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Amplitude-Phase Coupling and Average-Hamiltonian Cancellation in Figure-8-Driven Transmon Qubits: Theory and Phase-Resolved Experiments on Two Devices

Kosaku Tabuchi

Published Jul 6, 2026• 0 citations
Publisher

Abstract

We report phase-resolved measurements of amplitude-phase coupling (APC) in strongly driven superconducting transmon qubits, a symmetry-based mechanism analysis, and follow-up experiments on a second device. On a Rigetti Ankaa-3 transmon (1000 shots per point, five drive amplitudes, matched root-mean-square drive), the fitted optimal control phase $\phi^{*}(A)$ drifts by more than 3 rad across the explored amplitude range for both a rectangular reference pulse and a self-intersecting Figure-8 trajectory, and the phase contrast of the two waveform families exhibits a reproducible low-/high-amplitude crossover. Theoretically, we prove that in the resonant rotating frame of the three-level Duffing model a constant-phase rectangular drive cannot produce any phase-dependent population response: $C_{Rect}=0$ exactly. Any measured rectangular phase contrast is therefore a probe of physics outside this model (beyond-rotating-wave carrier-phase dependence, control-chain distortion, readout imperfections). The Figure-8 waveform breaks the symmetry intrinsically through its 2:1 Fourier content; a third-order resonance $\omega+\omega-2\omega=0$ mediated by the anharmonic level $|2\rangle$ predicts $C_{F8}\propto A^{6}N^{2}$ at weak drive, confirmed by direct numerical propagation (fitted log-log slope $5.97\pm0.02$; loop-number ratio 2434 vs. predicted 2500). On the successor device Cepheus-1-108Q, the predicted phase contrast lies below the binomial shot-noise floor at the accessible amplitudes, but the same dataset reveals a stronger signature of the underlying first-order average-Hamiltonian cancellation: the phase-averaged survival probability of the Figure-8 train remains pinned at $\langle P(|0\rangle)\rangle_{\phi}\approx0.985$ across all tested conditions, while the isenergetic rectangular reference sweeps the full range 0.24-0.93—an approximately 70-fold suppression of the Rabi-drive response. Together these results identify APC as a measurable control-landscape feature with distinct intrinsic and extrinsic components, and demonstrate a calibration-free, amplitude-insensitive idling property of Figure-8 pulse trains.

DOI
10.5281/zenodo.21212005

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Symmetry, average-Hamiltonian cancellation, and Rabi-drive suppression in a Figure-8–driven transmon: theory and follow-up experiments on a second device

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We previously reported amplitude–phase coupling (APC) in a strongly driven superconductingqubit on Rigetti Ankaa-3, quantified by the phase-contrast ratio R(A) = CF8(A)/CRect(A)between a self-intersecting Figure-8 trajectory and a rectangular reference pulse. Here wedevelop a symmetry-based theoretical framework and report a complementary follow-upcampaign on a second platform, Rigetti Cepheus-1-108Q, to test the predictions on a differentdevice. The theoretical analysis establishes that, in the rotating frame on resonanceand starting from the ground state, a constant-phase rectangular drive cannot produce anyphase-dependent population response in the three-level Duffing model. The Figure-8 waveformbreaks this symmetry intrinsically through its 2:1 Fourier content, and a third-orderresonance ω+ω−2ω = 0 mediated by anharmonic coupling to |2⟩ generates a phase contrastpredicted to scale as CF8 ∝ A6N2 in the weak-drive regime. Direct numerical propagationof the 50-loop pulse train confirms this scaling with a fitted log-log slope of 5.97 ± 0.02across N ∈ {1, 5, 10, 25, 50} and a loop-number ratio CF8(N=50)/CF8(N=1) ≈ 2434 (predictedN2 = 2500). The Cepheus follow-up dataset (96+72 executables under bare-circuitprotocol matching the Ankaa-3 main paper, 12 phase points, 500 shots/point) does notdirectly resolve the predicted A6 scaling because the predicted CF8 values across the accessibleamplitude range Ahw ∈ {0.01, 0.03, 0.05, 0.08} remain at or below the binomialshot-noise floor σ ∼ 0.022. A separate and stronger qualitative signature is, however, observed:the phase-averaged survival probability ⟨P(|0⟩)⟩ϕ remains pinned near 0.985 for theFigure-8 waveform across all eight (A,N) conditions tested, while the Rectangular referenceproduces strong amplitude-dependent and N-dependent Rabi-like oscillations sweeping⟨P(|0⟩)⟩ϕ across the full range 0.24–0.93. We interpret this as the experimental fingerprintof the average-Hamiltonian cancellation ¯H (1)F8 = 0 that lies upstream of the predicted A6N2scaling, and which is observable on real hardware at much smaller drive strengths than theunderlying A6 contrast itself.

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Journal of the Optical Society of America B•Dec 19, 2025•0 citations
We report phase-resolved measurements of amplitude-phase coupling (APC) in strongly driven superconducting transmon qubits, a symmetry-based mechanism analysis, and follow-up experiments on a second device. On a Rigetti Ankaa-3 transmon (1000 shots per point, five drive amplitudes, matched root-mean-square drive), the fitted optimal control phase ϕ∗(A)\phi^{*}(A)ϕ∗(A) drifts by more than 3 rad across the explored amplitude range for both a rectangular reference pulse and a self-intersecting Figure-8 trajectory, and the phase contrast of the two waveform families exhibits a reproducible low-/high-amplitude crossover. Theoretically, we prove that in the resonant rotating frame of the three-level Duffing model a constant-phase rectangular drive cannot produce any phase-dependent population response: CRect=0C_{Rect}=0CRect​=0 exactly. Any measured rectangular phase contrast is therefore a probe of physics outside this model (beyond-rotating-wave carrier-phase dependence, control-chain distortion, readout imperfections). The Figure-8 waveform breaks the symmetry intrinsically through its 2:1 Fourier content; a third-order resonance ω+ω−2ω=0\omega+\omega-2\omega=0ω+ω−2ω=0 mediated by the anharmonic level ∣2⟩|2\rangle∣2⟩ predicts CF8∝A6N2C_{F8}\propto A^{6}N^{2}CF8​∝A6N2 at weak drive, confirmed by direct numerical propagation (fitted log-log slope 5.97±0.025.97\pm0.025.97±0.02; loop-number ratio 2434 vs. predicted 2500). On the successor device Cepheus-1-108Q, the predicted phase contrast lies below the binomial shot-noise floor at the accessible amplitudes, but the same dataset reveals a stronger signature of the underlying first-order average-Hamiltonian cancellation: the phase-averaged survival probability of the Figure-8 train remains pinned at ⟨P(∣0⟩)⟩ϕ≈0.985\langle P(|0\rangle)\rangle_{\phi}\approx0.985⟨P(∣0⟩)⟩ϕ​≈0.985 across all tested conditions, while the isenergetic rectangular reference sweeps the full range 0.24-0.93—an approximately 70-fold suppression of the Rabi-drive response. Together these results identify APC as a measurable control-landscape feature with distinct intrinsic and extrinsic components, and demonstrate a calibration-free, amplitude-insensitive idling property of Figure-8 pulse trains.
We previously reported amplitude–phase coupling (APC) in a strongly driven superconductingqubit on Rigetti Ankaa-3, quantified by the phase-contrast ratio R(A) = CF8(A)/CRect(A)between a self-intersecting Figure-8 trajectory and a rectangular reference pulse. Here wedevelop a symmetry-based theoretical framework and report a complementary follow-upcampaign on a second platform, Rigetti Cepheus-1-108Q, to test the predictions on a differentdevice. The theoretical analysis establishes that, in the rotating frame on resonanceand starting from the ground state, a constant-phase rectangular drive cannot produce anyphase-dependent population response in the three-level Duffing model. The Figure-8 waveformbreaks this symmetry intrinsically through its 2:1 Fourier content, and a third-orderresonance ω+ω−2ω = 0 mediated by anharmonic coupling to |2⟩ generates a phase contrastpredicted to scale as CF8 ∝ A6N2 in the weak-drive regime. Direct numerical propagationof the 50-loop pulse train confirms this scaling with a fitted log-log slope of 5.97 ± 0.02across N ∈ {1, 5, 10, 25, 50} and a loop-number ratio CF8(N=50)/CF8(N=1) ≈ 2434 (predictedN2 = 2500). The Cepheus follow-up dataset (96+72 executables under bare-circuitprotocol matching the Ankaa-3 main paper, 12 phase points, 500 shots/point) does notdirectly resolve the predicted A6 scaling because the predicted CF8 values across the accessibleamplitude range Ahw ∈ {0.01, 0.03, 0.05, 0.08} remain at or below the binomialshot-noise floor σ ∼ 0.022. A separate and stronger qualitative signature is, however, observed:the phase-averaged survival probability ⟨P(|0⟩)⟩ϕ remains pinned near 0.985 for theFigure-8 waveform across all eight (A,N) conditions tested, while the Rectangular referenceproduces strong amplitude-dependent and N-dependent Rabi-like oscillations sweeping⟨P(|0⟩)⟩ϕ across the full range 0.24–0.93. We interpret this as the experimental fingerprintof the average-Hamiltonian cancellation ¯H (1)F8 = 0 that lies upstream of the predicted A6N2scaling, and which is observable on real hardware at much smaller drive strengths than theunderlying A6 contrast itself.
We report phase-resolved measurements revealingamplitude–phase coupling (APC) in a strongly drivensuperconducting qubit. Using 1000 shots per point acrossfive drive amplitudes on a Rigetti Ankaa-3 transmonqubit (Qubit 58; T1 = 36.7 μs, T2 = 22.2 μs), we comparea rectangular reference pulse and a self-intersectingFigure-8 trajectory under matched root-mean-squaredrive amplitude. From harmonic fits to the phasedependentsurvival probability, we extract the optimalcontrol phase ϕ∗(A) and show that it shifts by more than3 rad across the explored amplitude range for both pulsefamilies. To quantify phase sensitivity, we define thephase contrast C(A) = maxϕ ˆ F −minϕ ˆ F and the contrastratio R(A) = CF8/CRect. At low amplitude (A ≤ 0.6),the Figure-8 trajectory reduces phase contrast relativeto the rectangular reference (R = 0.48–0.57), whereas athigh amplitude (A ≥ 0.8) it increases it (R = 1.22–1.45),revealing a crossover in trajectory-dependent phase sensitivity.An independent replication acquired in a separatecalibration session at 100 shots per point preserves thesame low- and high-amplitude sign structure. Additionalpulse-train continuity tests show that internal timinggaps reduce phase contrast without measurably shiftingthe optimal phase, and that a fixed 4 ns gap showsno statistically significant dependence on its placementwithin the train. These observations identify APC as anexperimentally accessible feature of the measured controllandscape and show that fixed-phase calibration canbecome systematically biased across drive amplitudes inthe strongly driven regime.
Note (September 2026): This preprint is superseded by a new manuscript and dataset (doi:10.5281/zenodo.22803589); several of its claims are withdrawn there. [Update note, April 2026]A more comprehensive version of this work, with substantially increasedstatistics (12 phase points, 1000 shots/point, additional supplementaryexperiments including isenergetic and gap-insertion tests), has beenposted: "Phase-Resolved Observation of Amplitude-Phase Coupling in aStrongly Driven Superconducting Qubit" (DOI: [10.5281/zenodo.19816115]). The Berryconnection interpretation discussed in the present record has beenrevised in the subsequent work; readers interested in the mechanismanalysis should consult the accompanying mechanism paper(DOI: [10.5281/zenodo.19816695]).[End of update note] We report a differential measurement comparing the phase response of a superconducting transmonqubit under two control trajectories: a closed-loop Figure-8 pulse (non-zero area in the IQcontrol plane) and a constant-amplitude Rectangular pulse (zero area). Measurements were performedat five drive amplitudes on a Rigetti Ankaa-3 processor (qubit #25) via AWS Braket. Threetrajectory-dependent features are observed that are not trivially explained by Kerr nonlinearityalone: (1) progressive divergence of optimal control phase between trajectories, (2) trajectorydependentmodulation visibility with a crossover between low and high amplitudes, and (3) apparentsuppression of phase modulation for the Figure-8 trajectory at amplitude A = 0.8 whilethe Rectangular trajectory retains clear sinusoidal modulation. Observations 1 and 2 are robustand statistically clear. Observation 3 is suggestive but requires further statistical validation withincreased shot counts. These results are interpreted in the context of amplitude-phase coupling(APC) with measured coupling strength κ ≈ 0.9 rad/a u , previously reported in Ref. [1]. We proposefour follow-up experiments—including an isenergetic control test—to discriminate geometriccontributions from dynamic nonlinearities.
We present an experimental demonstration of a software-defined noise suppression protocolon a commercial superconducting quantum processor (Rigetti Ankaa-3 via AWS Braket).The protocol combines three layers: (1) Figure-8 control pulses whose 2:1 quadrature frequencyratio gives each loop a vanishing first Fourier moment, (2) node-synchronized readoutthat gates measurements at loop-closure times, and (3) adaptive phase tracking that compensatesfor environmental drift. Key experimental results include: a break-even thresholdexceeded with a +10 percentage-point improvement over baseline idle; phase-dependent suppressionyielding +82 percentage-point variation through optimal phase selection; and theexperimental observation of amplitude–phase coupling (APC), in which the optimal controlphase shifts by ≈ 0.9 rad per unit amplitude change. We report APC here as an operationalfeature of the measured control landscape; its microscopic interpretation in termsof average-Hamiltonian cancellation in a three-level Duffing model has been developed insubsequent work [2, 3]. The adaptive layer recovers fidelity from 13% to 85% autonomouslyafter environmental drift. These results demonstrate that the proposed protocol providesrobust, hardware-agnostic noise suppression without device modification.
Achieving fast gates and high-fidelity readout in superconducting quantum circuits often benefits from the use of large amplitude microwave drives. As the strength increases, unwanted effects such as excitation out of the logical states also become more prevalent. In transmon qubits, coherent controls relying on sub-resonant drives are fundamentally limited by the eventual appearance of strong hybridization in the spectrum. Understanding how this threshold depends on intrinsic device parameters is vital for the design and operation of circuits utilizing the transmon as a source of nonlinearity. In this work, we present an experimental study of 14 transmons with a wide range of anharmonicities, and use comparison with Floquet branch analysis and semiclassical simulations to establish a straightforward relationship between the critical value of the effective drive amplitude ηmax⁡η_{\max}ηmax​ and the ratio γ=ωq/αγ=ω_q/αγ=ωq​/α, where ωqω_qωq​ is the transmon frequency and −α-α−α is the anharmonicity. For the case where the transmon is used as a parametrically driven coupler, these results further allow us to estimate maximum parametric interaction rates, which informs the circuit design process for optimizing gate rates and fidelities.
Logic gates in superconducting quantum processors are implemented through precisely shaped microwave pulses, whose durations are constrained by both device coherence and the multilevel structure of artificial atoms. While single-qubit fidelities above 0.999 are routinely achieved, the physical execution time of these gates remains a major bottleneck for deep quantum circuits. Here we examine single-qubit control across adiabatic [where the rotating-wave approximation (RWA) is valid], intermediate, and diabatic driving regimes using both an ideal two-level model and a realistic weakly anharmonic transmon Hamiltonian. By sweeping pulse durations and drive strengths over experimentally relevant ranges, we compare performance trade-offs and identify the conditions under which ultrashort gates remain viable. Our simulations uncover a quantitative speed-limit mechanism intrinsic to weakly anharmonic superconducting qubits: leakage to higher levels rises sharply once the peak Rabi rate exceeds a fixed fraction of the anharmonicity. Across all pulse shapes and driving regimes studied, we find a consistent numerical threshold at Ω max /|Δ|≈0.15−0.20, where Ω max denotes the peak drive (Rabi) amplitude and Δ is the transmon anharmonicity. Beyond this ratio, gate fidelity deteriorates regardless of waveform engineering. This boundary, obtained from full multilevel time-domain simulations spanning adiabatic, intermediate, and diabatic regimes, constitutes a previously unreported characterization of the fundamental leakage barrier in transmon devices. Our results, therefore, provide the first unified quantitative map linking pulse duration, driving strength, and anharmonicity, establishing why realistic superconducting qubits cannot sustain picosecond-scale gates despite ideal two-level predictions, and offering a practical benchmark for next-generation architectures seeking shorter, high-fidelity control.