Robust 2-Qubit Gates in a Linear Ion Crystal Using a Frequency-Modulated Driving Force

In an ion trap quantum computer, collective motional modes are used to entangle two or more qubits in order to execute multiqubit logical gates. Any residual entanglement between the internal and motional states of the ions results in loss of fidelity, especially when there are many spectator ions in the crystal. We propose using a frequency-modulated driving force to minimize such errors. In simulation, we obtained an optimized frequency-modulated 2-qubit gate that can suppress errors to less than 0.01% and is robust against frequency drifts over $\pm 1\mathrm{kHz}$. Experimentally, we have obtained a 2-qubit gate fidelity of 98.3(4)%, a state-of-the-art result for 2-qubit gates with five ions.

Figure 1

Robust (violet, solid line) and nonrobust (blue, dash-dotted line) FM pulses for 2-qubit gate optimized for five ions, both with a gate time of $90\mu \mathrm{s}$. Green lines are experimental sideband frequencies, labeled 1–5, the first one being the common mode frequency. The pulses are designed to be symmetric in time. The dots and diamonds are the vertices of the frequency and represent the control parameters allowed to vary in our optimization algorithm.

Figure 2

Simulated PSTs with (a) no frequency error and (b) $-1\mathrm{kHz}$ sideband drift, using the FM pulses shown in Fig. 1. The end points for the robust pulse (circles) return to the starting point with the drift, whereas those for the nonrobust pulse (diamonds) fail to do so. The horizontal and vertical axes represent the quadratures $x_k\sim a_k^{†}+a_k$ and $p_k\sim i(a_k^{†}-a_k)$, respectively.

Figure 3

(a) State population and (b) parity scan of the two qubits after the optimized and robust 2-qubit gate shown in Fig. 1, indicating a fidelity of 98.3(4)%.

Figure 4

(a) Simulated gate error and (b) experimental even-parity populations of the two qubits after the gate for a range of detuning offsets. The robust gate has a significantly better performance than the nonrobust gate in both theory and experiment.

Figure 5

Optimized FM 2-qubit gate for 17 ions. The sideband frequencies (green) are obtained by simulations.