\mathrm{e}^{-\rmi \phi} {|1\rangle}^\mathrm{L}$) is obtained by applying $\Omega_{12}^-= 0$, $\Delta_{12}^+ \neq 0$ for a time $T = \frac{\phi}{\Delta_{12}^+}$. Pauli-$X$ rotations (plus a phase) are obtained for $0 \neq \Omega_{12}^- \in \mathbb{R}$, $\Delta_{12}^+ = 0$ and time $T = \frac{\pi}{\sqrt{2} \Omega_{12}^-}$. Note that to avoid errors in both cases, one should also set $\Omega_{12}^+ = \Delta_{12}^- = 0$ as will be discussed in section \[sec4\]. The Pauli $Y$ can be obtained as the $X$ just by using $i\Omega_{12}^- \in\ \mathbb{R}$, so that we will not discuss it further.

#### **Four atoms:** 

In this case, the way to do phase gates and rotations is the same as for the two atom case. However, in the case of the rotations, states within the computational subspace couple to auxiliary states for more than two atoms. In particular, the state ${|10\rangle}^\mathrm{L}$ is coupled to the auxiliary state ${|A_{1,3}A_{2,4}+A_{1,4}A_{2,3}\rangle}/\sqrt{2}$ for $\Omega_{12}^- \neq 0$ as shown in figure \[fig2\]. However, this transition can be made far off-resonance by setting $|\Delta_{34}^+| \gg |\Omega_{12}^-|$. This results in an additional error rate $\frac{|\Omega_{12}^-|^2}{2 \Delta_{34}^+}$ that will be considered when calculating the fidelity of the operation.[^1]

#### **For $N>4$ atoms:** 

Again in the case of rotations, transitions to states outside the 