle}/\sqrt{2}$.

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

In general (for $N>2$) the dimension of the DFS, ${ {N \choose N/2} }$ (for even $N$), is larger than the one of the computational subspace, $2^{N/2}$. Thus, one can split the projection onto the DFS, $\mathcal{P}$, into two orthogonal projections, i.e. one into the computational subspace $\mathcal{P}_\mathrm{CS}$ and its orthogonal counterpart $\mathcal{Q}_\mathrm{CS}$: $$\begin{aligned}
    \mathcal{P} =& \mathcal{P}_\mathrm{CS} + \mathcal{Q}_\mathrm{CS},\\
    \mathcal{P}_\mathrm{CS} =& \sum_{j\ \mathrm{odd}} {|0\rangle}^\mathrm{L}_j {\langle 0|} + {|1\rangle}^\mathrm{L}_j {\langle 1|}\,,\end{aligned}$$ such that the effective Hamiltonian can be written as follows $$H_\mathrm{eff} = 
      \mathcal{P}_\mathrm{CS} H_{\mathrm{las}} \mathcal{P}_\mathrm{CS} 
    + \left( \mathcal{P}_\mathrm{CS} H_{\mathrm{las}}   \mathcal{Q}_\mathrm{CS} + \mathrm{H.c.} \right) 
    + \mathcal{Q}_\mathrm{CS} H_{\mathrm{las}}  \mathcal{Q}_\mathrm{CS},$$ which separates the transitions within the computational \[auxiliary\] subspace $\mathcal{P}_\mathrm{CS} H \mathcal{P}_\mathrm{CS}$ \[$\mathcal{Q}_\mathrm{CS} H   \mathcal{Q}_\mathrm{CS}$\] and the coupling between these two subspaces: $\mathcal{P}_\mathrm{CS} H\mathcal{Q}_\mathrm{CS}$. This separation will be useful to argue that we can make operations within the DFS even in the situations with $N>4$ as we will show afterwards. For a general situation, it is easy to show that by projecting $H_{\mathrm{las}}$ into the computational subspace we obtain an effective evolution inside 