gate. A summary of the parameters for these gates can be found in table \[tab:GateSum\]. For completeness, we also give the parameters for other gates such as the Hadamard or SWAP gates. The former can be easily constructed because all single qubit rotations can be performed and the latter is constructed through the same idea as the controlled-$(-Z)$ gate. Due to the degeneracy of the DFS, the challenge lies in defining operations within the computational subspace, without populating the rest of the states within the DFS. In section \[sec4\], we revisit the problem and consider the effect of spontaneous emission and second order corrections to the Zeno dynamics that ultimately limit the fidelity of the operations.

[l|cccccc|c|c]{} Gate & $\Omega_{12}^-$ & $\Omega_{12}^+$ & $\Delta_{12}^+$ & $\Delta_{12}^-$ & $\Delta_{34}^+$ & $\Delta_{34}^-$ & $T$ & $\Delta_{n\geq5}$\
$ T_1$ & 0 & 0 & -$\Delta_\mathrm{T}$ & 0 & 0 & 0 & $\frac{\pi}{4 \Delta_\mathrm{T}}$& 0\
$ Z_1$ & 0 & 0 & $\Delta_\mathrm{Z}$ & 0 & 0 & 0 & $\frac{\pi}{\Delta_Z}$ & 0\
$ X_1$ & $\Omega_\mathrm{X} \in \mathbb{R}$ & 0 & 0 & 0 & $\Delta_\mathrm{D}$ & 0 & $\frac{\pi}{\sqrt{2} \Omega_\mathrm{X}}$ & $\Delta_\mathrm{D}$\
$ Y_1$ & $\Omega_\mathrm{Y} \in \rmi \mathbb{R}$ & 0 