Trapped-ion two-qubit gates with 99.99% fidelity without ground-state cooling

We introduce the "smooth gate": a novel entangling gate method for trapped-ion qubits where residual spin-motion entanglement errors are adiabatically eliminated by ramping the gate detuning. We demonstrate the power of this technique by performing electronically controlled two-qubit gates at Doppler temperature nbar~3.5, consistently estimating error rates at or below 1e-4. We further show that the error remains below 5e-4 for ions with average phonon occupation numbers of up to nbar~9.4 on the gate mode. These results show that trapped-ion quantum computation can be performed with high fidelity at temperatures above the Doppler limit, allowing for significantly faster and simpler device operation.