proposal and establish the general formalism that we use to characterize the operations. In section \[sec3\], we describe the logical qubits and computational subspace and show how to build a set of universal quantum gates in the ideal case, that is, without considering decay into other non-guided modes or deviations from Quantum Zeno dynamics [@zanardi97a; @lidar98a; @facchi02a]. Then, in sections \[sec4\] and \[sec5\], we analyze possible error sources, both analytically and numerically, including spontaneous emission and imperfect addressing for the different gates of our proposal. Finally, in section \[sec6\], we compare the scaling with the proposal of three-level atoms in leaky cavities already explored in the literature [@beige00a; @beige00b; @tregenna02a].

General set-up and formalism {#sec1}
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Set-up: waveguide QED
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The general set-up that we consider is depicted in figure \[fig1\]a; namely $N$ TLSs, $\{ |g\rangle_n$, $| e \rangle_n \}_{n = 1 \dots N}$, placed at positions $z_n$ and coupled to a 1d field with bosonic annihilation operators $a_q$. Due to the variety of implementations available nowadays, we will keep the discussion as general as possible without making further assumptions on the nature of the TLS and/or 1d waveguides.

![(a) TLS (in blue) trapped along a one-dimensional waveguide, addressed by 