Analytical blueprint for 99.999% fidelity X-gates on present superconducting hardware under strong driving

TL;DR

This paper analyzes the breakdown of the three-level model under strong driving in superconducting qubits and identifies dominant multi-photon leakage channels. It develops a recursive DRAG framework (R1D and R2D) that uses successive frame transformations to suppress both single- and two-photon leakage, supported by analytical time-ordering and Magnus-based calculations. The authors derive near-optimal prefactors for DRAG and demonstrate that, even with decoherence, single-qubit gates with infidelities below $10^{-5}$ are achievable in times as short as a few to ~7 ns, achieving practical ultra-fast control. The approach provides actionable pulse-design rules and calibration strategies, with experimental validation showing substantial gains in gate speed and fidelity on present superconducting hardware.

Abstract

Achieving very fast gates that undercut the natural limits set by decoherence requires going into the strong driving limit. Realizing single-qubit control predicted beyond semi-classical, time-dependent modeling has yet to be experimentally realized on superconducting and most other computing platforms. In this regime, the common model of dynamics within a three-level manifold breaks down, and instead, we see new quantum error channels growing abruptly with decreasing time. To identify these error processes we systematically calculate the effect of multi-photon transitions that occur out of the computational space. We then derive analytical formulas to suppress these effects, as well as amplitude and phase errors on the qubit space; we term these R1D for suppressing the $|0\rangle-|2\rangle$ transition and R2D when also suppressing $|1\rangle-|3\rangle$ leakage. We also answer long-standing questions about the optimal values of the DRAG prefactor as well as constant detuning, when accounting for time-ordering, and also show how to calibrate other prefactors for further performance improvement. Upon correcting these varied sources of error, we numerically demonstrate gate infidelities below $10^{-5}$ for a 7ns $π$-rotation when incorporating existing decoherence rates.

Figures (9)

• Figure 1: Analytical pulse performance for a transmon qubit. a) Low-lying energy structure of a transmon system represented as a particle in a cosine potential, where different opacity refers to the importance of higher energy levels in the performance of the gate. b) Structure of the energy levels of the transmon system in the rotating frame with respect to the transition $\omega_{10}$. The dashed arrows connecting different states represent single-photon transitions, while dotted dashed connects states through two-photon processes. c) Gate infidelity for implementing an X gate under different pulses: black lines corresponds to Hann pulse, blue line is for DRAG and green and violet stand for recursive pulses R1D and R2D, respectively. These pulses are simulated analytically without calibration. R2D outperforms every method and achieves a gate fidelity of 99.99 % at 9 ns and a fidelity of 99.999% at 11.8 ns. d) Histogram with the final population of the relevant leakage states for different pulses when implementing an X gate at 6ns. The main source of error in DRAG can be addressed by using R1D, which will suppress 2 photon transitions to $|2\rangle$. However ,for R1D the leakage to $|3\rangle$ is a major source of error. This can suppressed by an order of magnitude by R2D. The simulations were performed by modeling the transmon as an anharmonic oscillator with anharmonicity $\Delta_{2} = -2\pi \times 0.225$ GHz.
• Figure 2: Gate performance under different pulse protocols. Gate infidelity as a function of Gate time for analytical linear DRAG (blue), linear DRAG using the predictions for $\alpha$, $\beta$ and $\delta_c$ ( DRAG P, orange), linear DRAG with optimized parameters (DRAG N, green) and R2D with optimized parameters, $\alpha_{12}$, $\alpha_{02}$, $\alpha_{13}$, $\beta$ and $\delta_c$ (R2D N, purple). Despite the mismatch between the predictions for $\alpha$ for linear DRAG and the optimized values, we obtain a significant improvement on the fidelity of the gate demonstrating that parametrized constants can eliminate higher order errors. The parameterized R2D pulse outperforms all others for all gate times considered, achieving a fidelity of 99.999% at 6.70 ns, a $\approx$ 25% improvement versus standard DRAG (which reaches 99.999% fidelity at 8.93 ns.)
• Figure 3: Amplitude and Constant Detuning as a function of $\alpha$ a) Comparison between the optimized values of constant detuning $\delta_c$ for the DRAG pulse and the analytical approximation, for different values of $\alpha$. The analytical expression can accurately predict the values of constant detuning. The longer the gate time, the better the prediction. b) Comparison between the optimized values of $\beta$ for the DRAG pulse and the analytical approximation, for different values of $\alpha$. Once again, the analytical expression can accurately predict the values of constant detuning.
• Figure 4: Analytically vs Numerically optimized $\alpha$'s parameter. Comparison between the optimized values of $\alpha$ for linear DRAG pulse and the analytical expression of $\alpha$, as a function of gate time T. Although our expression captures the general behaviour of $\alpha$, we find large discrepancies, most likely due to higher orders.
• Figure 5: Numerically optimized $\alpha$'s parameters. Optimal values of $\alpha_{12}$, $\alpha_{02}$ and $\alpha_{13}$ for the R2D pulse as a function of gate time T. These results generally match the ones obtained in experiment GAO2025, validating the theory.