Robust Quantum Optimal Control with Trajectory Optimization

The ability to engineer high-fidelity gates on quantum processors in the presence of systematic errors remains the primary barrier to achieving quantum advantage. Quantum optimal control methods have proven effective in experimentally realizing high-fidelity gates, but they require exquisite calibration to be performant. We apply robust trajectory optimization techniques to suppress gate errors arising from system parameter uncertainty. We propose a derivative-based approach that maintains computational efficiency by using forward-mode differentiation. Additionally, the effect of depolarization on a gate is typically modeled by integrating the Lindblad master equation, which is computationally expensive. We employ a computationally efficient model and utilize time-optimal control to achieve high-fidelity gates in the presence of depolarization. We apply these techniques to a fluxonium qubit and suppress simulated gate errors due to parameter uncertainty below ${10}^{-7}$ for static parameter deviations of the order of 1%.

Figure 1

(a) Flux pulses for the numerical gates (dark blue) and the analytic gates (light pink). (b) The $T_1$ interpolation function used in optimization. Circle markers indicate measured $T_1$ times. Noncircle markers are plotted at the time-averaged absolute flux and the time-averaged $T_1$ time for each pulse. (c) Cumulative gate errors due to depolarization as a function of the number of gates applied. Cumulative gate errors for the numerical $Z/2$ and $Y/2$ gates are indistinguishable. Inset shows log-scaled cumulative gate errors for small gate counts.

Figure 2

(a) Flux pulses for $Z/2$ gates robust to qubit frequency detunings constructed with the analytic ($A$), sampling ($S$), unscented sampling ($U$), and the first- and second-order derivative methods ($D1,D2$). The flux pulses shown for the sampling, unscented sampling, and derivative methods are optimized for twice the gate time of the analytic gate. (b) Single-gate error at a one-percent qubit frequency detuning as a function of the gate time. Missing data points represent gates with a gate error greater than $5\times {10}^{-5}$. (c) Single-gate error as a function of the qubit frequency detuning. The gate errors for the analytic and first-order derivative methods are shown for gate times that are multiples of $1/4f_q\sim 18\mathrm{ns}$. The gate errors for the two methods are indistinguishable at the gate time $18\mathrm{ns}$.

Figure 3

(a) Flux pulses for $X/2$ gates robust to flux noise constructed with the analytic ($A$), sampling ($S$), unscented sampling ($U$), and the first- and second-order derivative methods ($D1$, $D2$). (b) Cumulative gate error due to $1/f$ flux noise for successive gate applications. The cumulative gate errors for the sampling, unscented sampling, and the derivative methods are indistinguishable.