Robust Entanglement Gates for Trapped-Ion Qubits

High-fidelity two-qubit entangling gates play an important role in many quantum information processing tasks and are a necessary building block for constructing a universal quantum computer. Such high-fidelity gates have been demonstrated on trapped-ion qubits; however, control errors and noise in gate parameters may still lead to reduced fidelity. Here we propose and demonstrate a general family of two-qubit entangling gates which are robust to different sources of noise and control errors. These gates generalize the renowned Mølmer-Sørensen gate by using multitone drives. We experimentally implemented several of the proposed gates on ${}^{88}{\mathrm{Sr}}^{+}$ ions trapped in a linear Paul trap and verified their resilience.

Figure 1

Phase-space trajectories and population evolution. (a) Phase-space trajectories of MS (red) and the heart-shaped Cardioid(1,2) (orange) and Cardioid(2,3) (green) gates in the absence of errors. All three gates start and end at (0,0) and enclose an area of $\pi /2$. (b) Same as (a), but with a 6% gate timing error. The MS gate no longer closes the trajectory, while the cardioid gates are almost unaffected. (c) Evolution of the MS gate. In this and the following figures, population data (connected circles) of the SS (blue), DD (red) and $\mathrm{SD}+\mathrm{DS}$ (orange) states are obtained by averaging over 625 realizations, yielding at most 2% projection error, and are compared to the analytic solution (solid line) with no fitting parameters. The gate time, $t=T$, is seen as the SS and DD populations are at 50%, while the $\mathrm{SD}+\mathrm{DS}$ population vanishes. (d) Cardioid(2,3) gate showing a flat response of the populations around the gate time, indicating increased robustness to timing errors. (e) Antioid(2,3) gate employing the same tones and power as in the Cardioid(2,3) gate, however with uniform phase. We observe a narrow quadratic response around the gate time.

Figure 2

Finite-temperature gates. Here ions are Doppler cooled to $\overline{n}\approx 9.8$. Color coding is identical to that of Fig. 1. (a) Antioid(2,3) gate population evolution. A narrow change around $t=T$ marks ion entanglement. Gate fidelity is very sensitive to timing errors. (b) Cardioid(2,3,7,8) gate population evolution. Using four tones increases the robustness, resulting in a wide feature around gate time.

Figure 3

Off-resonance carrier coupling. (a) Population evolution of the MS and Cardioid(2,3) gates, showing data (circles) and analytical solution (solid lines). Here $\mathrm{\Omega }/\nu \sim 5\%$, which, according to Eq. (8), corresponds to 2% infidelity for the MS gate and 0.1% for the Cardioid(2,3) gate. This infidelity is seen by the $\mathrm{SD}+\mathrm{DS}$ populations not vanishing at the gate time. (b) Zoom-in scan of $\mathrm{SD}+\mathrm{DS}$ populations at the gate start. Fast carrier oscillations are easily observed for the MS gate (yellow points). These are heuristically fitted to off-resonance Rabi oscillations $A{\sin }^2(\sqrt{{\mathrm{\Omega }}^2+{\nu }^2}t)$ (yellow line). For the cardioid no such oscillations are seen. (c) Zoom-in scan of $\mathrm{SD}+\mathrm{DS}$ populations around the gate time. The carrier oscillations are less distinct, since here projection noise is on par with the oscillation amplitude. The MS gate populations oscillate around 8% at gate time, while the cardioid populations oscillate around 2%, which corresponds directly to an increased gate fidelity. The 2% cardioid infidelity is due to other imperfections and not carrier coupling.

Figure 4

CarNu(2,3,7) gate. (a) Phase-space trajectory of MS (red) and CarNu(2,3,7) (green) gates for an ideal case (light) and for a combined 6% timing error and 6% normal-mode frequency error (dark). Clearly, the CarNu(2,3,7) gate displays robustness, as its trajectory is almost unaffected. (b) Population evolution of CarNu(2,3,7). Color coding is the same as in Fig. 1. The plot shows a flat response of the state populations around the gate time, indicating increased robustness to timing errors.

Figure 5

Gate fidelities, showing data (filled circles) and a rescaled analytical solution (solid lines). (a) Gate fidelity vs timing error. The MS (red) and Antioid(2,3) (orange) fidelities scale quadratically with the timing error, with the antioid showing the narrowest response. The CarNu(2,3,7) (blue) and Cardioid(2,3,7) (purple) fidelities have fourth- and sixth-order dependences on the timing error, respectively, and are therefore more resilient. (b) Gate fidelity vs normal-mode frequency errors. Here optimization is not order by order; rather, the prefactor of the quadratic term is minimized. The CarNu(2,3,7) (blue) shows the flattest response compared to the other entangling schemes. The inset shows gate purity, for which the optimization is order by order. The CarNu(2,3,7) purity has a fourth-order dependence on the error, while the other gates scale quadratically.