of the DFS. We use this effective laser coupling to control the atomic state of the ensemble. Besides, there is a second order correction resulting from $H_{\mathrm{las}}$ that will be relevant for the analysis of the error probability of our proposal as we show in section \[sec4\].

-   The contribution of the emission of photons to other radiative modes different from the guided mode of the waveguide that we embed into a single decay rate, $\Gamma^*$ and describe through the Liouvillian $$\mathcal{L}_\mathrm{pert}[\rho]=\mathcal{L}_{*}[\rho]=\sum_n \frac{\Gamma^*}{2}(\sigma_{ge}^n\rho\sigma_{eg}^n-\rho \sigma_{ee}^n+\mathrm{H.c.}).
             \label{eq:SpEm}$$ This contribution is relevant for the error analysis of the gates in section \[sec4\].

Universal Single- and Two-Qubit Gates {#sec3}
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In this Section, we show how to engineer a set of universal gates, i.e., defined by any arbitrary single-qubit rotation and a controlled gate [@nielsen_book00a], using the effective evolution $H_{\mathrm{eff}}$ within the DFS that appear in our waveguide QED setup. Firstly, due to the large degeneracy of the DFS, we need to define a set of logical qubits that will expand our computational subspace. Then, we show how to choose $\{\Omega_n,\Delta_n\}$ such that they define a set of universal one and two-qubit gates, namely, the phase and Pauli-$X$ (and $Y$) gate and the controlled-$(-Z)$ 