Stability thresholds and calculation techniques for fast entangling gates on trapped ions

Fast entangling gates have been proposed for trapped ions that are orders of magnitude faster than current implementations. We present here a detailed analysis of the challenges involved in performing a successful fast gate. We show that the rotating wave approximation is stable with respect to pulse numbers: the time scale on which we can neglect terms rotating at the atomic frequency is negligibly affected by the number of pulses in the fast gate. In contrast, we show that the laser pulse instability does give rise to a pulse-number-dependent effect; the fast gate infidelity is compounded with the number of applied imperfect pulses. Using the dimensional reduction method presented here, we find bounds on the pulse stability required to achieve two-qubit gate fidelity thresholds.