relevant figures of merit of the system, e.g., $P_{\oned}$, that in previous works [@beige00a; @beige00b; @tregenna02a] was mainly done through numerical analysis; *iv)* discuss the consequences of imperfect addressing on the fidelity of the gates.

In this work, we show an implementation of universal quantum gates by using $N$ TLS strongly coupled to 1d photon-like reservoirs. First, we show that by *pairing* the TLS, we can define decoherence free-qubits in the singlet (i.e., antisymmetric) states of each pair. The combination of these singlets form the so-called *computational subspace* where we define our operations. Then, we explicitly show how to build single qubit (e.g., phase gates and Pauli-$X$ gates) and two-qubit (e.g., controlled $Z$-gates) operations within the computational subspace without coupling to the other states in the DFS. By using Liouvillian perturbation theory, we obtain analytical expressions for the scaling of the fidelities of the operation ($1-F\propto 1/\sqrt{P_\oned}$) and estimate the error when increasing the number of atoms. Finally, we revisit the problem of the implementation with $\Lambda$ systems in leaky cavities [@beige00a; @beige00b; @tregenna02a] and show how both lead to the similar scaling.

The paper is divided as follows: in section \[sec1\], we introduce the set-up where we implement our 