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Sensitivity-Adapted Closed-Loop Optimization for High-Fidelity Controlled-Z Gates in Superconducting Qubits

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A sensitivity-adaptive closed-loop optimizer, combined with signal pre-distortion, pushes controlled-Z gate error to 0.09(10)% in a 64 ns pulse on fixed-frequency superconducting qubits, reaching average gate fidelities above 99.9%.

desk verdict A solid experimental calibration study with a real engineering payoff, but the 99.9% fidelity headline overreaches its own error bar and the unmeasured coupler leakage keeps that claim from being established. read the letter →

arxiv 2412.17454 v1 pith:2ZDDL4OK submitted 2024-12-23 quant-ph

Niklas J. Glaser , Federico A. Roy , Ivan Tsitsilin , Leon Koch , Niklas Bruckmoser , Johannes Schirk , João H. Romeiro , Gerhard B. P. Huber
show 8 more authors
Florian Wallner Malay Singh Gleb Krylov Achim Marx Lasse Södergren Christian M. F. Schneider Max Werninghaus Stefan Filipp
This is my paper · ORCID
classification quant-ph PACS 03.67.Lx85.25.-j
keywords superconductingqubitscontrolled-Zgatetunablecouplerclosed-loopoptimizationrandomizedbenchmarkingpulseshapingflux-linedistortioncorrectionCMA-ES
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Long, fast two-qubit gates are the usual fidelity bottleneck in superconducting quantum processors, and this paper claims a practical route past it: a closed-loop optimizer that tunes the coupler-flux pulse directly on the device, using a randomized-benchmarking cost function whose sensitivity is continuously adapted as the candidates improve. On a pair of fixed-frequency superconducting qubits (transmons) coupled through a flux-tunable coupler, the method calibrates a 64 ns Fourier-series controlled-Z pulse — the standard entangling gate that gives the $|11\rangle$ state an extra $\pi$ phase — defined by only a handful of parameters, to a gate error of 0.09(10)%, an average fidelity above 99.9%. The second ingredient is systematic correction of the flux line: inspecting the shapes the optimizer converged to revealed microsecond-scale low-pass distortions, and compensating them with on-device pre-distortion dropped the error of the same pulse family from about 0.6% to 0.09%. Three pulse parametrizations (Gaussian-square, Fourier-series, and piecewise-constant-slope) are compared, and the pipeline is offered as a general tune-up and recalibration method for superconducting processors.

What carries the argument

The central object is the flux pulse together with the feedback loop that shapes it. The gate itself is produced by the conditional frequency shift $\xi$, tuned by threading external flux $\Phi_{\mathrm{ext}}$ through the coupler's SQUID loop so that the $|11\rangle$ state accumulates a $\pi$ phase; how fast the pulse traverses the avoided crossing at $\Phi^{\star}$ decides how much population leaks to the $|1,01\rangle$ and $|2,00\rangle$ states and whether Landau-Zener-Stückelberg interference returns it. The feedback loop is a CMA-ES optimizer (covariance-matrix-adaptation evolution strategy) minimizing an ORBIT cost function built from randomized-benchmarking sequences, with the sensitivity $S = dE/d\epsilon$ kept near its maximum by increasing the number of Clifford gates $N$ whenever the mean cost drops below 0.2. Pulse shapes are parametrized as Gaussian-square, Fourier-series, or piecewise-constant-slope (PiCoS) forms; the Fourier envelope is $\Phi(t) = A \sum_{n=1}^{N} \lambda_n \left(1 - \cos\left(2\pi n (t - t_p/2)/\tau_w\right)\right)$ with few coefficients. The last ingredient is distortion correction: an extended cryoscope protocol (a pulse-based measurement of the flux-line step response) measures the coupler's response through the fixed-frequency qubit, and four exponential IIR filters plus a 72-tap FIR filter running in real time on the arbitrary waveform generator compress the flux-line response time from 2.5 µs to 10 ns.

What would settle it

Measure the coupler-state population after the optimized 64 ns Fourier pulse: a device whose tunable coupler is coupled to its own readout resonator could reveal, in a Ramsey or population-measurement sequence, whether a non-negligible fraction of the population sits in the $|1,01\rangle$ or $|2,00\rangle$ states at the end of the pulse. If a substantial fraction is found and does not coherently return to the qubit subspace, the reported 0.09(10)% underestimates the true gate error; if the coupler is found empty, the interleaved-randomized-benchmarking number is a faithful measure of gate fidelity. A leakage-sensitive variant of randomized benchmarking that tracks these states would serve the same purpose without new hardware.

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Extended reading notes

Core claim

The paper's central claim is that closed-loop optimization with an adaptive cost function, combined with signal pre-distortion, reaches average controlled-Z gate fidelities above 99.9%: reoptimized 64 ns Fourier-series pulses achieve $\epsilon_{\mathrm{Fourier,C}} = 0.09(10)\%$ gate error. The gate is implemented by adiabatically sweeping a tunable coupler's frequency so that the conditional frequency shift $\xi$ — the extra phase accumulated by the $|11\rangle$ state relative to the other computational states — integrates to $\pi$, while Landau-Zener-Stückelberg interference recovers population that leaks through avoided crossings during the outbound and return passages. The paper compares three pulse parametrizations: a Gaussian-square pulse reaching about 4.79% error; a Fourier-series pulse at 0.62(15)% error before correction and 0.09(10)% after pre-distortion; and a piecewise-constant-slope (PiCoS) pulse at 0.25(9)% before correction and 0.21(9)% at a shorter 20 ns pulse plus a 12 ns buffer after correction. The authors interpret the converged pulse shapes as encoding the hardware's systematic errors, in particular low-pass filtering of the flux line with time constants up to 2.5 µs, and show that correcting this transfer function in real time on the waveform generator is what unlocks the sub-0.1% regime. The same closed-loop scheme is proposed for tune-up and recalibration of superconducting quantum processors.

Load-bearing premise

The central number assumes that interleaved randomized benchmarking reports the true gate error even though the experiment never directly checks for population leaking into the coupler: if population leaves the two qubits during the 64 ns pulse and returns only during the measurement, the reported error could be smaller than the actual gate error, a limitation the paper states explicitly in Section IV.

Editorial extensions

If this is right

  • If the 0.09(10)% error holds, the CZ gate ceases to be the fidelity bottleneck on this architecture: a 64 ns two-qubit gate at 99.9% average fidelity on fixed-frequency transmons with a tunable coupler is practical, and the same closed-loop recipe can be applied to every qubit pair on a chip.
  • The adaptive sensitivity rule removes the need to preset the benchmarking depth: the optimizer starts at short sequences suitable for poor candidates and lengthens them as candidates improve, so one run carries the gate from roughly 50% error to the 0.1% regime without manual retuning.
  • Recalibration after parameter drift can reuse the converged covariance matrices, and the identified distortion model shortens later optimizations (from 12 h to 4.5 h in the demonstrated case), supporting the paper's proposal of interleaving optimization with algorithmic tasks to track drift in real time.
  • The parametrization comparison indicates that control complexity and hardware correction are partly interchangeable: before pre-distortion the 21-parameter PiCoS pulse beats the simpler Fourier pulse, while after pre-distortion the simpler Fourier pulse reaches the lowest error, suggesting that adding correction hardware can buy simpler pulses.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The headline number's own uncertainty is large: at 0.09(10)%, the measured error is within one standard deviation of zero, so the claim of 'above 99.9%' would be sharpened by a longer interleaved randomized benchmarking run or by an independent fidelity estimate that does not rely on the same leakage-blind decay model.
  • Because the paper's stated limitation is that coupler states cannot be read out, a decisive follow-up is to run the identical optimization on a device where the coupler has its own dispersive readout; if population is found to linger in the coupler after the 64 ns pulse, both the 0.09% number and the leakage-free interpretation of the converged pulse would need revision.
  • The converged pulse shapes are described as encoding the flux-line transfer function, so one could formalize the pipeline to output, in a single run, both the optimal pulse and an empirical distortion model usable for open-loop design on future chips — turning what is here a two-step procedure (optimize, then characterize) into a single self-characterizing calibration.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper reports closed-loop, sensitivity-adaptive CMA-ES optimization of controlled-Z gates on fixed-frequency transmons coupled through a tunable coupler. Three pulse parametrizations (Gaussian-square, Fourier-series, and PiCoS) are compared using ORBIT/RB-derived costs, and the authors identify flux-line distortions with an extended cryoscope protocol, apply IIR/FIR pre-distortion, and report a final interleaved randomized benchmarking error of 0.09(10)% for a 64 ns Fourier-series pulse, summarized as "average gate fidelities above 99.9%". The appendices provide device parameters, leakage and ZZ-phase characterization, optimization details and runtime analysis, and a description of the distortion-correction filter model.

Significance. If the headline error is unbiased, this is a useful practical demonstration: a 64 ns CZ gate with roughly 0.1% error on fixed-frequency transmons with a tunable coupler, obtained by combining black-box closed-loop optimization with independent line-distortion characterization. The strengths of the paper are the explicit sensitivity-adaptive cost schedule, the systematic comparison across parametrizations, and the cryoscope-based IIR/FIR correction with an independently measured step response. The main caveat is that the fidelity metric is qubit-only interleaved randomized benchmarking, with an unmeasured coupler-leakage channel, so the "above 99.9%" claim is not yet fully established.

major comments (2)
  1. [Section IV, final paragraph; Fig. 4(c)] The headline claim rests on interleaved randomized benchmarking, which records only qubit states. The manuscript explicitly states that the coupler states are not directly measurable on the current device and that numerical simulations identify coupler states as the dominant leakage channel, and the earlier Fourier-series pulse already shows leakage signatures in IRB. For the final Fourier,C pulse, only "minimal indications of leakage" are reported, without a quantitative bound or a leakage-robust analysis. Since leaked coupler population that returns to |0> at readout is counted as a successful RB outcome, an unquantified leakage of order 0.1% would directly move the reported 0.09(10)% error across the 99.9% boundary. Please provide either a leakage-robust RB estimate, a numerical master-equation bound on coupler population during the 64 ns pulse, or an explicitly weakened claim that does not assert fidelity above 99.9%.
  2. [Section V, first and last paragraphs; Section IV, Fig. 4(c)] The statement "average gate fidelities above 99.9%" is not supported by the reported IRB error epsilon_Fourier,C = 0.09(10)%. With the stated uncertainty, the interval extends to about 0.19% error (99.81% fidelity) or beyond, so the data do not demonstrate that the fidelity is above 99.9%. Please report the uncertainty model used for the IRB estimates and either soften the claim to "about 99.9%" or provide a confidence-level statement for exceeding the 99.9% threshold.
minor comments (4)
  1. [Table II] The row labels in Table II appear transposed relative to the text: the row with np=8 and 0.52(15)% should be the Fourier-series pulse, the row with np=21 and 0.25(9)% should be the PiCoS pulse, the row with np=8 and 0.09(10)% should be Fourier,C, and the row with np=21 and 0.21(9)% should be PiCoS,C.
  2. [Section III and Section IV] The parameter count for the Fourier-series decomposition is inconsistent: Section III states eight parameters, Section IV says "five Fourier-components", and Table II (as printed) lists np=21 for the Fourier row; please state the count consistently in all three places.
  3. [Fig. 3 caption] The caption contains the duplicated phrase "as shown is shown in Fig. 3(c)", and the shading change used to indicate sensitivity updates is not explained in the caption.
  4. [Section IV, paragraph on PiCoS pulse] The sentence reporting the PiCoS 20 ns pulse with a 12 ns buffer would benefit from explicitly stating whether the 0.21(9)% error refers to the total gate duration or only the active pulse width, since the table lists tp=20 ns and tb=12 ns separately.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the optimization and benchmarking share an RB-family metric, but the pulse shapes are experimental calibration results rather than predictions, and the distortion correction rests on an independent cryoscope measurement.

full rationale

The paper's central claims are experimental: closed-loop CMA-ES optimization of parametrized flux pulses and interleaved randomized benchmarking of the resulting gates. No equation in the paper reduces a purported prediction to a fitted input. The cost used during optimization is an ORBIT/RB sequence fidelity, and the final reported error is also obtained from interleaved RB; this shared metric family is standard practice for closed-loop tune-up followed by benchmarking, not a circular derivation, because the optimizer genuinely searches over pulse parameters and the final error is a measured characterization of the implemented gate. The sensitivity-adaptive schedule is a heuristic for choosing the RB sequence length and is not used as evidence for the final fidelity. The flux-line distortion correction is supported by an independent time-domain cryoscope measurement of the step response, with fitted IIR/FIR parameters that are not derived from the final gate error. The main weakness identified in the manuscript is the lack of direct coupler-state readout, so leakage into the coupler is not experimentally verified; that is a validity or correctness risk for the 99.9% claim, but it is not circularity. Self-citations appear in the method lineage, notably for ORBIT-based cost functions and leakage signatures, but the ORBIT protocol originates in the external work of Kelly et al. and the leakage-detection interpretation is also supported by external references, so the self-citations are not load-bearing in a way that forces the conclusion. Overall, the derivation chain is self-contained experimental calibration and characterization, with no fitted parameter renamed as a prediction and no definitional circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the fitted pulse parameters, the IIR/FIR distortion model, and the RB fidelity model. No new physical entities are introduced.

free parameters (3)
  • IIR filter amplitudes A_i and time constants τ_i = A = [-0.021, -0.012, -0.393, +0.595], τ = [846, 151, 36.0, 21.6] ns
    Fitted to the measured coupler flux step response and used for pre-distortion; listed in Table III.
  • Optimized pulse parameters (node amplitudes, Fourier coefficients, width, amplitude, Z-rotation phases) = Not tabulated; shown in Fig. 4
    The central result is obtained by fitting these parameters to minimize the RB cost; they are the object of the optimization.
  • Cost-function hyperparameters: sensitivity threshold 0.2, Clifford count N, population P=4+4log(Np), M=80, shots=128 = N=2 to 12; P=13 for 21 parameters; threshold 0.2
    Chosen by hand to balance sensitivity and runtime; the adaptive schedule relies on these.
assumptions (5)
  • domain assumption Hamiltonian Eq. (1) with 3 qubit levels and 4 coupler levels accurately describes the device
    Used for numerical diagonalization and leakage simulations; parameters from Table I.
  • domain assumption Randomized benchmarking fidelity follows F(N)=A F_C^N + B with F_C = F_SQ^8.5 * F_CZ^1.5 and uncorrelated errors
    This model underlies the ORBIT cost function and the sensitivity schedule in Appendix C1.
  • domain assumption Adiabatic and Landau-Zener-Stückelberg models correctly predict leakage recovery
    Motivates pulse shaping in Section II; leakage channels into the coupler are not directly verified.
  • domain assumption The measured qubit-frequency response to flux pulses via Q1 accurately represents the coupler's flux response
    Extended cryoscope in Appendix D infers coupler response through qubit Q1; calibration relies on this mapping.
  • ad hoc to paper IIR/FIR filter model (Eqs. D1, D2) captures all relevant flux-line distortion
    The correction is fitted to the observed step response; residual uncorrected distortions are discussed but assumed small after correction.

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Cite this review

Pith. "Pith review of Sensitivity-Adapted Closed-Loop Optimization for High-Fidelity Controlled-Z Gates in Superconducting Qubits." pith.science (2026). https://pith.science/paper/2ZDDL4OK

@misc{pith2026241217454,
  author       = {Pith},
  title        = {Pith review of: Sensitivity-Adapted Closed-Loop Optimization for High-Fidelity Controlled-Z Gates in Superconducting Qubits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2ZDDL4OK}},
  note         = {Machine review of arXiv:2412.17454}
}
read the original abstract

Achieving fast and high-fidelity qubit operations is crucial for unlocking the potential of quantum computers. In particular, reaching low gate errors in two-qubit gates has been a long-standing challenge in the field of superconducting qubits due to their typically long duration relative to coherence times. To realize fast gates, we utilize the hybridization between fixed-frequency superconducting qubits with a strongly interacting coupler mode that is tunable in frequency. To reduce population leakage during required adiabatic passages through avoided level crossings, we employ a sensitivity-adaptive closed-loop optimization method to design complex pulse shapes. We compare the performance of Gaussian-square, Fourier-series, and piecewise-constant-slope (PiCoS) pulse parametrizations and are able to reach 99.9 % controlled-Z gate fidelity using a 64 ns long Fourier-series pulse defined by only seven parameters. These high-fidelity values are achieved by analyzing the optimized pulse shapes to identify and systematically mitigate signal-line distortions in the experiment. To improve the convergence speed of the optimization we implement an adaptive cost function, which continuously maximizes the sensitivity. The demonstrated method can be used for tune-up and recalibration of superconducting quantum processors.

Figures

Figures reproduced from arXiv: 2412.17454 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Lumped-element circuit diagram of the system [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Population leakage and retrieval. Simulated popu [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Exemplary optimization curves for a PiCoS pulse [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Optimized pulse-shape parametrizations and interleaved randomized benchmarking results comparing (a, c) the Fourier [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Experimental setup and schematic of the wiring for [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Leakage induced by the CPHASE gate pulse measured from the (a) [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Sensitivity [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Illustration of the execution flow of the quantum cir [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Breakdown of the time required per evolution dur [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Measurement of the coupler flux response. (a) [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Relative increase in gate errors when using [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

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Forward citations

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