were due to the larger dimension of the DFS with respect to the computational space. In this section, we take into account other sources of errors that will be present in most of the implementations, namely, i) errors coming from spontaneous emission to other modes, with rate $\Gamma^*$, included through ${\cal L}_{*}\left[\rho\right] $ as in equation \[eq:SpEm\]; ii) errors from deviations from the Zeno Hamiltonian, attributed to photons emitted to the waveguide from the small population present in the states outside the DFS; iii) errors that may arise from an imperfect control of the laser parameters $\{\Omega_n,\Delta_n\}$. In what follows, we assume to work in a regime with $P_{\oned}\gg 1$, such that the following parameter hierarchy can be satisfied: $\Gamma^*\ll ||H_\mathrm{eff}|| \ll \Gamma_\oned$.

This section discusses, for each gate separately, the numerical results and their analytical approximation of the fidelity between the theoretical final (goal) state, ${|\psi_f\rangle}$, and the real atomic state, $\rho$, obtained after the gate operation, i.e., $F = \langle \psi_\mathrm{f} | \rho | \psi_\mathrm{f} \rangle^{1/2}$. The numerical results are obtained by solving the master equation in second order perturbation theory (see equation \[eq:effemastereq\]). We have checked numerically that this is a good approximation in the parameter ranges 