Correction Formulas for the M{\o}lmer-S{\o}rensen Gate Under Strong Driving

TL;DR

This paper derives explicit analytical correction formulas for the M{46}lmer-S{46}rensen gate errors caused by the Lamb-Dicke approximation, enabling improved gate fidelity through calibration and pulse shaping.

Contribution

It provides the first explicit fourth-order analytical formulas for gate errors, surpassing previous approximations, and demonstrates methods to significantly reduce these errors.

Findings

Fourth order Magnus expansion terms are essential for accurate error estimation.

Analytical renormalization and pulse shaping can dramatically improve gate fidelity.

Explicit formulas enable better calibration and error mitigation in ion trap quantum computing.

Abstract

The M{\o}lmer-S{\o}rensen gate is a widely used entangling gate for ion platforms with inherent robustness to trap heating. The gate performance is limited by coherent errors, arising from the Lamb-Dicke (LD) approximation and sideband errors. Here, we provide explicit analytical formulas for errors up to fourth order in the LD parameter, by using the Magnus expansion to match numerical precision, and overcome significant, orders-of-magnitude underestimation of errors by previous theory methods. We show that fourth order Magnus expansion terms are unavoidable, being in fact leading order in LD, and are therefore critical to include for typical experimental fidelity ranges. We show how these errors can be dramatically reduced compared to previous theory by using analytical renormalization of the drive strength, by calibration of the Lamb-Dicke parameter, and by the use of smooth pulse shaping.