Half-Integer Spectral Zeros for Leakage Suppression in Fast Transmon Pulses

arXiv:2605.10578 · cond-mat.mes-hall · Submitted 2026-05-11 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: Today's paper: "Half-Integer Spectral Zeros for Leakage Suppression in Fast Transmon Pulses".

Mira: The gist The exact half-integer spectral-zero condition τα = n + 1/2 for a symmetric flat-top envelope with cosine ramps arises from destructive interference between the contributions of the rising…

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So we're looking at this paper right now, "Half-Integer Spectral Zeros for Leakage Suppression in Fast Transmon Pulses." Mira, what’s the big picture here when you look at that title?

Mira: It sounds very specific to pulse shaping and leakage control in transmon systems. The idea is that by designing the pulse envelope correctly, you can force certain spectral zeros to appear at these half-integer values of ramp duration times anharmonicity, tau alpha = n + one/two <ref:2605.10578#pg1>.

Kai: So it's about using those exact spectral zeros to predict where you get those endpoint leakage minima in fast pulses. It sounds like a way to target the problem without just running huge simulations for every possible drive parameter.

Lev: From my side, the implication is that if we can find this analytical skeleton for the pulse envelope, it might give us a solid starting point instead of just guessing shapes that work on paper.

Kai: Exactly. It’s about moving from a brute-force simulation approach to something more constrained by fundamental physics in how we shape these pulses.

The paper's summary: Mira: The authors explain that this condition, tau alpha = n + one/two arises because of destructive interference between the rising and falling edges of a symmetric flat-top envelope with cosine ramps <ref:2605.10578#pg1>. This is what predicts those endpoint leakage minima when you’re in the weak-drive regime.

Kai: So, they're saying that for this specific pulse family, the zeros are exactly where those leakage points show up in simulations. It connects the shape of the envelope directly to the expected failure points of a gate.

Lev: And they mention that when you look at their simulation results on four-level Duffing models, they see these minima collapsing onto a sequence at six total pulse durations, specifically between fifty and one hundred nanoseconds, with a root mean square deviation of zero point zero three four in units of tau alpha less than one.

Kai: That's the numerical validation part—they’re seeing these predicted minima in their simulations on real-world hardware representative devices, which is pretty solid for this kind of theoretical prediction.

Mira: But they also clarify what this actually predicts: it specifically addresses endpoint leakage and doesn't tackle transient exposure yet. They are focusing only on those endpoint populations, which is an important distinction when we talk about gate fidelity and error correction.

The paper's improvements: Kai: The paper points out a couple of things they think are useful for the next steps, especially in how we characterize these pulse shapes. They introduce an independent ramp timescale tau to remove the degeneracy between the adiabaticity measures and pulse area.

Mira: That’s a big one. They show that within any self-similar one-scale envelope family, where you have (t) = zero f(t/sigma), the non-adiabaticity parameter becomes exactly degenerate with the pulse area, eta ref times theta = C shape <ref:2605.10578#pg2>.

Lev: So by introducing that independent timescale tau, they can set a specific duration for the ramp while keeping the rotation angle constant. It decouples things, which is crucial if you want to design a control strategy where you can scan in one parameter without messing up another.

Kai: That means they are treating a scan in eta as just a scan in pulse area, which makes it much cleaner for designing robust control pulses that don't accidentally introduce unwanted non-adiabaticity from other factors.

Mira: They also show that the resulting half-integer sequence describes the minima for /alpha less than or equal to zero point zero five over the range they tested, but they flag a failure point where at /alpha = zero point two one five, their prediction fails by a factor of five hundred fifty in leakage.

Conclusion: Kai: So, to wrap up, this paper gives us an exact closed-form sequence, tau alpha = n + one/two for the symmetric flat-top envelope with cosine ramps <ref:2605.10578#pg1>. This sequence has no fitted parameters at all, and because the cancellation of leakage is independent of the plateau length, you don't need to worry about that parameter when calculating the total pulse duration.

Mira: The main implication is that this half-integer sequence provides an exact spectral skeleton for this pulse family in the weak-drive picture. It predicts where those endpoint leakage minima should be, and it doesn't rely on fitting any parameters to get there.

Lev: For us building real hardware, the displacement of those dynamical minima from the bare half-integer sequence is governed by the population-weighted pulse geometry, which they call (t) (t). That means we need to consider that population weighting term when we try to target these minima in practice.

Kai: It's a way to get a solid prediction for endpoint leakage without needing full multilevel dynamics simulations for every single parameter set. We get a leading-order prediction from the envelope alone.

Mira: The work also points out that the displacement of those minima is not determined by the drive strength, but by this first-order population weight, which is independent of the overall drive amplitude.

A. M. Tishin

Lomonosov Moscow State University · Moscow Institute of Physics and Technology

cond-mat.mes-hall

Submitted: 2026-05-11

Updated: 2026-08-14

Comments: 17 pages, 5 figures

Journal ref: Next Materials, 13 (2026) 103674

DOI: 10.1016/j.nxmate.2026.103674

Code: https://github.com/alexandermtishin-art/Half-Integer-Spectral-Zeros-for-Leakage-Suppressionin-Fast-Transmon-Pulses

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 89/100

The gist: The gist The exact half-integer spectral-zero condition τα = n + 1/2 for a symmetric flat-top envelope with cosine ramps arises from destructive interference between the contributions of the rising

Key concepts

Spectral Zeros ($ au|\alpha|=n+1/2$): These are specific points where the Fourier amplitude of the pulse envelope exactly vanishes. For a flat-top pulse with cosine ramps, this occurs when the product of ramp duration ($ au$) and anharmonicity (${\alpha}$) equals $n+1/2$, where $n$ is an integer. This exact null condition is a key feature predicted by the pulse shape alone.
Endpoint Leakage: This refers to leakage that occurs specifically at the very end of a fast quantum pulse. The paper shows that the half-integer spectral zeros predict minima in this leakage regime because destructive interference between the rising and falling edges of the pulse envelope cancels out these unwanted leakage contributions.
Independent Ramp Timescale ($ au$): The researchers introduced an independent timescale ($ au$) to remove a degeneracy between two different ways of measuring adiabaticity. In simpler terms, this allowed them to isolate the effect of the ramp speed on pulse quality, ensuring that the spectral zeros observed are due to the pulse shape itself rather than being dependent on other pulse parameters like total duration.
Population Weighting: This is a correction factor used in numerical simulations. While the ideal half-integer sequence is exact, real-world dynamics show small deviations. Population weighting—using the instantaneous excited-state amplitude multiplied by the envelope shape—is shown to capture these residual dynamical minima, reducing positional errors significantly.

Terminology

Calibration — 1× in this paper · explained in 7 episodes
Superconducting Qubits — 1× in this paper · explained in 2 episodes

Summary

The gist The exact half-integer spectral-zero condition τα = n + 1/2 for a symmetric flat-top envelope with cosine ramps arises from destructive interference between the contributions of the rising and falling edges, which predicts endpoint-leakage minima in fast transmon pulses.

Methodology and Key Concepts

The paper investigates how exact spectral zeros predict endpoint-leakage minima in fast transmon pulses using a driven Duffing transmon model truncated to four levels<ref:2605.10578#pg4> The mechanism underlying such nulls has recently been resolved into two distinct control objectives: endpoint leakage and transient leakage exposure<ref:2605.10578#pg4>. Endpoint leakage is governed by the drive spectrum evaluated at the anharmonicity, whereas transient leakage exposure depends instead on spectral weight across a finite band<ref:2605.10578#pg4>.

The analysis relies on establishing an independent ramp timescale to remove degeneracy between dimensionless adiabaticity measures<ref:2605.10578#pg5>. Within any self-similar one-scale envelope family Ω(t) = Ω0f(t/σ), the non-adiabaticity parameter is exactly degenerate with pulse area, ηref·θ = Cshape, so an independent ramp timescale τ is introduced to remove the degeneracy<ref:2605.10578#pg6>. A flat-top envelope with cosine ramps of duration τ at each edge and total duration T has ηref = π(T − τ)/(2τθ), so that at fixed T and fixed θ the non-adiabaticity is set by τ alone while the rotation angle is held exactly constant<ref:2605.10578#pg7>.

Analytical Result of Spectral Zeros

The condition τα = n + 1/2 follows analytically from the envelope alone<ref:2605.10578#pg10>. The Fourier transform of the flat-top envelope is written as F(omega̅) = exp(iomega̅T/2) G(omega̅) with G real, and the zeros of F are the zeros of G<ref:2605.10578#pg10>. Direct integration shows that at τα = (2n + 1)π, the first factor vanishes, A = 0, and the second term of B vanishes with cos omega̅τ = −1, leaving B = 2/omega̅<ref:2605.10578#pg10>. This condition is exactly τα = n + 1/2<ref:2605.10578#pg10>. The surviving terms sin(omega̅a) and sin(omega̅a + omega̅τ) are the contributions of the rising and falling edges, separated in phase by omega̅τ<ref:2605.10578#pg10>. This condition is thus destructive interference between the leakage amplitudes generated at the two edges<ref:2605.10578#pg10>.

Numerical Validation and Weak-Drive Regime

Four-level Duffing simulations with parameters representative of IBM Heron r2 devices collapse the minima onto the sequence at six total pulse durations from 50 to 100 ns with a root mean square deviation of 0.034 in units of τα<ref:2605.10578#pg9>. For four durations resolved on a finer grid, inclusion of the first-order population weighting reduces the positional deviation from 0.0234 to 0.0096<ref:2605.10578#pg9>. The residual displacement of the dynamical minima from the bare half-integer sequence is set by population weighting rather than by drive strength<ref:2605.10578#pg10>. This first-order weight is the product Ω(t)Θ(t), and its shape is independent of the overall drive amplitude<ref:2605.10578#pg10>.

The half-integer sequence describes the minima for Ω/α ≤ 0.05 over the range tested<ref:2605.10578#pg12>. However, at Ω/α = 0.215, the prediction fails by a factor of 550 in leakage, which serves as a negative control on the perturbative interpretation<ref:2605.10578#pg10>.

Experimental Accessibility and Limitations

The predicted operating window is 0.40 ns wide at the ten-fold level, against a control-electronics sample granularity of dt = 1/4.5 GHz = 0.222 ns, so the operating point is addressable in timing terms in principle<ref:2605.10578#pg15>. The measured drive-amplitude calibration biases of −1.84 % and +1.37 % correspond to equivalent fractional shifts in η of about 1.4–1.8 %, at which the computed leakage minimum is perturbed by less than 10 %, so the condition is not fragile with respect to the dominant measured systematic<ref:2605.10578#pg10>. The transfer function of the control chain is not modelled here; it smears sharp ramps and has been identified elsewhere as a dominant error source for gates below 10 ns, so the quoted window is an upper bound on the achievable tolerance<ref:2605.10578#pg15>.

Conclusion

The closed-form sequence τα = n + 1/2 for the symmetric flat-top envelope with cosine ramps is derived in the resonant weak-drive picture, and it provides an exact spectral skeleton for this pulse family<ref:2605.10578#pg16>. The half-integer sequence contains no fitted parameters, and because the cancellation is independent of the plateau length the total pulse duration does not enter<ref:2605.10578#pg14>. The displacement of the dynamical minima is governed by the population-weighted pulse geometry<ref:2605.10578#pg16>.

The numerical scripts that generate every computed result and figure reported here, the resulting data in comma-separated form, the hardware execution records with their job identifiers, and the Supplementary Information are openly available in the accompanying GitHub repository<ref:2605.10578#pg16>. The repository contains the four-level Duffing model and the leakage scan<ref:2605.10578#pg16>. The hardware records are provided as a calibration log; they do not contain a measurement of the leakage minima, which were not measured on hardware<ref:2605.10578#pg16>.

Competing interests The author declares no competing interests<ref:2605.10578#pg17>. Declaration of Generative AI in Scientific Writing The author used AI-based tools (Google Gemini, ChatGPT and Claude) to improve the manuscript's language and to assist with the generation of Python scripts<ref:2605.10578#pg17>. After using these tools the author reviewed and edited the content as needed and takes full responsibility for the content of the publication<ref:2605.10578#pg17>.

References 1 Chen, Z., Kelly, J., Quintana, C., et al. Measuring and suppressing quantum state leakage in a superconducting qubit<ref:2605.10578#pg17>. References 2 Gambetta, J. M., Motzoi, F., Merkel, S. T., and Wilhelm, F. K. Analytic control methods for high-fidelity unitary operations in a weakly nonlinear oscillator<ref:2605.10578#pg17>. References 3 Motzoi, F., Gambetta, J. M., Rebentrost, P., and Wilhelm, F. K. Simple pulses for elimination of leakage in weakly nonlinear qubits<ref:2605.10578#pg17>. References 4 Hyyppä, E., Vepsäläinen, A., Papič, M., et al. Reducing leakage of single-qubit gates for superconducting quantum processors using analytical control pulse envelopes<ref:2605.10578#pg17>. References 5 Yang, H., Liu, F., Wang, W., Fei, Y., and Shan, Z. Separate control of transient leakage exposure and endpoint leakage in fast transmon gates<ref:2605.10578#pg17>. References 6 Türkpençe, D., and Çakmak, S. Anharmonicity-limited gate-time boundaries in transmon single-qubit quantum gates<ref:2605.10578#pg17>. References 7 Schutjens, R., Abu Dagga, F., Egger, D. J., and Wilhelm, F. K.

Improvements for AI systems

textbfTheoretically Constrained Pulse Design for Leakage Suppression: Enabling New Control Paradigms with Exact Spectral Zeros in Transmon Pulses. This allows AI systems to design pulse envelopes that possess exact spectral zeros at halfinteger values of the product of ramp duration and anharmonicity, τα = n + 1/2, providing a leading-order prediction for endpoint-leakage minima without needing full multilevel dynamics simulations for every parameter set.
textbfDynamical Minimum Prediction Based on Population Weighting: Enhancing Gate Optimization by Incorporating First-Order Population Effects. The AI system can predict the displacement of the dynamical minima from the bare half-integer sequence using a first-order population weighting term, reducing positional deviation from 0.0234 to 0.0096 in simulations, allowing for more accurate pulse targeting in weak-drive limits where drive amplitude is reduced.
textbfAdaptive Control Coordinate Selection: Improving Robustness by Separating Pulse Parameters. The AI can utilize the envelope-degeneracy relation ηref·θ = Cshape to establish an independent ramp timescale, allowing the system to treat a scan in η is a scan in pulse area, thus decoupling non-adiabaticity from other pulse parameters for more robust control strategy formulation.