Resilient Entangling Gates for Trapped Ions

Constructing a large-scale ion trap quantum processor will require entangling gate operations that are robust in the presence of noise and experimental imperfection. We experimentally demonstrate how a new type of Mølmer-Sørensen gate protects against infidelity caused by heating of the motional mode used during the gate. Furthermore, we show how the same technique simultaneously provides significant protection against slow fluctuations and mis-sets in the secular frequency. Since this parameter sensitivity is worsened in cases where the ions are not ground-state cooled, our method provides a path towards relaxing ion cooling requirements in practical realizations of quantum computing and simulation.

Figure 1

(a) Energy level diagram showing multitone gate fields detuned from the correct gate detuning $\delta$ by ${\mathrm{\Delta }}_r$ and ${\mathrm{\Delta }}_b$ for the red and blue sidebands, respectively. MTMS gates provide protection against errors of the form ${\mathrm{\Delta }}_r={\mathrm{\Delta }}_b=\mathrm{\Delta }$. (b) Phase space trajectories for one- (red), two- (blue), and three-tone (green) gates. Unlike the single-tone case, the average displacement $⟨\alpha (t)⟩=0$ for two or more tones. In addition, as the number of tones increases, $⟨|\alpha (t){|}^2⟩$ becomes smaller. The effect of this reduction in squared displacement is to reduce the adverse effect of heating on gate fidelity. (c) For a single-tone gate, a symmetric detuning error ($\delta /\mathrm{\Delta }=0.05$) results in both incomplete loops in phase space, causing error due to the residual entanglement between the spin and motional states of the qubit, and incorrect phase accumulation. For the two- and three-tone gates, the loops come much closer to completion (visually indistinguishable from closed loops), effectively eliminating residual qubit-motion entanglement as a contribution to the infidelity, and although not visually obvious, the phase picked up also becomes closer to the ideal. Both of these effects lead to a reduction in sensitivity to symmetric detuning errors.

Figure 2

Infidelities due to heating are reduced by moving from a single- to a two-tone MS gate, shown in red and blue, respectively. Both gates are of duration $\tau =3.4\mathrm{ms}$. Solid lines are the results of a numerical simulation, and three experimental points are shown for each type of gate where the heating rate has been increased artificially through noise injection. Fidelities are normalized to the fidelity at the single-tone fidelity measured at the lowest heating rate [0.94(2)] to account for other errors [29]. The two-tone gate requires a higher peak Rabi frequency for a given gate duration. Numerical simulation of a faster single-tone gate with this higher Rabi frequency is shown in a dashed black line, showing the two-tone gate is still superior.

Figure 3

The effect of symmetric detuning error is significantly reduced by moving to two tones, shown here for a gate duration of $\tau =3.4\mathrm{ms}$. The experimental fidelities for nonzero nominal detuning errors are calculated with respect to the Bell state defined by zero nominal detuning, to account for any phase shift induced by the error. Solid lines are from numerical simulations, but with an offset applied to detuning error to account for any error in determining the trap frequency. Experimentally determined fidelities are normalized to the single-tone fidelity with zero nominal detuning error [0.96(2)] to account for other infidelities. The symmetric detuning offset was fitted using the single-tone theory curve to account for the uncertainty in the initial setting of experimental parameters.

Figure 4

Parity curves at an initial motional mode of $\overline{n}=53(4)$ for a single- (red) and two-tone (blue) gate after ions have been only Doppler cooled. A significant improvement in contrast is seen for the two-tone gate, since the dependence of gate infidelity on $\overline{n}$ to first order in $\mathrm{\Delta }$ is eliminated by the use of multiple tones.