Toward a M{\o}lmer S{\o}rensen Gate With .9999 Fidelity

Realistic fault-tolerant quantum computing at reasonable overhead requires two-qubit gates with the highest possible fidelity. Typically, an infidelity of $\lesssim 10^{-4}$ is recommended in the literature. Focusing on the phase-sensitive architecture used in laboratories and by commercial companies to implement quantum computers, we show that even under noise-free, ideal conditions, neglecting the carrier term and linearizing the Lamb-Dicke term in the Hamiltonian used for control-pulse construction for generating M{\o}lmer-S{\o}rensen XX gates based on the Raman scheme are not justified if the goal is an infidelity target of $10^{-4}$. We obtain these results with a gate simulator code that, in addition to the computational space, explicitly takes the most relevant part of the phonon space into account. With the help of a Magnus expansion carried to the third order, keeping terms up to the fourth order in the Lamb-Dicke parameters, we identify the leading sources of coherent errors, which we show can be eliminated by adding a single linear equation to the phase-space closure conditions and subsequently adjusting the amplitude of the control pulse (calibration). This way, we obtain XX gates with infidelities $< 10^{-4}$.