The main infidelity of the dark-state evolution in Fig. 3 is due to nonadiabatic errors and the $B_2$ interaction terms. The nonadiabatic errors are minimal at the beginning, middle, and end of the Gaussian pulse where $\dot{\mathrm{\Omega }}=0$. Effects of $B_2$ are present in the first and third two-photon excitations in Toffoli and fan-out gates, respectively, making the effect benign in the fan-out gate due to the minor population while causing a visible deviation in the Toffoli gate.

In the analytical error estimates in Sec. 4, we used the maximum interaction strength, i.e., the interaction of neighboring sites, because the averaged gate error is mainly affected by the qubit configurations with large intracomponent interactions. This conservative estimate yields better agreement with lattice simulations than, e.g., the distance averaged interaction strength.

The polarizability is not the same for different Rydberg levels, and the typical Rydberg scaling laws are not applicable in the presence of external fields. However, we may summarize the main effect of, e.g., reducing the principal quantum numbers: For lower $n$ values, a stronger electric field is required to tune the appropriate Rydberg pairs into resonance. Higher decay rates especially affect the Toffoli gate due to its higher time-integrated population of the Rydberg excited state, while the second rotation error decreases, and higher optical-transition dipole moments and hence achievable Rabi frequency permit faster near-adiabatic operation. Finally, the critical distance explained in Appendix pp4 would be smaller to account for the weaker interaction coefficients. After optimization of the physical parameters, we hence expect similar performance with moderate changes of the principal numbers.