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High-power readout of a transmon qubit using a nonlinear coupling

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A purely nonlinear coupling keeps transmon readout faithful and QND at hundreds of photons.

desk verdict Solid experimental demonstration of high-power QND readout with nonlinear cosφ coupling; the 300-photon plateau is credible qualitatively, but the photon-number axis is calibrated only to 200 photons, so the highest-power numbers should be taken with a grain of salt. read the letter →

arxiv 2507.03642 v2 pith:HQOMONOJ submitted 2025-07-04 quant-ph cond-mat.mes-hall

Cyril Mori , Vladimir Milchakov , Francesca D'Esposito , Lucas Ruela , Shelender Kumar , Vishnu Narayanan Suresh , Waël Ardati , Dorian Nicolas
show 8 more authors
Giulio Cappelli Arpit Ranadive Gwenael Le Gal Martina Esposito Quentin Ficheux Nicolas Roch Tomás Ramos Olivier Buisson
This is my paper · ORCID
classification quant-phcond-mat.mes-hall PACS 85.25.Cp03.67.Lx
keywords transmonmoleculecosφ-couplingQNDreadouthigh-powercross-KerrcouplingsuperconductingqubitscriticalphotonnumberPurcellprotection
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

The paper claims that qubit readout through a purely nonlinear $\cos\varphi$ coupling, implemented by a transmon molecule, keeps high fidelity and quantum non-demolition (QND) behavior at readout powers where the standard dispersive scheme degrades. In this sample, single-shot fidelity reaches 99.21% at roughly 89 photons in the readout mode, and the QNDness stays at 96.7% at that power and above 96% up to about 300 photons. The authors explain the stability by deriving a new critical photon number, $\bar{n}_r^\mathrm{crit} = 377$, set by the low phase-drop approximation used to reduce the $\cos\varphi$ coupling to a cross-Kerr term, rather than by the perturbative detuning criterion of transverse coupling. If correct, the scheme lifts the standard power ceiling on readout signal-to-noise ratio and does so without Purcell filtering, since the qubit is decoupled from the cavity by symmetry.

What carries the argument

The load-bearing element is the purely nonlinear coupling term $-2E_J[\cos(\hat{\varphi}_q)-1][\cos(\hat{\varphi}_a)-1]$, which comes from galvanic coupling between two transmon modes and contains no linear term. After the low phase-drop approximation and rotating-wave approximation it yields the nonperturbative cross-Kerr coupling, and every transition it induces conserves parity in both transmon excitations and readout photons. A third resonator, transversely coupled to the ancilla and decoupled from the qubit by symmetry, hybridizes into polariton readout modes; choosing the more linear polariton gives the reduced dispersive Hamiltonian and protects the qubit from Purcell decay. The paper's new high-power scale is the critical photon number $\bar{n}_r^{\mathrm{crit}} = \frac{1}{2\sin^2\theta}\sqrt{\frac{E_J(1+2L_J/L_a)}{E_{Ca}}} = 377$ photons, marking where the fourth-order Taylor expansion of $\cos(\hat{\varphi}_a)$ stops being valid.

What would settle it

Compare an independent photon-number calibration, such as transmitted-power measurement or sideband thermometry, against the Stark-shift scale between 200 and 400 photons; a disagreement would rescale $\bar{n}_r^{\mathrm{crit}}$ and the QNDness-versus-power map, while a match would confirm the 377-photon limit.

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Extended reading notes

Core claim

The central claim is that the transmon molecule's $\cos\varphi$-coupling produces a nonperturbative cross-Kerr interaction of the form $2\chi_{qr}\,\hat{q}^\dagger\hat{q}\hat{c}_r^\dagger\hat{c}_r$, so the readout Hamiltonian looks like dispersive readout but without any large-detuning approximation. Consequently, the standard critical photon number $\bar{n}_\mathrm{std}^\mathrm{crit} = \Delta^2/(2g)^2$ does not limit this scheme; the limiting assumptions are instead the rotating-wave approximation, linearization of the readout mode, and the low phase-drop Taylor expansion. For this sample the last gives $\bar{n}_r^{\mathrm{crit}} = 377$ photons, which coincides with the abrupt rise in QNDness errors visible in the measured power map. The device achieves $T_1 = 124.5\,\mu\mathrm{s}$ without any Purcell filter, whereas the equivalent transverse-coupling readout would have $T_1 \approx 800\,\mathrm{ns}$ and a critical photon number of only 26.

Load-bearing premise

The photon-number scale assumes the AC-Stark shift remains linear in readout power above the 200-photon range where it was experimentally verified, so the 300-photon and 377-photon numbers rest on an extrapolation.

Editorial extensions

If this is right

  • Readout power can be pushed to roughly 100 photons while keeping fidelity above 99% and QNDness near 97%, so signal-to-noise ratio is no longer limited by dispersive-shift collapse.
  • A transmon molecule can deliver high-fidelity readout without a Purcell filter, with $T_1 = 124.5\,\mu\mathrm{s}$ compared with about $800\,\mathrm{ns}$ for an equivalent transverse-coupling readout.
  • QNDness errors stay below 4% up to about 300 photons, supporting repeated-measurement protocols, pre-selection, and active feedback at much higher power than standard readout allows.
  • The derived 377-photon limit gives a concrete design target: increasing it by tuning the hybridization angle $\theta$, the ancilla charging energy, or the Josephson energy would push the scheme further into the high-power regime.

Reading between the lines

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

  • If the AC-Stark photon-number calibration is independently verified above 200 photons, the 377-photon critical number becomes a quantitative design rule; if that calibration drifts, the quoted power scale shifts but the qualitative robustness of the readout likely remains.
  • The parity-conservation property of the $\cos\varphi$ coupling suggests that measurement-induced state transitions are strongly suppressed compared with transverse readout; a systematic census of transitions above 300 photons would test this directly.
  • Because the qubit is Purcell-protected by symmetry, the scheme could plausibly be extended to multiplexed multi-qubit readout where several high-power measurements coexist on one chip, though this would require a dedicated architecture study.
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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 / 5 minor

Summary. The paper reports an experimental study of a transmon-molecule readout scheme in which the qubit is coupled to a readout mode through a purely nonlinear cos(φ) coupling rather than through the usual transverse linear coupling. The authors report a post-selected readout fidelity of 99.21 ± 0.01% at an average readout photon number of 89, a two-pulse QNDness of 96.7 ± 0.2% at the same power, and QNDness errors below 4% for powers corresponding to up to 300 photons. They also derive a critical photon number associated with the low phase-drop approximation, obtaining nbar_r^crit = 377 photons for their sample parameters, and they argue that this value explains the observed onset of non-QND errors at high power. The manuscript includes detailed appendices on the circuit Hamiltonian, hybridization, fabrication, setup, and calibration, as well as a comparison with an equivalent standard dispersive readout scheme.

Significance. If the results hold, this is a significant experimental advance for superconducting qubit readout: it demonstrates that a purely nonlinear coupling can provide high-fidelity, high-QND readout at photon numbers where standard dispersive readout degrades, and it does so without Purcell filtering. The measurement protocols are standard and the headline numbers come with error bars. A notable strength is that the theoretical critical photon number is not fitted to the QNDness curve but is evaluated from independently characterized circuit parameters, and the manuscript provides extensive supporting details on fabrication, setup, and calibration. The work should be of interest to the circuit-QED and quantum-computing communities, both for the specific transmon-molecule platform and for the general lesson that nonlinear couplings can extend the high-power readout regime.

major comments (2)
  1. [Appendix F and Fig. 3(b)] The photon-number axis used for the high-power QNDness map in Fig. 3(b) is calibrated via the AC-Stark shift using nbar_r(P) = Delta-omega_q(P)/(2 chi_qr). The last sentence of Appendix F states that this linear calibration was experimentally verified only up to 200 photons and then extrapolated to higher powers. Because the quantitative claims 'errors below 4% up to 300 photons' and 'the onset near 400 photons coincides with nbar_r^crit = 377' are read off this extrapolated axis, the linearity assumption is load-bearing. If the AC-Stark shift saturates or deviates from linearity above 200 photons, the photon-number scale is overestimated and the apparent coincidence with the theoretical ncrit would shift. I recommend extending the calibration into the 300-400 photon range, or, if that is not possible, explicitly marking the extrapolated region in Fig. 3 and softening the quantitative wording of the plateau and onset claims.
  2. [Section VI, Eq. (A14)] The theoretical critical photon number nbar_r^crit = nbar_r^lowphi = 377 is presented as a quantitative prediction that coincides with the measured onset of QNDness errors. Equation (A14) is evaluated using EJ, La, ECa, and sin^2(theta), but Table II does not specify whether these are measured values, design targets, or extracted fit parameters, nor does it give their uncertainties. To make the claimed agreement convincing, the manuscript should state the provenance of each input and propagate its uncertainty to nbar_r^crit, or at least give a sensitivity estimate. As written, the good agreement between 377 and the observed onset could be fortuitous, especially given the calibration caveat in Appendix F.
minor comments (5)
  1. [Section IV.A] When the readout power is first quoted as '89 photons', the text refers the reader to Appendix F for the calibration; it would be helpful to state already at that point that the calibration is linear only up to 200 photons and that higher-power data are extrapolated.
  2. [Fig. 7 caption] The caption says 'ovelayed black line' and the main text says 'shifts quite linearly as a function of power until it reaches about 1.7 GHz'; this wording is ambiguous because it is not clear whether 1.7 GHz refers to the shifted qubit frequency or to the total shift. Please clarify the sentence and correct the typo.
  3. [Section VI, Eq. (A10)] The derivation of nbar_r^RWA states that the perturbative term 'reaches the same probability amplitude as the non-perturbative term,' but the threshold criterion is not precisely defined. Please state the exact condition used to obtain Eq. (A10).
  4. [Appendix A.4] The equivalent standard readout scheme uses a transverse coupling g_eq/(2pi) = 515 MHz, which is a very large coupling that would also produce strong dressing beyond the dispersive approximation; the comparison is illustrative, but it should be labelled as such so that the derived nbar_std^crit = 26 is not overinterpreted.
  5. [Appendix G] The sentence introducing Eq. (G2) says 'the dephasing induced by both of the readout mode and the unused lower polariton, which is noted with the index l'; this is grammatically awkward and should be rephrased for clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the headline fidelities and QNDness values are direct measurements, and the 377-photon critical number is derived from independently determined circuit parameters, not fitted to the QNDness data; the acknowledged AC-Stark extrapolation is a calibration risk, not a circular step.

full rationale

The paper's central quantitative claims (99.21% post-selected fidelity at 89 photons, 96.7% QNDness, QNDness errors below 4% up to 300 photons) are direct experimental measurements of a fabricated device. They are not derived from the theory, so they cannot reduce by construction to the theory's inputs. The only theoretical prediction is the critical photon number nbar_r^crit = 377, derived in Appendix A.3c from the low-phase-drop condition via Eq. (A14) using EJ, LJ/La, ECa, and sin^2(theta); these are spectroscopically and geometrically determined circuit parameters, not values fitted to the QNDness error map. The paper explicitly frames the comparison as qualitative ('qualitatively explained', 'coincides with an abrupt increase'), which is a post-hoc consistency check rather than a fitted prediction. The genuine weakness is the Appendix F photon-number calibration nbar(P) = Delta_omega_q(P)/(2 chi_qr), which the manuscript itself states was experimentally verified only up to 200 photons and then extrapolated to higher power; this could shift the quantitative location of the 300-photon plateau and the 377-photon onset if the AC-Stark shift becomes nonlinear above 200 photons, but it is a calibration extrapolation risk, not a logical circularity. The self-citations to [11,12,16] supply the circuit Hamiltonian, the prior low-power transmon-molecule demonstration, and the QNDness protocol; these are published, parameter-free derivations and are not invoked as a uniqueness theorem to forbid alternative explanations. No load-bearing step in the derivation chain is equivalent to its own input by definition, so no specific circular step is identified under the quote-and-reduction standard.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central theoretical prediction (ncrit=377) is an expression evaluated at the circuit parameters (EJ, La, ECa, θ), none of which are fitted to the QNDness data. The paper's starting Hamiltonian and several approximations are inherited from prior work by the same group, which is the main dependent content. The photon-number calibration assumption (linear AC-Stark shift) is the principal unverified input in the experiment.

assumptions (6)
  • domain assumption The transmon molecule circuit Hamiltonian (Eq. A1), including the cosφ coupling term, is taken from Refs [11,12] as the starting point.
    The derivation of the cross-Kerr readout model and the high-power limits builds on this Hamiltonian without rederiving it from the full circuit Lagrangian.
  • domain assumption Low phase-drop approximation ⟨φ_q,a²⟩≪1, used to Taylor expand the cosφ coupling to fourth order.
    This is the approximation whose breakdown defines ncrit=377; it is itself a modeling assumption about the operating regime.
  • domain assumption Rotating-wave approximation is applied to the cosφ coupling and the ancilla-cavity coupling.
    The effective cross-Kerr Hamiltonian Eq. (2) and the polariton description discard counter-rotating terms; the paper separately derives an RWA critical photon number (2068).
  • domain assumption The ancilla mode is approximated as linear when forming polaritons, and the unused lower polariton is assumed to remain in vacuum.
    This allows reduction to the two-mode Hamiltonian Eq. (2); nonlinearity of the readout mode is later accounted for separately via the bifurcation threshold.
  • domain assumption The qubit mode is decoupled from the resonator by dipole-moment orthogonality (symmetry), so no direct transverse coupling or Purcell decay is present in the ideal model.
    The design is based on this symmetry; residual couplings are minimized by the circular geometry and evidenced by T1=124.5 μs, but the ideal Hamiltonian assumes exact decoupling.
  • domain assumption Photon number is extracted from the AC-Stark shift as nbar = Δωq/(2χqr), assuming χqr remains constant with power.
    Verified up to 200 photons (Appendix F) and extrapolated to the 300-photon plateau; the extrapolation is an unverified assumption at the highest reported powers.

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Cite this review

Pith. "Pith review of High-power readout of a transmon qubit using a nonlinear coupling." pith.science (2026). https://pith.science/paper/HQOMONOJ

@misc{pith2026250703642,
  author       = {Pith},
  title        = {Pith review of: High-power readout of a transmon qubit using a nonlinear coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HQOMONOJ}},
  note         = {Machine review of arXiv:2507.03642}
}
abstract

The field of superconducting qubits is constantly evolving with new circuit designs. However, when it comes to qubit readout, the use of simple transverse linear coupling remains overwhelmingly prevalent. This standard readout scheme has significant drawbacks: in addition to the Purcell effect, it suffers from a limitation on the maximal number of photons in the readout mode, which restricts the signal-to-noise ratio (SNR) and the Quantum Non-Demolition (QND) nature of the readout. Here, we explore the high-power regime by engineering a nonlinear coupling between a transmon qubit and its readout mode. Our approach builds upon previous work by Dassonneville et al. [Physical Review X 10, 011045 (2020)], on qubit readout with a non-perturbative cross-Kerr coupling in a transmon molecule. We demonstrate a readout fidelity of 99.21% with 89 photons utilizing a parametric amplifier. At this elevated photon number, the QND nature remains high at 96.7%. Even with up to 300 photons, the QNDness is only reduced by a few percent. This is qualitatively explained by deriving a critical number of photons associated with the nonlinear coupling, yielding a theoretical value of $\bar{n}_r^\text{crit} = 377$ photons for our sample's parameters. These results highlight the promising performance of the transmon molecule in the high-power regime, establishing it as a compelling platform for high-fidelity qubit readout.

Figures

Figures reproduced from arXiv: 2507.03642 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Scanning electron microscope image of a trans [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Pulse sequence for the readout fidelity measure [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Pulse sequence for the QNDness measurement. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Full microwave setup [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a),(b) Optical microscopy pictures of the rectangu [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Reward functions used for optimizing the circuit’s [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. AC-Stark shift measurement: a spectroscopy of the [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

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Forward citations

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Reference graph

Works this paper leans on

60 extracted references · 42 canonical work pages · cited by 4 Pith papers

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    as well as the current circular design. All the modes are simulated for1 Joule of stored energy so that the re- sults are comparable. Both simulated designs also have the same resonant frequencies: here, the qubit mode is around 2 GHz and the ancilla mode is around3.5 GHz. From the simulations, we can compute the field distribu- tion and the total average...

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    Transmon molecule Hamiltonian The Hamiltonian for the transmon molecule circuit [green circuit in Fig. 1(c)] is derived in [11, 12]. It is flux-dependent due to the superconducting loop formed by the two Josephson junctions and the inductance and takes the following form: ˆHtm(Φext) = 4ECq ˆn2 q − EJq cos ( ˆφq) + 4ECa ˆn2 a −2EJ " cos ( ˆφa) − LJ La(Φext...

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    Equivalent transverse-coupling readout scheme To highlight the benefits of the cos φ-coupling, the sample can be compared to an equivalent transversely- coupled-transmon. The standard readout scheme uses the Hamiltonian ˆH eq std/ℏ = ωeq q ˆq† ˆq + αeq q 2 ˆq† ˆq† ˆq ˆq + ωeq c ˆc†ˆc + geq x ˆqˆc† + ˆq†ˆc , (A15) where ωeq q is the transmon’s resonant fre...

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    Full microwave setup The full microwave setup is shown in Fig. 4. The cav- ity control and qubit control pulses are generated by a Quantum Machines OPX+ and up-converted by a Quan- tum Machines Octave. The two control signals are then combined and sent to the cavity through a common mi- crowave line. The cavity’s output signal is amplified at base tempera...

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    Cavity characteristics The 3D resonator used is a rectangular cavity made of OFHC copper, with dimensions 5 × 24.5 × 35 mm. We use its TE101 mode for the transmon molecule read- out scheme. The cavity can be probed in transmission 4 K 300 K 30 mK -20 dB-20 dB-20 dBEcco. Ecco. -20 dB -6 dB -10 dB HEMT TWPA -20 dBEcco. Ecco. -10 dB -3 dB Oct. Oct. ~ ~ Oct. ...

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Pith tools

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