Universal pulses for superconducting qudit ladder gates

TL;DR

This work tackles the challenge of fast, high-fidelity qudit control in superconducting transmon ladders by constructing universal, analytic pulses via a recursive DRAG framework. By reducing the dynamics to a four-level effective model that captures dominant leakage channels, the authors derive DRAG2 and DRAG4 corrections that suppress both single- and multi-photon leakage while mitigating phase and amplitude errors. The approach delivers substantial fidelity gains and faster gate times (e.g., π/2 gates reduced to ~10 ns, with target errors around 10^-4) and reveals a universal scaling of the quantum speed limit with the ladder nonlinearity Δ_k, independent of level-specific details. These results provide a scalable path to high-fidelity qudit operations across various hardware and leakage channels, with practical calibration guidance and applicability to other ladder-structured quantum systems.

Abstract

Qudits, generalizations of qubits to multi-level quantum systems, offer enhanced computational efficiency by encoding more information per lattice cell, avoiding costly swap operations and providing even exponential speedup in some cases. Utilizing the $d$-level manifold, however, requires high-speed gate operations because of the stronger decoherence at higher levels. While analytical control methods have proven effective for qubits in achieving fast gates with minimal control errors, their extension to qudits is nontrivial due to the increased complexity of the energy level structure arising from additional ancillary states. In this work, we present a universal pulse construction for generating rapid, high-fidelity unitary rotations between adjacent qudit levels, thereby providing a prescription for any gate in $SU(d)$. Control errors in these operations are effectively analyzed within a four-level subspace, including two leakage levels with approximately opposite detuning. By identifying the optimal degrees of freedom, we derive concise analytical pulse schemes that suppress multiple control errors and outperform existing methods. Remarkably, our approach achieves consistent coherent error scaling across all levels, approaching the quantum speed limit independently of parameter variations between levels. Validation on transmon circuits demonstrates significant improvements in gate fidelity for various qudit sizes aiming for $10^{-4}$ error. This method provides a scalable solution for improving qudit control and can be broadly applied to other quantum systems with ladder structures or operations involving multiple ancillary levels.

Figures (11)

• Figure 1: Energy structure of driving a two-level transition in a qudit system. (a) Typical energy structure of a transmon system. Energy levels and error transitions for (b) the ground and first excited states, and (c) general ladder transition between higher levels in the rotating frame. (d) The number of levels that can be used as a qudit quantum register as a function of the anharmonicity. The upper bound is set by the decoherence introduced by charge fluctuations. The detailed discussion can be found in \ref{['sec:decoherence']}.
• Figure 2: Properties of transmon qudits. a) Detuning between the target subspace and the leakage levels in the rotating frame. The qubit frequency and anharmonicity of the ground state are $5$ GHz and $-100$ MHz, corresponding to $E_J/E_C\approx 355$. For $k=0$, $\Delta_k=\alpha.$ b) The small energy gap between the two leakage levels $\ket{k-1}$ and $\ket{k+2}$ for the first five transitions. This is much smaller than $\Delta_k$, leading to the energy structure shown in \ref{['fig:qudit energy levels']}. The grey vertical line marks the parameters used in (a).
• Figure 3: Control error in driving ladder transitions in a transmon qudit. a) Estimated error budget of driving a $\ket{1}\leftrightarrow\ket{2}$$\pi$ rotation using a Hann pulse with an anharmonicity of $\alpha/(2\pi)=-200$ MHz. The definitions of different errors are given in Appendix \ref{['sec:error defintion']}. b) The leakage error calculated via Eq. \ref{['eq:leakage']} in a regime where the single-derivative DRAG faces limitations and offers no improvement, even with an optimized DRAG coefficient. Parameters used are $\lambda_{k-1}=\lambda_k=\lambda_{k+1}=1$, $\delta_{k-1,k+2}=0$ and $t_\textnormal{f}=15$ns. The pulse is a single-derivative DRAG shape $\Omega-ia\dot{\Omega}/\Delta_k$, with $\Omega$ the standard Hann pulse.
• Figure 4: Gate infidelity as a function of duration for different drive schemes driving the $\ket{1} \leftrightarrow \ket{2}$ transition, with $\alpha/(2\pi)=-200$ MHz and $\omega_{10}/(2\pi)=5$ GHz. The DRAG2 pulse is defined in \ref{['eq: DRAG2 corrections1', 'eq: DRAG2 corrections2']} and the DRAG4 pulse in \ref{['eq: DRAG4 corrections1', 'eq: DRAG4 corrections2']}. The solid line represents the analytically derived DRAG pulse and the dashed line represents gate error with calibrated DRAG coefficients.
• Figure 5: Controlling the ladder transitions in a transmon qudit. a) The $\pi/2$ gate error for different ladder transitions $\ket{k}\leftrightarrow\ket{k+1}$ for $\alpha/(2\pi)=-200, -100, -50$ MHz using a fixed gate duration of 8, 15 and 30 ns respectively. These values of anharmonicity correspond to $E_J/E_C\approx$100, 355 and 1331, respectively. b) The minimum gate time required to achieve infidelity of $10^{-4}$ for different drive schemes and different hardware parameters for a $\pi/2$ gate. The gate duration is multiplied by the corresponding leakage level separation $\Delta_k$, resulting in overlapping outcomes across different hardware parameters and qudit levels.