be homogeneous if the positions of the atom-like systems are chosen properly [@lehmberg70a; @lehmberg70b], in contrast to 2d or 3d system. This collective dissipation leads to the emergence of subradiant states that form the so-called Decoherence-Free Subspace (DFS) [@zanardi97a; @lidar98a].

Previous works have already considered how to use the DFS of two atoms trapped in leaky cavities to design one and two-qubit gates using three-level $\Lambda$-type schemes [@beige00a; @beige00b; @tregenna02a], where two atomic hyperfine levels are used to encode the qubit. In the light of the variety of systems available nowadays that allows to engineer robust one-dimensional DFS [@joannopoulos_book95a; @laucht12a; @goban13a; @yu14a; @tiecke14a; @sollner14a; @arcari14a; @goban15a; @young15a; @vetsch10a; @goban12a; @mitsch14a; @petersen14a; @beguin14a; @chang07a; @dzsotjan10a; @gonzaleztudela11a; @bermudez15a; @koppens11a; @huidobro12a; @christensen11a; @martinmoreno15a; @mlynek14a], which may couple to different types of quantum emitters, e.g., atoms, quantum dots, NV centers or superconducting qubits, it is interesting to revisit the problem and fill some of the gaps that have not been considered so far, namely,: *i)* how to encode decoherence-free qubits using only two-level systems (TLS) (as $\Lambda$-schemes might not be available for all platforms); *ii)* extend the proposal to systems with more than two atoms; *iii)* analyze the scaling of the fidelities with the 