Correction formulas for the Molmer-Sorensen gate

The widely used Molmer-Sorensen gate's performance is limited by coherent errors, arising from the Lamb-Dicke (LD) approximation and sideband errors. We show that fourth order Magnus expansion terms are unavoidable and substantial in magnitude, being in fact leading order in LD, and are therefore critical to include for typical target gate fidelities. We provide explicit analytical formulas for errors up to 4th order in the LD parameter, by using the Magnus expansion to match numerical precision, while the standardly used 2nd (or 3rd) order drastically misrepresents the effective Rabi frequency. We show how these errors can be partially compensated using analytical renormalization of the drive strength, by calibration of the Lamb-Dicke parameter, and by the use of smooth pulse shaping.