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REVIEW 3 major objections 4 minor 1 cited by

Fluxonium as a control qubit for bosonic quantum information

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The fluxonium can serve as a cavity control qubit that reaches a 1 MHz dispersive shift with vanishing self-Kerr, a regime the paper argues is out of reach for transmons.

desk verdict Solid first fluxonium-ancilla strong-dispersive control experiment, but the headline K≈0 design is a numerical extrapolation that needs independent validation before it carries the paper. read the letter →

arxiv 2505.23641 v2 pith:QTW3VQT7 submitted 2025-05-29 quant-ph cond-mat.mes-hall

Ke Nie , J. Nofear Bradford , Supriya Mandal , Aayam Bista , Wolfgang Pfaff , Angela Kou This is my paper
classification quant-phcond-mat.mes-hall
keywords fluxoniumbosonicquantuminformationdispersivecouplingself-KerrnonlinearitySNAPgateWignertomographysuperconductingresonatorstransmonKerrbound
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

This paper argues that the fluxonium, a superconducting qubit with a long lifetime and a flux-tunable energy structure, can act as the control qubit for a microwave cavity storing quantum information. The aim is to get a large qubit-cavity coupling so operations are fast, without the cavity inheriting the qubit's nonlinearity, which normally distorts stored states. The authors built planar fluxonium-resonator devices, reached the strong-dispersive regime with a shift of about 1 MHz per photon, and showed that the cavity's inherited self-Kerr nonlinearity can be tuned through zero with flux. Using this control they prepared Fock states and superpositions with fidelities limited by the planar cavity's short lifetime. With simulations matched to data, they then predict a fluxonium design with 1 MHz dispersive shift and essentially zero self-Kerr, a combination they argue is forbidden for transmon qubits.

What carries the argument

The central object is the fluxonium-resonator Hamiltonian $\hat{H}/\hbar = 4E_C\hat{n}^2 + \tfrac{1}{2}E_L\hat{\phi}^2 - E_J\cos(\hat{\phi}-\phi_{\mathrm{ext}}) + \tilde{\omega}\hat{a}^\dagger\hat{a} - ig\hat{n}(\hat{a}-\hat{a}^\dagger)$, which in the dispersive regime reduces to the effective cavity-QED Hamiltonian $\hat{H}_{\mathrm{eff}}/\hbar = \Omega|e\rangle\langle e| + \omega\hat{a}^\dagger\hat{a} + \chi\hat{a}^\dagger\hat{a}|e\rangle\langle e| + \tfrac{K}{2}\hat{a}^\dagger\hat{a}^\dagger\hat{a}\hat{a}$. The fluxonium's external flux tunes its level structure, providing an in situ knob that changes $\chi$ and $K$ independently; the work uses this knob to find flux points where $K$ changes sign and crosses zero. Numerical diagonalization of the circuit Hamiltonian, benchmarked against measured qubit and resonator spectra, supplies the predicted designs and the higher-order nonlinearity checks.

What would settle it

Fabricate the numerically optimized fluxonium circuit of Figure 5, bias to the predicted zero-Kerr flux point, and measure the storage-mode self-Kerr by photon-number-dependent Ramsey interferometry; the central claim fails if the measured $|K|/2\pi$ at $\chi/2\pi \approx 1$ MHz is much larger than the roughly 300 Hz resolution of the present experiments, or if the bare-dressed state overlap drops noticeably below one.

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Extended reading notes

Core claim

The paper's central claim is that a fluxonium, unlike a transmon, can be engineered to combine a large dispersive shift with a vanishing self-Kerr nonlinearity. The authors support this with experiments on two planar devices: a storage resonator coupled to a fluxonium with $\chi/2\pi = 1.013$ MHz and $K/2\pi = 3.6$ kHz at half flux, a measured sign change of $K$ with flux bias, and quantitative agreement between measured $\chi$ and $K$ and numerical diagonalization of the full circuit Hamiltonian. They then use the benchmarked model to find fluxonium parameters with $\chi/2\pi = 1$ MHz and $K \approx 0$ while the overlap of bare and dressed states stays near one. This is contrasted with a transmon bound $|K|/2\pi \gtrsim (\chi/2\pi)^2/(2.12\,\mathrm{GHz})$ derived from the formulas $K \approx -E_C(g/\Delta)^4$ and $\chi \approx -2g^2E_C/[\Delta(\Delta-E_C)]$ together with the requirement of low qubit-cavity hybridization and practical $E_C$ limits.

Load-bearing premise

The zero-Kerr design assumes the fitted circuit parameters and flux bias stay fixed while measurements run; two-level-system defects observed in both devices shift qubit parameters by about 10% on timescales of days, which could move the zero-Kerr point and make the canceled nonlinearity reappear.

Editorial extensions

If this is right

  • A fluxonium-resonator system can be used for universal bosonic control with strong dispersive coupling, as demonstrated by preparation and characterization of the states $|1\rangle$ and $(|0\rangle-|1\rangle)/\sqrt{2}$.
  • The flux-tunable zero crossing of the storage-mode self-Kerr provides an in situ operating point where photon-number-dependent dephasing is suppressed, with the experiment placing an upper bound $K/2\pi \approx 300$ Hz at the zero-Kerr flux point.
  • The numerically optimized fluxonium with $\chi/2\pi = 1$ MHz and $K \approx 0$ lies outside the transmon bound $|K|/2\pi \gtrsim (\chi/2\pi)^2/(2.12\,\mathrm{GHz})$ while keeping qubit-cavity hybridization low.
  • Higher-order nonlinearities of the low-$K$ design are negligible in simulation up to about 20 photons, so bosonic encodings with larger photon numbers are not limited by the cavity's inherited nonlinearity.
  • With a cavity lifetime of 1 ms and a fluxonium dephasing time of 50 $\mu$s, simulated SNAP-gate incoherent errors fall to about 1%, showing that the prototype's modest fidelity reflects its short planar cavity lifetime rather than a fundamental limitation.

Reading between the lines

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

  • Editorial: If the zero-Kerr bias point remains stable over time, fluxonium could restore static dispersive coupling as a simple control route for bosonic error correction, removing the need for weak-coupling or parametric schemes used to protect transmon ancillas.
  • Editorial: The transmon bound is derived under a low-hybridization assumption and practical choices of $E_C$, so it is a design boundary rather than a fundamental limit; other multilevel qubits with tunable anharmonicity may also cross it.
  • Editorial: A natural next test is to place the predicted low-$K$ fluxonium in a high-$Q$ 3D cavity and check whether $K$ stays below the roughly 300 Hz resolution of the present measurement at the predicted flux point, and whether the zero-Kerr point drifts with two-level-system fluctuations.
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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

3 major / 4 minor

Summary. The paper reports experiments coupling a fluxonium qubit to a superconducting storage resonator in the strong-dispersive regime, measuring the resulting self-Kerr nonlinearity of the resonator, and demonstrating bosonic control via SNAP gates and Wigner tomography. It also presents a numerical design study arguing that a fluxonium can achieve a large dispersive shift (χ/2π ≈ 1 MHz) with vanishing self-Kerr and low qubit-cavity hybridization, a regime claimed to be inaccessible to transmons because of a derived bound. Two devices with different storage frequencies are characterized as a function of flux, and their χ and K curves are compared with numerical simulations of the circuit Hamiltonian.

Significance. If the predicted K≈0 design is validated, the work would establish fluxonium as a promising control ancilla for bosonic codes, offering tunable linearity beyond transmon capabilities. The experimental core is solid and carefully reported: strong-dispersive coupling with photon-number splitting, cavity Ramsey and Q-function extraction of K, a two-device comparison, and master-equation-matched Wigner tomography with a detailed error budget. The numerical modeling uses open-source packages (scQubits, QuTiP) and the appendices give enough detail to reproduce the simulations. The paper is honest about known limitations, including the short planar-cavity lifetime and TLS-induced drift. However, the central K≈0 claim currently rests on a model whose K predictions have not been validated out-of-sample and on an experimental zero-crossing that is only an upper bound; this gap must be closed before the main claim can be fully accepted.

major comments (3)
  1. [Appendix A4 / Fig. 3] The model validation is partly circular: Appendix A4 states that the circuit parameters (EC, EL, EJ, bare frequencies, coupling strengths) were 'tuned to simultaneously match spectroscopy data and measured χ and K.' Since K is itself a fitting target, the agreement between simulated and measured K curves in Fig. 3(e)-(f) is not an out-of-sample test of the model's predictive power for K. This matters because the K≈0 design point in Fig. 5 is obtained from the same Hamiltonian. Please provide an out-of-sample test (for example, fit to spectroscopy and χ only and then predict K across flux for both devices), or report the covariance matrix of the fitted parameters and propagate it to Fig. 5 to show that the K=0 operating point, and the chosen point 100 MHz away from it, are robust within the fitting uncertainty.
  2. [Appendix D1 / Fig. 14] The experimental evidence for 'vanishing K' is an upper bound, not a null result at the design point. At the zero-crossing flux Φ_ext ≈ 0.456 Φ0, the cavity Ramsey data place only an upper bound K/2π ≲ 300 Hz (Appendix D1), and the large-χ, K≈0 parameter set in Fig. 5 is numerical only. The text should distinguish more sharply between the measured zero-crossing (K consistent with zero at the resolution of this device) and the simulated design point. As written, the abstract and Section VI imply that elimination of cavity self-Kerr is experimentally established; please soften to 'predicted' or provide a direct K measurement at a fluxonium design point with comparable χ.
  3. [Appendix A4 / Appendix B5 / Fig. 5] No uncertainty or sensitivity analysis is given for the Fig. 5 prediction. Appendix A4 explicitly notes that uncertainties for the circuit parameters were not computed, and Appendix B5 documents a TLS that shifts χ by about 10% over days, plus time-dependent T2 fluctuations in device B. Because the K=0 crossing is a sensitive function of the fluxonium level structure, parameter drift or fitting error could move the zero point and reintroduce a sizable K. Please add a sensitivity analysis of K and the dressed-state overlap to variations in EC, EL, EJ, g, and flux, and discuss whether the K≈0 point is stable under the observed TLS-induced drift.
minor comments (4)
  1. [Appendix C4 / Fig. 13 caption] The caption states the example point as Δ/2π = −2.25 MHz, while the main text (Section VI) says the point is chosen with a detuning of 100 MHz from the K=0 case; please correct the unit or the value so the two statements are consistent.
  2. [Fig. 5(c) / Table II] The literature points in Fig. 5(c) mix measured and simulated K values, as indicated in Table II, but the plot symbols do not distinguish these two categories; adding distinct markers would make it clear which points are direct measurements and which are simulations.
  3. [Appendix C3 / Eq. (C12)] The bound in Eq. (C12) is derived from a heuristic constraint set (EJ ≥ 50 EC, qubit frequency below 10 GHz) and should be described consistently as a design-rule estimate rather than a fundamental limit; the main text already uses 'estimated bound' in places, but the abstract's phrasing 'forbidden' is stronger than what the derivation supports.
  4. [Fig. 14(d)] The text notes that the measured resonance is not as sharply peaked as the simulation, but the simulation curve is not shown in the panel; displaying it would make the comparison and the stated extraction caveat easier to assess.

Circularity Check

1 steps flagged · score 4.0 of 10

Fig. 3's χ/K 'prediction' is partly retrospective because K was a fit target; the Fig. 5 zero-K design remains an extrapolation, but independent spectroscopy and Wigner data keep the core result substantive.

  1. fitted input called prediction [Appendix A 4 (Error Analysis); cf. Section IV (Flux dependence of χ and K), Fig. 3 caption]
    "In Table I, the fluxonium circuit parameters, bare resonator frequencies, and their coupling strengths were tuned to simultaneously match spectroscopy data and measured χ and K. ... Using the extracted circuit parameters, we then numerically computed the expected flux dependence of χ and K. The comparison with the measured χ and K is shown in Figure 3(c) and (e). ... in excellent agreement with our numerical predictions."

    Because χ and K were explicitly included as targets of the parameter fit, the simulated χ and K curves in Fig. 3 are not independent out-of-sample predictions; their agreement with the measured values is partly a property of the fit. The paper's wording, 'excellent agreement with our numerical predictions,' overstates the predictive content. Since the Fig. 5 fluxonium design with K ≈ 0 is produced by the same scQubits Hamiltonian model, the experimental support for that design is weakened: the model's 'predictability' for K was benchmarked using K as a fit input. The circularity is partial because the model is also constrained by independent fluxonium and resonator spectroscopy and by independent master-equation agreement with Wigner/SNAP dynamics.

full rationale

The paper's core experimental contributions — strong-dispersive fluxonium-resonator coupling, measurement of inherited Kerr, and SNAP-based bosonic control — are self-contained and do not reduce to their inputs. The main circularity concern is localized to the benchmarking claim for the numerical model. Appendix A4 states that the circuit parameters were 'tuned to simultaneously match spectroscopy data and measured χ and K.' Consequently, the agreement shown for the flux-dependent χ and K curves in Fig. 3 is partly retrospective: K and χ were not withheld from the fit, so the curves are not clean predictions. This weakens the statement that the model's 'predictability' of nonlinearities is established by that comparison. The Fig. 5 prediction that a fluxonium can reach a large χ with vanishing K is a forward numerical optimization over circuit parameters, not a restatement of the fitted K data, so it is not circular by construction. The transmon bound is derived from external formulas in Blais et al. and from published transmon data, not from the authors' own fitted values. No load-bearing self-citation chain or uniqueness argument is present. Given the independent constraints from fluxonium/resonator spectroscopy and from Wigner and SNAP master-equation simulations, I assess the circularity as moderate and localized, not as a wholesale reduction of the central claim to its inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new physical entities are postulated. The main cost is the fitted circuit parameters and the truncation assumptions; the K=0 fluxonium is a parameter design, not a new interaction. The observed TLS fluctuations are environmental, not invented by the paper.

free parameters (4)
  • Device A circuit parameters (EC, EL, EJ, ω̃, g) = EC/2π=1.142 GHz, EL/2π=0.559 GHz, EJ/2π=3.645 GHz, ω̃/2π=5.3575 GHz, g/2π=64 MHz
    Fit to simultaneously match spectroscopy data and measured χ and K (Appendix A4). Uncertainties not reported.
  • Device B circuit parameters = EC/2π=1.101 GHz, EL/2π=0.564 GHz, EJ/2π=3.890 GHz, ω̃/2π=7.3269 GHz, g/2π=106 MHz
    Same fitting procedure as device A, including measured χ and K.
  • Proposed K≈0 fluxonium parameters = EC/2π=1.19 GHz, EL/2π=556 MHz, EJ/2π=3.04 GHz
    Hand-chosen design point from numerical optimization to reach K=0 with χ/2π=1 MHz (Fig. 5).
  • Coupling g at each detuning for Fig. 13 = tuned to keep |χ|/2π = 1 MHz
    Normalization choice in the χ/K comparison; g is re-tuned per detuning, not fixed by a physical constraint.
assumptions (5)
  • domain assumption Circuit Hamiltonian Eq. (1) (fluxonium plus two resonator modes, capacitive coupling) is the correct model of the device.
    Used to extract all parameters and to simulate χ and K; standard fluxonium model.
  • domain assumption Effective dispersive Hamiltonian Eq. (2) truncated to two fluxonium levels and leading self-Kerr K.
    Used for all control simulations; higher-order Kp are checked only for the proposed device (Fig. 13), and device B shows significant higher-order terms above ~10 photons.
  • domain assumption Markovian Lindblad master equation with collapse operators √κ a, √Γ↓ |g><e|, √Γ↑ |e><g|, √(Γφ/2)(|e><e|-|g><g|) captures decoherence.
    Standard open-quantum-system assumption; TLS fluctuations documented in Appendix B5 are not included in the model.
  • domain assumption Transmon parameter bounds EJ ≥ 50EC and qubit frequency below 10 GHz for the χ/K bound.
    State-of-practice bounds used to evaluate the constant 2120 MHz in the transmon K bound (Appendix C3).
  • standard math Hilbert space truncations in numerical diagonalization (charge basis ±15, storage 5 Fock states in Eq. C10 for the hybridization check; larger truncations for spectrum fits).
    Numerical approximations; convergence is not demonstrated in the paper.

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Pith. "Pith review of Fluxonium as a control qubit for bosonic quantum information." pith.science (2026). https://pith.science/paper/QTW3VQT7

@misc{pith2026250523641,
  author       = {Pith},
  title        = {Pith review of: Fluxonium as a control qubit for bosonic quantum information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QTW3VQT7}},
  note         = {Machine review of arXiv:2505.23641}
}
read the original abstract

Bosonic codes in superconducting resonators are a hardware-efficient avenue for quantum error correction and benefit from favorable error hierarchies provided by long-lived cavities compared to typical superconducting qubits. The required coupling to an ancillary control qubit, however, can negate these benefits by inducing highly detrimental effects such as excess decoherence and undesired nonlinearities. An important question is thus whether a cavity-qubit coupling can be realized that offers readout and control capabilities without spoiling the cavity. Here, motivated by its long lifetime and design flexibility of its Hamiltonian, we experimentally investigate the fluxonium as a control qubit for superconducting cavities. We couple a fluxonium qubit to a superconducting resonator in the strong-dispersive regime and use it to measure the coherence and inherited nonlinearities of the resonator. We then demonstrate universal control by preparing and characterizing resonator Fock states and their superpositions, with fidelities limited by resonator decay in our planar prototype device. Finally, we use the predictability of the resonator's inherited nonlinearities to show numerically that the fluxonium can reach cavity-coupling regimes that eliminate undesirable cavity nonlinearities. These results demonstrate the potential of the fluxonium as a high-performance bosonic control qubit for superconducting cavities.

Figures

Figures reproduced from arXiv: 2505.23641 by the authors.

Figure 1
Figure 1. FIG. 1. Fluxonium-resonator system. (a) Schematic com [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Hamiltonian parameters as function of external flux. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Bosonic control using the fluxonium. (a) Pulse sequence for the preparation and characterization of Fock states in the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (12 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Relation of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Measurement and cryogenic setup. The readout tone [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Resonator spectroscopy of the two devices. (a) Readout resonator spectroscopy and (b) Storage resonator spectroscopy [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Half-flux characterization of device B. (a) Pulse se [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Displacement calibration of device A. (a) Pulse se [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Evidence of TLS in Device A. (a) Fluxonium two [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Time-domain measurements of device B over time. [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Higher order Kerr in the fluxonium devices. (a) [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. (a) Detuning extracted from the Ramsey data [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Q function under Kerr evolution. (a) Pulse se [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Photon-number selective power Rabi calibration. [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Wigner function comparison. (a) Target Wigner [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]

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Reference graph

Works this paper leans on

80 extracted references · 65 canonical work pages · cited by 1 Pith paper

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    Device fabrication The devices were fabricated with a Ta baselayer on a sapphire substrate. The Ta was patterned via pho- tolithography and wet etching to form the ground plane, Parameter Device A Device B EC /2π 1.142 GHz 1.101 GHz EL/2π 0.559 GHz 0.564 GHz EJ /2π 3.645 GHz 3.890 GHz NJ J 116 116 ˜ω/2π 5.3575 GHz 7.3269 GHz ˜ωr/2π 6.4615 GHz 6.8435 GHz g...

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    A schematic of the measurement setup and device wiring is shown in Fig

    Cryogenic setup The devices were cooled to 10 mK and measured in an Oxford Instruments Triton 500dilution refrigerator. A schematic of the measurement setup and device wiring is shown in Fig. 6. A global superconducting coil was used for flux biasing, with a Yokogawa GS200 serving as the current source

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    The full system Hamiltonian is given 7 300 K 4 K HEMT 0.1 K qubit control and readout 10 mKmu-metal 1020 20 20 DAC DAC DAC DAC ADC ADC supercon

    Spectroscopic Characterization Here we describe the full system Hamiltonian and present spectroscopic characterization of the readout and storage resonators. The full system Hamiltonian is given 7 300 K 4 K HEMT 0.1 K qubit control and readout 10 mKmu-metal 1020 20 20 DAC DAC DAC DAC ADC ADC supercon. rf coax circulator, Cu body LP filter, X GHz cutoff Ecc...

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    In Table I, the fluxo- nium circuit parameters, bare resonator frequencies, and their coupling strengths were tuned to simultaneously match spectroscopy data and measured χ and K

    Error Analysis Here, we briefly describe the fitting uncertainty asso- ciated with the system parameters. In Table I, the fluxo- nium circuit parameters, bare resonator frequencies, and their coupling strengths were tuned to simultaneously match spectroscopy data and measured χ and K. While we did not compute uncertainties for this procedure, ob- taining ...

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    Qubit initializa- tion is necessary due to the low transition frequency of 8 (a) (b) (c) (d) FIG

    Qubit initialization In this section, we describe the procedure used to pre- pare the qubit in its ground state at half flux and provide details on the initialization efficiency. Qubit initializa- tion is necessary due to the low transition frequency of 8 (a) (b) (c) (d) FIG. 7. Resonator spectroscopy of the two devices. (a) Readout resonator spectroscopy...

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    In the ideal case, the stor- age resonator is measured via the qubit with a constant readout background

    Readout background correction Here, we describe our general procedure used to correct for measurement background. In the ideal case, the stor- age resonator is measured via the qubit with a constant readout background. Resonator cross talk and qubit de- cay after measurement-based cooling, however, cause a skewed background. For the reported experiments w...

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    Depending on whether photon- number splitting is observable in the qubit spectrum, dif- ferent calibration methods were employed

    Device A displacement calibration Here, we describe the calibration procedures used to relate the drive amplitude to the displacement in the stor- age mode for device A. Depending on whether photon- number splitting is observable in the qubit spectrum, dif- ferent calibration methods were employed. At the flux sweet spots, where the fluxonium qubit is fir...

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    Due to a nearby spuri- ous mode, the storage mode of device B couples more strongly to the transmission line than intended, result- ing in a reduced lifetime for the storage mode

    Device B half-flux characterization In this section, we discuss the characterization of de- vice B at half flux, which requires different methods from those used for device A. Due to a nearby spuri- ous mode, the storage mode of device B couples more strongly to the transmission line than intended, result- ing in a reduced lifetime for the storage mode. C...

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    System fluctuations due to TLS During our measurements of both devices, we directly observed signatures of two-level systems (TLSs), includ- ing fluctuations in system parameters attributable to their presence. For Device A at half flux, we observed a splitting in the fluxoniu...

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    (1) using the scQubits [57, 58] Python package

    System Hamiltonian To simulate the system spectrum, we numerically di- agonalized the full Hamiltonian given in Eq. (1) using the scQubits [57, 58] Python package. From the spectrum, we extracted the qubit and resonator frequencies, as well as the dispersive shift χ and the se...

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    Master equation simulation To incorporate decoherence and energy relaxation, we simulated the system dynamics by numerically solving the Lindblad master equation, ∂ ˆρ ∂t = −i[ ˆHeff,rot+ ˆHdrive,rot, ˆρ]+ X k [LkρL† k− 1 2 {L† kLk, ˆρ}], (C5) where ˆρ is the system density op...

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