Experimental Demonstration of a Brachistochrone Nonadiabatic Holonomic Quantum-Gate Scheme in a Trapped Ion

Abstract

Nonadiabatic holonomic quantum computation (NHQC) offers intrinsic resilience to certain control imperfections. However, conventional nonadiabatic holonomic protocols are constrained by the fixed-pulse-area condition, which limits flexibility and prolongs duration of small-angle gates. Here we experimentally demonstrate a universal brachistochrone nonadiabatic holonomic quantum gate scheme in a trapped 40Ca+ ion, and realized the construction of pX gate under the conventional NHQC, brachistochrone NHQC (BNHQC) and composite BNHQC (CBNHQC) protocols. By characterizing the performance of gate performance in the presence of dissipation, Rabi-frequency errors and detuning errors, we show that BNHQC and CBNHQC outperform conventional NHQC, and BNHQC can offer a favorable balance between operation speed and robustness. It further shows that keeping high fidelity and strong robustness need decrease the accumulated population of excited state in the evolution process. These results highlight nonadiabatic holonomic computation as a practical route toward fast and robust quantum gates in trapped-ion platforms.

Figures (4)

• Figure 1: Experimental demonstration of universal quantum gate in a trapped ion. (a) Schematic for realizing a universal single quantum gate: (left) the energy-level and (right) the trajectories of realizing a conventional dynamical gate (blue) and a geometric gate based on NHQC (red) on the Bloch sphere. (b) Set-up of realizing a universal single quantum gate in a single trapped $^{40}\mathrm{Ca}^{+}$ ion. (c)-(e) Experimentally measured population evolution for the $\sqrt{X}$ gate implemented by NHQC, BNHQC, and CBNHQC, respectively, (f) The evolution of fidelity with respect to the ideal target state. Here the initial state is $\lvert g\rangle$, the Rabi frequency $\Omega/2\pi=47.1$ kHz, the effective decay rate of auxiliary state $\kappa=5$ kHz and the error bars indicating the statistical standard deviation of the experimental data are obtained by 20000 measurements for each data point.
• Figure 2: The experimentally measured histogram charts for the real and imaginary parts of process matrix $\chi$ of the $\sqrt{X}$ gate under different control protocols, where the data around 0 and 0.5 are enlarged to display the difference among different gate schemes and distinguish small errors of the experimental data. Here the parameters selected same as Fig.\ref{['fig1']}.
• Figure 3: (a) The accumulated population in the auxiliary excited state $\lvert a\rangle$. (b) The comparison of robustness against the dissipation rate $\kappa$ based on the experimental fidelities of the output states after undertaking the $\sqrt{X}$ gate. Here the parameters are selected same as Fig.\ref{['fig1']}.
• Figure 4: Robustness of different schemes against the control errors, where (a) for resonance frequency error $\Delta$ of qubit and (b) for the Rabi frequency error $\delta_{\Omega}$ of the qubit. Here the Rabi frequency $\Omega/2\pi=33.3$ kHz and the dissipation rate $\kappa=66.7$ kHz.