Minimal Time Robust Control for Two Superconducting Qubits

Abstract

High-fidelity quantum gates are crucial for achieving fault-tolerant quantum computing; however, decoherence significantly reduces gate fidelities during long operation times. Although optimal control techniques can theoretically minimize these operation times, they often neglect realistic uncertainties in system parameters. In this work, we demonstrate that by using robust optimal control strategies, the cross-resonance gate in superconducting systems can be operated within 64 ns, achieving fidelities of F > 0.99 while maintaining robustness against up to 10% uncertainty in a single parameter. Alternatively, by extending the control time to 71 ns, we achieve fidelities of F > 0.999 with robustness against up to 3% uncertainty. Our results identify the minimal control times attainable with experimentally feasible pulses and system parameters, as well as the maximum allowable static parameter error for high-fidelity operations. Furthermore, we demonstrate simultaneous robustness against both static and time-dependent errors by generating 100 ns control pulses (F > 0.99) that maintain robustness against 10% static parameter error and time-dependent parameter fluctuations two orders of magnitude stronger than typical experimental noise. These findings demonstrate a viable open-loop strategy for implementing fast, high-fidelity quantum gates in the presence of realistic system uncertainties that would otherwise degrade conventional control pulses.

Figures (4)

• Figure 1: Robust Cross-Resonance Gates Performance. The top panel plots the cross-resonance gate infidelity ($\log_{10}(1 - \mathcal{F})$) achieved by the SCP optimization algorithm as a function of the target gate time ($1\,\text{ns}$ to $100\,\text{ns}$), where each curve corresponds to an optimization targeting robustness against a specific maximum static uncertainty in the $J$ coupling ($\Delta J/J$), ranging from 0% (standard optimization, black line) up to $\pm 10\%$ (colored lines, see legend). For each gate time and uncertainty target, the optimization was run multiple times with different random initial pulses; the plotted infidelity at each $T$ is the average over random starts of the worst-case fidelity across the three sampled couplings. Background shading indicates target fidelity regions: $\mathcal{F} > 0.99$ (red, infidelity $< 10^{-2}$) and $\mathcal{F} > 0.999$ (light blue, infidelity $< 10^{-3}$). The minimum pulse duration for $\mathcal{F} > 0.99$ fidelity (optimized for $\pm 10\%$ robustness) is $64\,\text{ns}$. For $\mathcal{F} > 0.999$ fidelity (optimized for $\pm 3\%$ robustness), the minimum duration is $71\,\text{ns}$. The bottom panels show the infidelity ($\log_{10}(1 - \mathcal{F})$) of two representative pre-optimized pulses when subjected to a range of errors in the $J$ coupling: the left panel shows the performance of a pulse optimized for a $T = 64\,\text{ns}$ gate time targeting $\pm 10\%$ robustness in $\Delta J/J$, achieving the desired $\mathcal{F} > 0.99$ fidelity within this uncertainty range, while the right panel shows a pulse optimized for $T = 71\,\text{ns}$ targeting $\pm 3\%$ uncertainty, achieving $\mathcal{F} > 0.999$ within its range. The solid line indicates the infidelity of the best optimized pulse simulated at the corresponding coupling error, while the shaded area represents the infidelity range across an ensemble of optimized pulses (from the other starting points) tested at that error. The vertical dotted lines mark the $\pm$ error range for which each pulse was specifically optimized.
• Figure 2: Performance of Cross-Resonance Gates Optimized for Simultaneous Robustness ($T=100\,\text{ns}$). This figure shows the gate infidelity, $\log_{10}(1-\mathcal{F})$, of two representative $T=100\,\text{ns}$ pulses explicitly optimized to ensure simultaneous robustness against static coupling errors and time-dependent parameter fluctuations. The left panel shows the performance of a pulse optimized for $\pm 10\%$ static error in $\Delta J/J$ alongside realistic experimental noise, achieving the desired $\mathcal{F} > 0.99$ fidelity. The right panel shows a pulse optimized for the same $\pm 10\%$ static error but targeting robustness against a strong-noise environment, where parameter fluctuations are two orders of magnitude larger. (Both noise environments are defined in Table \ref{['tab:noise_conditions_full']}.) The solid line shows the infidelity of these robust pulses in a noise-free environment, while the shaded area represents the infidelity range across 100 unique, time dependent error realizations. The vertical dotted lines mark the $\pm$ static error range for which each pulse was specifically optimized.
• Figure 3: Example of a Robust Cross-Resonance Control Pulse. This pulse was obtained from the SCP optimization, achieving a fidelity of $\mathcal{F} \ge 0.999$ with a duration of $71$ ns while maintaining robustness against $\pm 3\%$ static uncertainty in the $J$ coupling.
• Figure 4: Population Dynamics of Robust Cross-Resonance Gates. This figure shows the time evolution of populations (in the computational subspace and the third and fourth levels) during the 71 ns cross-resonance gate obtained from SCP optimizations. The subplots illustrate the dynamics for gates optimized against varying levels of static $J$ coupling uncertainty ($\Delta J/J$): the left panel corresponds to a 0% target (standard), the middle to $\pm 3\%$, and the right to $\pm 10\%.$