considered throughout this manuscript. To obtain the analytical approximations, we used the effective non-hermitian Hamiltonian that can be obtained from the second order master equation (see appendix A).

Phase Shift Gate
----------------

![Infidelities of single and two qubit gates for $N=4$ atoms. (a)-(d) Infidelity of a $\pi/2$ phase shift on the first logical qubit of four atoms on the state ${|10+00\rangle}^\mathrm{L}/\sqrt{2}$ for (a) $\Delta_{12}^- = 0$ and $\Delta_{34}^+ = 0$, (b) $\Delta_{12}^- = 0.1 \Delta_{12}^+$ and $\Delta_{34}^+ = 0$, (c) $\Delta_{12}^- = \Delta_{12}^+$ and $\Delta_{34}^+ = 0$, (d) $\Delta_{12}^- = \Delta_{12}^+$ and $\Delta_{34}^+ = 20 \Delta_{12}^+$. (e) Infidelity of a single qubit Pauli-X gate on the first qubit of four atoms on the state ${|00\rangle}^\mathrm{L}$ for $\Delta_{34}^+ = 10 \Omega_{12}^-$. (f) Infidelity of the controlled-$(-Z)$ gate for four atoms (2 qubits) acting on the state ${|10+11\rangle}^\mathrm{L}/\sqrt{2}$. The black lines in (d)-(f) represent the scaling of the coupling strength $\Delta_{12}^-$, $\Omega_{12}^-$ and $\Delta_{12}^-$, respectively, for the minimal infidelity with $P_\oned^{-1/2}$.[]{data-label="fig3"}](figure3v1.pdf){width="98.00000%"}

For the phase shift gate we must set $\Omega_n = 0$ for all $n$ and $\Delta_n = \Delta_\mathrm{D} \gg \Delta_{12}^-$ for all $n \geq 3$ to avoid errors from transitions to auxiliary states. By choosing $\Delta_1 = \Delta_2$, i.e., $\Delta_{12}^-=0$, no 