Suppressing leakage and maintaining robustness in transmon qubits: Signatures of a trade-off relation

TL;DR

The paper tackles the challenge of designing high-fidelity quantum gates that operate within a computational subspace embedded in a larger Hilbert space while facing unknown static perturbations. It derives a fidelity-susceptibility-based robustness measure and defines leakage and robustness cost functionals, $J_L$ and $J_R$, and implements a two-stage optimization to separately minimize target fidelity, robustness, and leakage. Applying this framework to a transmon qubit with piecewise-constant controls and DRAG benchmarks, the authors demonstrate high-fidelity, robust single-qubit gates and reveal an inherent trade-off between leakage suppression and robustness when attempting to optimize all criteria simultaneously. The results inform practical gate design for multilevel superconducting qubits, highlighting resource demands (time, anharmonicity) and fundamental limits imposed by subspace controllability on achieving universal robustness and leakage minimization together.

Abstract

We study the problem of optimally generating quantum gates in a logical subspace embedded in a larger Hilbert space, where the dynamics is also affected by unknown static imperfections. This general problem is widespread across various emergent quantum technology architectures. We derive the fidelity susceptibility in the computational subspace as a measure of robustness to perturbations, and define a cost function that quantifies leakage out of the subspace. We tackle both effects using a two-stage optimization where two cost functions are minimized in series. Specifically, we apply this framework to the generation of single-qubit gates in a superconducting transmon system, and find high-fidelity solutions robust to detuning and amplitude errors across various parameter regimes. We also show control pulses which maximize fidelity while minimizing leakage at all times during the evolution. However, finding control solutions that address both effects simultaneously is shown to be much more challenging, indicating the presence of a trade-off relation.

Figures (7)

• Figure 1: Outline of the relevant control problem. (a) Diagram of possible evolutions of the system (green lines) from the initial unitary $U_{\rm init}=1$ to the target one $U_{\rm tar}$. (b) Workflow of two-stage optimization. Inputs and output control pulse vectors show by green arrows and optimisation stages in blue boxes.
• Figure 2: Optimized cost functions (a) target $J_U$ and (b) robustness $J_R$ against gate time $T$ for single stage (blue) and two-stage (black/red) optimization. Red dots correspond to choosing $V=\hat{n}$ as the perturbation operator, while black triangles to $V=\hat{q}$. For two-stage optimizations we have fixed the threshold at $\varepsilon_U = 10^{-4}$. Other parameters are set at $\alpha/\Omega=-2$, $\delta/\Omega=-0.5$.
• Figure 3: Optimized cost functions (a) target $J_U$ and (b) robustness $J_R$ against gate time $T$ for different value of anharmonicity $\alpha$. Results are shown for $V=\hat{n}$, $\delta/\Omega=-0.5$. In the optimization the threshold is set at $\varepsilon_U = 10^{-4}$.
• Figure 4: Average gate fidelity, as defined in Eq. (\ref{['eq:fidP']}), for various control protocols in presence of a static perturbation $\lambda V$. Each column corresponds to a different choice of $V$, and $\tilde{\lambda}=\lambda/\hbox{Tr}_P(V^2)$ is the rescaled perturbation strength. Blue: target-only optimization with $T/T_{\Omega}=0.6$. Red: target & robustness optimization with $T/T_{\Omega}=1.3$. Threshold in two-stage optimization is fixed at $\varepsilon_U = 10^{-5}$. Black dotted lines show the performance of simple versions of DRAG pulses with evolution time $T/T_{\Omega}=1.3$, and setting $\sigma/T_{\Omega}=0.369$ (see \ref{['app:DRAG']} for the definition of these parameters; values were chosen to optimize the fidelity at zero perturbation). Other parameters were set to $\alpha/\Omega=-2$ and $\delta/\Omega=-0.5$ (for the optimal control simulations).
• Figure 5: Optimized cost functions (a) target $J_U$ and (b) leakage $J_L$ against gate time $T$ the two-stage optimization target & leakage. We have fixed the threshold at $\varepsilon_U = 10^{-4}$. Other parameters are set at $\alpha/\Omega=-2$, $\delta/\Omega=-0.5$.