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Native nonlinear qubit-readout couplings neither eliminate drive-induced leakage nor reliably suppress it; without careful engineering of auxiliary modes they often make leakage worse, and leakage rates still swing by orders of magnitude wi

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2026-07-12 13:53 UTC pith:S43VQPCH

load-bearing objection Solid experimental warning that mediated nonlinear readout couplings re-crowd the multiphoton landscape and can make leakage swing by >10 imes for a <7% frequency shift; the two-cooldown comparison is the only real soft spot. the 2 major comments →

arxiv 2606.16055 v2 pith:S43VQPCH submitted 2026-06-14 quant-ph

Readout-Induced Leakage in Superconducting Circuits with Nonlinear Couplings

Sumeru Hazra , Wei Dai , Daniel K. Weiss , Pranav D. Parakh , Luigi Frunzio , Michel H. Devoret This is my paper
classification quant-ph
keywords superconducting circuitsqubit readoutdrive-induced leakagemultiphoton resonancesnonlinear couplingscosine-cosine interactionFloquet analysisPurcell protection
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Superconducting qubit readout is limited by drive-induced multiphoton transitions that push the qubit into leakage states, creating correlated errors that damage quantum error correction. Nonlinear couplings such as cosine-cosine interactions look attractive because they promise intrinsic Purcell protection and stricter selection rules than ordinary linear hybridization. This paper shows that, in real devices, those couplings alone do not remove the unwanted transitions and frequently introduce new parasitic channels through auxiliary or package modes. Floquet simulations and pump-probe spectroscopy map a dense landscape of resonances whose strength remains highly sensitive to the precise readout frequency. Two experiments on the same device that differ by less than 7% in resonator frequency exhibit more than an order-of-magnitude difference in measured leakage. The practical message is that the advertised benefits of nonlinear readout appear only when every relevant auxiliary mode is deliberately placed and parasitic modes are eliminated.

Core claim

In realistic superconducting circuits, native nonlinear qubit-readout couplings (ideal cosine-cosine, balanced cross-Kerr, or mediated cosine-cosine) neither eliminate nor systematically suppress drive-induced multiphoton leakage. Auxiliary modes required to realize the nonlinear interaction enlarge the Hilbert space and reintroduce dense families of allowed resonances; leakage rates can still change by more than an order of magnitude when the readout frequency is shifted by less than 7%.

What carries the argument

Floquet steady-state branch analysis of the driven multi-mode Hamiltonian, quantified by the hybridization parameter Θ that flags multiphoton resonances as avoided crossings between dressed computational and leakage states; this is used both to compare idealized coupling schemes and to identify the joint qubit-mediator transitions observed in spectroscopy.

Load-bearing premise

The two successive cooldowns that change only the intended readout-resonator frequency leave every other device parameter (junction aging, package modes, TLS bath, dispersive shift, linewidth) sufficiently unchanged that the observed order-of-magnitude leakage difference can be attributed solely to the multiphoton landscape predicted by the Floquet model.

What would settle it

Repeat the leakage-benchmarking experiment on a single cooldown while continuously tuning the readout frequency across the same ~7% window (or fabricate an otherwise identical device with a tunable resonator) and check whether leakage still jumps by more than tenfold exactly where the Floquet map predicts the [2,3:3] resonance.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Device design for nonlinear readout must treat auxiliary-mode frequencies and anharmonicities as first-class optimization parameters, not afterthoughts.
  • Single-frequency leakage characterization is insufficient; any claim of improved QND performance requires a frequency-sweep map of multiphoton resonances.
  • Package and parasitic modes must be identified and detuned or suppressed before the selection-rule advantages of cosine-cosine coupling can be realized.
  • Readout-frequency placement windows free of low-order joint resonances can be opened by deliberate linearization or frequency engineering of the mediator mode.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same frequency-crowding problem will appear in any multi-mode circuit that relies on mediated nonlinear interactions (parametric gates, beamsplitters, or couplers), not only in readout.
  • High-frequency readout (ωr/ωq ≳ 5–10) may still be the simplest practical route to sparse multiphoton landscapes even for nonlinear couplings, because matrix elements to high-lying well states vanish.
  • Automated Floquet-plus-HFSS co-design loops that jointly place qubit, mediator, resonator and package modes could become standard before tape-out of nonlinear-readout chips.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper argues that native nonlinear qubit–readout couplings (ideal cosine-cosine, balanced cross-Kerr, and mediated cosine-cosine) do not automatically eliminate or suppress drive-induced multiphoton leakage relative to linear hybridization. Floquet simulations of the four model Hamiltonians show that ideal cosine-cosine yields the sparsest resonance landscape via selection rules, yet realistic mediation by auxiliary modes reintroduces dense parasitic channels. Pump-probe spectroscopy on a dimon device maps allowed versus symmetry-forbidden transitions under symmetric/asymmetric drives, and a repeated-readout leakage benchmark on two cooldowns (readout frequencies differing by <7 %) demonstrates leakage rates that change by more than an order of magnitude when a predicted multiphoton resonance is avoided. The authors conclude that the benefits of nonlinear couplings appear only after deliberate spectral engineering of auxiliary and parasitic modes.

Significance. Leakage is a dominant correlated error for surface-code QEC; any architecture that claims intrinsic Purcell protection or stricter selection rules must therefore be stress-tested against multiphoton resonances. The work supplies a concrete, falsifiable methodology—Floquet branch analysis of the full multi-mode Hamiltonian, symmetry-resolved DUST spectroscopy, and frequency-swept leakage benchmarking—that can be applied to other nonlinear-coupling proposals. The experimental demonstration that a <7 % shift in readout frequency changes leakage by >10 imes is immediately actionable for device design. Strengths include the systematic comparison of four Hamiltonians (Fig. 2), the clear experimental separation of allowed versus forbidden processes (Fig. 3), and the quantitative Table I that links a specific Floquet resonance to measured leakage.

major comments (2)
  1. [Sensitivity to the choice of readout frequency; Table I and Fig. 4] The load-bearing experimental claim that leakage varies by orders of magnitude for a <7 % frequency change rests on the comparison of Exp 1 (ω_r/2π = 7.513 GHz) and Exp 2 (7.025 GHz) across successive cooldowns (Table I, Fig. 4). While χ and κ are kept nominally matched and Floquet predicts a [2,3:3] resonance only at the first frequency, the SM reports qubit-frequency aging of tens of MHz (6.271 → 6.209 GHz) and notes that package/TLS environments can drift. Without a quantitative bound on residual aging or mode-drift contributions (e.g., re-running Floquet with the aged parameters or additional DUST maps on both cooldowns), the attribution of the entire >10 imes improvement solely to the intended multiphoton landscape remains incompletely controlled.
  2. [Impact of auxiliary modes; Fig. 3 and SM] Transition labels [x,y:n] in Fig. 3 and the SM spectroscopy are obtained by fitting EJ, ECJ, ECs to the low-lying spectrum and then performing Floquet branch analysis. The SM itself notes that higher levels deviate because of higher harmonics of the potential and hybridization with cavity modes. A short sensitivity analysis (how much do the resonance loci move under plausible parameter variations or inclusion of the next harmonic) would confirm that the dominant auxiliary-mode channels remain correctly identified and that the denser landscape of Fig. 2(d) is robust.
minor comments (4)
  1. [References throughout] Citation numbering in the main text and SM is inconsistent (multiple distinct papers share the label [11], [19], etc.). A clean renumbering would improve readability.
  2. [Fig. 2] Fig. 2 color scale and hybridization-parameter definition Θ(jt) are clear, but the caption could explicitly state that the color bar is logarithmic so that the relative density of resonances is immediately visible.
  3. [SM, Readout settings] In the SM, the active-reset cost function (S25) uses an integral to 10τ; a one-sentence justification for the upper limit would help reproducibility.
  4. [Impact of auxiliary modes] The shorthand [q,m:d] is introduced late; defining it once in the main text near Fig. 3 would avoid forcing the reader to the SM.

Circularity Check

0 steps flagged

No circularity: Floquet multiphoton landscapes and independent two-frequency leakage measurements do not reduce to their inputs by construction.

full rationale

The paper’s load-bearing chain is (i) Floquet hybridization maps for four explicit circuit Hamiltonians (linear, balanced cross-Kerr, ideal cos–cos, mediated cos–cos), (ii) pump–probe DUST spectroscopy that maps auxiliary-mode-enabled resonances on a dimon device, and (iii) a two-cooldown leakage benchmark at readout frequencies differing by ≲7% (Table I, Fig. 4). Resonance conditions follow from energy matching of dressed states under the driven Hamiltonians (SM Eqs. S2–S8); device parameters (EJ, EC, mode frequencies) are fitted to undriven spectroscopy in the usual circuit-QED way and then used only to label observed features and to place the Floquet window—they do not force the measured leakage rates. The order-of-magnitude leakage difference between Exp 1 and Exp 2 is an independent experimental observable extracted from correlation decay under a standard incoherent leakage model (Eq. 1), not a quantity reconstructed from the fit. Self-citations to the authors’ prior dimon and DUST papers supply the device architecture and spectroscopy methods; they are not uniqueness theorems or load-bearing premises that make the central claim true by definition. No self-definitional loop, fitted-input-as-prediction, or ansatz-smuggling step appears. Score 0 is therefore appropriate.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard circuit-QED Hamiltonians, Floquet theory, and the experimental identification of multiphoton resonances. Free parameters are the usual fitted circuit energies and the two chosen readout frequencies; no new physical entities are postulated. Domain assumptions include the validity of the displaced-frame approximation, the incoherent leakage model used for RILB fitting, and the attribution of cooldown-to-cooldown differences primarily to resonator frequency.

free parameters (3)
  • EJ, ECJ, ECs (dimon circuit energies) = EJ=16.52 GHz, ECJ=0.3221 GHz, ECs=0.775 GHz
    Fitted to the measured level spectrum of the qubit-mediator system; used as input to all Floquet simulations that identify the multiphoton resonances.
  • Readout frequencies Exp1 / Exp2 = 7.513 GHz and 7.025 GHz
    Deliberately chosen (7.513 GHz vs 7.025 GHz) to sit on vs off a predicted [2,3:3] resonance; the leakage contrast is the central experimental claim.
  • Mediator frequency and anharmonicity = ωm/2π ≈ 4.605 GHz, αm/2π ≈ -102 to -110 MHz
    Measured and used to set the spectral location of joint-excitation channels; small shifts move the resonance landscape.
axioms (4)
  • domain assumption Floquet steady-state analysis correctly identifies multiphoton resonances as branch swaps in the driven spectrum.
    Standard in the DUST literature; used throughout Figs. 2 and 4 and the Supplementary Materials.
  • domain assumption Displaced-frame approximation (neglecting quantum fluctuations of the readout resonator) is sufficient for the resonance landscape.
    Invoked when deriving the effective Hamiltonians H'cc, H'D, H' for the four coupling schemes.
  • domain assumption Incoherent leakage model (average L↑, L↓ rates, no coherence between computational and leakage subspaces) adequately describes the repeated-readout correlation decay.
    Used to extract leakage and seepage from the RILB sequence (Eq. 1 / S42).
  • standard math Selection rules of ideal cos φq cos φr coupling forbid odd-parity qubit transitions.
    Follows from the parity of the interaction operator; used to interpret the sparse landscape of Fig. 2(c).

pith-pipeline@v1.1.0-grok45 · 31570 in / 3036 out tokens · 23244 ms · 2026-07-12T13:53:31.607300+00:00 · methodology

0 comments
read the original abstract

In superconducting circuits, drive-induced unwanted transitions limit the readout power, thereby constraining readout speed and fidelity. When such transitions excite the qubit into leakage states, they produce correlated errors that are particularly harmful for quantum error correction. Native nonlinear qubit-readout resonator coupling is a promising alternative to conventional linear hybridization because it provides intrinsic Purcell protection and stricter selection rules for multiphoton processes. In realistic devices, however, we show that such a coupling alone neither eliminates nor necessarily suppresses drive-induced transitions. Instead, if not appropriately engineered, these couplings often worsen the situation by introducing additional parasitic processes. Moreover, the rates of these unwanted transitions remain sensitive to the choice of readout frequency, regardless of the coupling mechanism. We demonstrate that readout-induced leakage can thus vary by orders of magnitude even when readout frequencies differ by less than ~7%. Our results establish that the benefits of native nonlinear couplings are realized only through informed device design, including the spectral placement of relevant auxiliary modes and elimination of parasitic ones.

Figures

Figures reproduced from arXiv: 2606.16055 by Daniel K. Weiss, Luigi Frunzio, Michel H. Devoret, Pranav D. Parakh, Sumeru Hazra, Wei Dai.

Figure 1
Figure 1. Figure 1: Drive-induced leakage in superconducting cir [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Multiphoton resonance landscape visualized by the sum of displaced state overlaps [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impact of auxiliary modes and residual asymmetry. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Readout-induced leakage in two experiments with [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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Reference graph

Works this paper leans on

71 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    D. P. DiVincenzo, Fortschritte der Physik48, 771 (2000)

    2000

  2. [2]

    Ristè, C

    D. Ristè, C. C. Bultink, K. W. Lehnert, and L. DiCarlo, Phys. Rev. Lett.109, 240502 (2012)

    2012

  3. [3]

    Y. Zhou, Z. Zhang, Z. Yin,et al., Nature Communica- tions 12, 5924 (2021)

    2021

  4. [4]

    L. Ding, M. Hays, Y. Sung, B. Kannan, J. An, A. Di Paolo, A. H. Karamlou, T. M. Hazard, K. Azar, D. K. Kim, B. M. Niedzielski, A. Melville, M. E. Schwartz, J. L. Yoder, T. P. Orlando, S. Gustavsson, J. A. Grover, K. Serniak, and W. D. Oliver, Phys. Rev. X 13, 031035 (2023)

    2023

  5. [5]

    R. Li, K. Kubo, Y. Ho, Z. Yan, Y. Nakamura, and H. Goto, Phys. Rev. X14, 041050 (2024)

    2024

  6. [6]

    F. m. c. Swiadek, R. Shillito, P. Magnard, A. Remm, C. Hellings, N. Lacroix, Q. Ficheux, D. C. Zanuz, G. J. Norris, A. Blais, S. Krinner, and A. Wallraff, PRX Quan- tum 5, 040326 (2024)

    2024

  7. [7]

    P. A. Spring, L. Milanovic, Y. Sunada, S. Wang, A. F. van Loo, S. Tamate, and Y. Nakamura, PRX Quantum 6, 020345 (2025)

    2025

  8. [8]

    P. D. Kurilovich, T. Connolly, C. G. L. Bøttcher, D. K. Weiss, S. Hazra, V. R. Joshi, A. Z. Ding, H. Nho, S. Di- amond, V. D. Kurilovich, W. Dai, V. Fatemi, L. Frun- zio, L. I. Glazman, and M. H. Devoret, High-frequency readout free from transmon multi-excitation resonances (2025), arXiv:2501.09161 [quant-ph]

    Pith/arXiv arXiv 2025

  9. [9]

    Google Quantum AI and Collaborators, Nature638, 920 (2025)

    2025

  10. [10]

    Gambetta, W

    J. Gambetta, W. A. Braff, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A76, 012325 (2007)

    2007

  11. [11]

    D. Sank, Z. Chen, M. Khezri, J. Kelly, R. Barends, B. Campbell, Y. Chen, B. Chiaro, A. Dunsworth, A. Fowler, E. Jeffrey, E. Lucero,et al., Phys. Rev. Lett. 117, 190503 (2016)

    2016

  12. [12]

    Khezri, A

    M. Khezri, A. Opremcak, Z. Chen, K. C. Miao, M. McEwen, A. Bengtsson, T. White, O. Naaman, D. Sank, A. N. Korotkov, Y. Chen, and V. Smelyanskiy, Phys. Rev. Appl.20, 054008 (2023)

    2023

  13. [13]

    Verney, R

    L. Verney, R. Lescanne, M. H. Devoret, Z. Leghtas, and M. Mirrahimi, Phys. Rev. Appl.11, 024003 (2019)

    2019

  14. [14]

    M. F. Dumas, B. Groleau-Paré, A. McDonald, M. H. Muñoz Arias, C. Lledó, B. D’Anjou, and A. Blais, Phys. Rev. X 14, 041023 (2024)

    2024

  15. [15]

    X. Xiao, J. Venkatraman, R. G. Cortiñas, S. Chowdhury, and M. H. Devoret, Phys. Rev. Appl.24, 044026 (2025)

    2025

  16. [16]

    W. Dai, S. Hazra, D. K. Weiss, P. D. Kurilovich, T. Con- nolly, H. K. Babla, S. Singh, V. R. Joshi, A. Z. Ding, P. D. Parakh, J. Venkatraman, X. Xiao, L. Frunzio, and M. H. Devoret, Phys. Rev. X16, 011011 (2026)

    2026

  17. [19]

    Y. Lu, A. Maiti, J. W. Garmon, S. Ganjam, Y. Zhang, J. Claes, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Nature Communications 14, 5767 (2023)

    2023

  18. [21]

    Didier, J

    N. Didier, J. Bourassa, and A. Blais, Phys. Rev. Lett. 115, 203601 (2015)

    2015

  19. [22]

    J. M. Gambetta, A. A. Houck, and A. Blais, Phys. Rev. Lett. 106, 030502 (2011)

    2011

  20. [23]

    Dassonneville, T

    R. Dassonneville, T. Ramos, V. Milchakov, L. Planat, E. Dumur, F. Foroughi, J. Puertas, S. Leger, K. Bharad- waj, J. Delaforce, C. Naud, W. Hasch-Guichard, J. J. García-Ripoll, N. Roch, and O. Buisson, Phys. Rev. X 10, 011045 (2020)

    2020

  21. [24]

    C. A. Potts, R. C. Dekker, S. Deve, E. W. Strijbis, and G. A. Steele, Phys. Rev. Lett.134, 153603 (2025). 7

    2025

  22. [25]

    All these nonlinear coupling schemes intrinsically elimi- nate Purcell decay of the qubit mode

  23. [26]

    Y. Ye, J. B. Kline, S. Chen, A. Yen, and K. P. O’Brien, Science Advances10, eado9094 (2024)

    2024

  24. [28]

    Wang, F.-M

    C. Wang, F.-M. Liu, H. Chen, Y.-F. Du, C. Ying, J.-W. Wang, Y.-H. Huo, C.-Z. Peng, X. Zhu, M.-C. Chen, C.-Y. Lu, and J.-W. Pan, Phys. Rev. Lett.135, 060803 (2025)

    2025

  25. [29]

    Beaulieu, J.-Z

    G. Beaulieu, J.-Z. Chen, M. Scigliuzzo, O. Benhayoune- Khadraoui, A. A. Chapple, P. A. Spring, A. Blais, and P. Scarlino, Fast, high-fidelity transmon readout with in- trinsic purcell protection via nonperturbative cross-kerr coupling (2026), arXiv:2601.04975 [quant-ph]

    arXiv 2026

  26. [30]

    K. V. Salunkhe, S. Kundu, S. Das, J. Deshmukh, M. P. Patankar, and R. Vijay, Applied Physics Letters 126, 254001 (2025)

    2025

  27. [31]

    Hazra, W

    S. Hazra, W. Dai, T. Connolly, P. D. Kurilovich, Z. Wang, L. Frunzio, and M. H. Devoret, Phys. Rev. Lett. 134, 100601 (2025)

    2025

  28. [33]

    C. Mori, F. D’Esposito, A. Petrescu, L. Ruela, S. Ku- mar, V. N. Suresh, W. Ardati, D. Nicolas, G. Cap- pelli, A. Ranadive, G. L. Gal, M. Esposito, Q. Ficheux, N. Roch, and O. Buisson, PRX Quantum (2026)

    2026

  29. [35]

    Readout-Induced Leak- age in Superconducting Circuits with Nonlinear Cou- plings

    See Supplemental Materials for “Readout-Induced Leak- age in Superconducting Circuits with Nonlinear Cou- plings” (2026)

    2026

  30. [36]

    Sheldon, M

    S. Sheldon, M. Sandberg, H. Paik, B. Abdo, J. M. Chow, M. Steffen, and J. M. Gambetta, Applied Physics Letters 111, 222601 (2017)

    2017

  31. [37]

    Hazra, D

    S. Hazra, D. K. Weiss, W. Dai, L. Frunzio, and M. H. Devoret (2026), manuscript in preparation

    2026

  32. [38]

    Ranadive, M

    T.Roy, S.Kundu, M.Chand, S.Hazra, N.Nehra, R.Cos- mic, A. Ranadive, M. P. Patankar, K. Damle, and R. Vi- jay, Phys. Rev. Appl.7, 054025 (2017)

    2017

  33. [39]

    Pfeiffer, M

    F. Pfeiffer, M. Werninghaus, C. Schweizer, N. Bruck- moser, L. Koch, N. J. Glaser, G. B. P. Huber, D. Bunch, F. X. Haslbeck, M. Knudsen, G. Krylov, K. Liegener, A. Marx, L. Richard, J. H. Romeiro, F. A. Roy, J. Schirk, C. Schneider, M. Singh, L. Södergren, I. Tsit- silin, F. Wallner, C. A. Riofrío, and S. Filipp, Phys. Rev. X 14, 041007 (2024)

    2024

  34. [40]

    Singh, G

    S. Singh, G. Refael, A. Clerk, and E. Rosenfeld, PRX Quantum 6, 040304 (2025)

    2025

  35. [41]

    D. T. McClure, H. Paik, L. S. Bishop, M. Steffen, J. M. Chow, and J. M. Gambetta, Phys. Rev. Appl.5, 011001 (2016)

    2016

  36. [42]

    Readout-Induced Leakage in Superconducting Circuits with Nonlinear Couplings

    A. Chatterjee, J. Schwinger, and Y. Y. Gao, Phys. Rev. Appl. 23, 054057 (2025). Supplementary Materials for “Readout-Induced Leakage in Superconducting Circuits with Nonlinear Couplings” MODEL HAMIL TONIAN FOR NONLINEAR COUPLINGS In this section, we define the model Hamiltonian for each of the different nonlinear coupling schemes described in the main tex...

    2025

  37. [43]

    true positive

    The qubit has not leaked, and the following holds for the last two (m and m−1) detection cycles: in each cycle, either no error occurs, or both an SNR error and a Pauli error occur simultaneously. The probability of this “true positive” event is Ptp1 = [(1−ϵSNR)(1−ϵσ)+ ϵSNRϵσ]2(1−PL) = F2 ge(1−PL) where we define the readout fidelity:Fge ∶= (1−ϵSNR)(1−ϵσ)+ ϵSNRϵσ

  38. [44]

    true positive

    The qubit has not leaked, and each of the two cycles has either an SNR error or a Pauli error (but not both). The probability of this “true positive” event is Ptp2 = [ϵσ(1−ϵSNR)+ ϵSNR(1−ϵσ)]2(1−PL) = (1−Fge)2(1−PL)

  39. [45]

    false positive

    The qubit has leaked into non-computational states, but the readout outcome falls on opposite sides of the threshold in the two successive readout cycles. Denoting the probability that it falls on the same side as∣0⟩by α= P(0∣ψL), the “false positive” probability is, Pfp = PL[α(1−α)+(1−α)α]= 2PLα(1−α) The total probability of obtaining a correlated outcom...

  40. [46]

    The probability is Ptn1 = F2 ge(1−PL)

    The qubit has not leaked, and in the two most recent detection cycles (m and m−1), either no readout or Pauli error occurs, or both errors occur simultaneously. The probability is Ptn1 = F2 ge(1−PL)

  41. [47]

    The probability is Ptn2 = (1−Fge)2(1−PL)

    The qubit has not leaked, and each detection cycle has either a readout or a Pauli error (but not both). The probability is Ptn2 = (1−Fge)2(1−PL)

  42. [48]

    false negative

    The qubit has leaked into non-computational states, and the readout outcomes fall on the same side of the threshold in successive measurements. The “false negative” probability is: Pfn = PL[α2+(1−α)2] Hence, the total probability of getting a correlated outcome forI inputs is: P I corr(m) =[F2 ge+(1−Fge)2](1−PL(m)) + PL(m)[α2+(1−α)2] (S40) Since the input...

  43. [49]

    Didier, J

    N. Didier, J. Bourassa, and A. Blais, Phys. Rev. Lett.115, 203601 (2015)

    2015

  44. [50]

    A. A. Chapple, A. McDonald, M. H. Muñoz Arias, M. Lachapelle, and A. Blais, Phys. Rev. Appl.24, 034026 (2025)

    2025

  45. [51]

    C. Mori, F. D’Esposito, A. Petrescu, L. Ruela, S. Kumar, V. N. Suresh, W. Ardati, D. Nicolas, G. Cappelli, A. Ranadive, G. L. Gal, M. Esposito, Q. Ficheux, N. Roch, and O. Buisson, PRX Quantum (2026)

    2026

  46. [52]

    Hazra, W

    S. Hazra, W. Dai, T. Connolly, P. D. Kurilovich, Z. Wang, L. Frunzio, and M. H. Devoret, Phys. Rev. Lett.134, 100601 (2025)

    2025

  47. [53]

    A. A. Chapple, O. Benhayoune-Khadraoui, S. Richer, and A. Blais, Phys. Rev. Lett.135, 256002 (2025)

    2025

  48. [54]

    D. K. Weiss, W. DeGottardi, J. Koch, and D. G. Ferguson, Phys. Rev. Res.3, 033244 (2021)

    2021

  49. [55]

    Goldstein, Classical Mechanics (Addison-Wesley, 1980) pp

    H. Goldstein, Classical Mechanics (Addison-Wesley, 1980) pp. 250–253

    1980

  50. [56]

    J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A76, 042319 (2007)

    2007

  51. [57]

    Z. K. Minev, Z. Leghtas, S. O. Mundhada, L. Christakis, I. M. Pop, and M. H. Devoret, npj Quantum Information7, 131 (2021)

    2021

  52. [58]

    Rehammar and S

    R. Rehammar and S. Gasparinetti, IEEE Transactions on Microwave Theory and Techniques71, 3075 (2023)

    2023

  53. [59]

    W. Dai, S. Hazra, D. K. Weiss, P. D. Kurilovich, T. Connolly, H. K. Babla, S. Singh, V. R. Joshi, A. Z. Ding, P. D. Parakh, J. Venkatraman, X. Xiao, L. Frunzio, and M. H. Devoret, Phys. Rev. X16, 011011 (2026)

    2026

  54. [60]

    J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett.95, 210503 (2005)

    2005

  55. [61]

    P. V. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant, B. Burkett, R. Barends, K. Arya, B. Chiaro, Y. Chen, A. Dunsworth, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, T. Huang, E. Jeffrey, E. Lucero, J. Y. Mutus, O. Naaman, C. Neill, C. Quintana, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, S. Boixo, R. Babbush, V. N. Smelyanski...

    2018

  56. [62]

    M. Chen, J. C. Owens, H. Putterman, M. Schäfer, and O. Painter, Science Advances10, eado6240 (2024)

    2024

  57. [63]

    Lisenfeld, A

    J. Lisenfeld, A. Bilmes, A. Megrant, R. Barends, J. Kelly, P. Klimov, G. Weiss, J. M. Martinis, and A. V. Ustinov, npj Quantum Information 5, 1 (2019)

    2019

  58. [64]

    J. H. Cole, C. Müller, P. Bushev, G. J. Grabovskij, J. Lisenfeld, A. Lukashenko, A. V. Ustinov, and A. Shnirman, Applied Physics Letters 97, 252501 (2010)

    2010

  59. [65]

    J. J. Burnett, A. Bengtsson, M. Scigliuzzo, D. Niepce, M. Kudra, P. Delsing, and J. Bylander, npj Quantum Information 5, 1 (2019)

    2019

  60. [66]

    Winkel, R

    M.Spiecker, P.Paluch, N.Gosling, N.Drucker, S.Matityahu, D.Gusenkova, S.Günzler, D.Rieger, I.Takmakov, F.Valenti, P. Winkel, R. Gebauer, O. Sander, G. Catelani, A. Shnirman, A. V. Ustinov, W. Wernsdorfer, Y. Cohen, and I. M. Pop, Nature Physics 19, 1320 (2023)

    2023

  61. [67]

    C. Mori, V. Milchakov, F. D’Esposito, L. Ruela, S. Kumar, V. N. Suresh, W. Ardati, D. Nicolas, G. Cappelli, A. Ranadive, G. L. Gal, M. Esposito, Q. Ficheux, N. Roch, T. Ramos, and O. Buisson, High-power readout of a transmon qubit using a nonlinear coupling (2025), arXiv:2507.03642 [quant-ph]

    Pith/arXiv arXiv 2025

  62. [68]

    F. Yan, Y. Sung, P. Krantz, A. Kamal, D. K. Kim, J. L. Yoder, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Engineering framework for optimizing superconducting qubit designs (2020), arXiv:2006.04130 [quant-ph]. S-16

    Pith/arXiv arXiv 2020

  63. [69]

    D. K. Weiss, Floquet: Identifying nonlinear resonances in quantum systems due to parametric drives,https://github. com/dkweiss31/floquet (2024)

    2024

  64. [70]

    Willsch, D

    D. Willsch, D. Rieger, P. Winkel, M. Willsch, C. Dickel, J. Krause, Y. Ando, R. Lescanne, Z. Leghtas, N. T. Bronn,et al., Nature Physics 20, 815 (2024)

    2024

  65. [71]

    J. Kim, M. Hays, I. T. Rosen, J. An, H. Zhang, A. Goswami, K. Azar, J. M. Gertler, B. M. Niedzielski, M. E. Schwartz, T. P. Orlando, J. A. Grover, K. Serniak, and W. D. Oliver, Nature Physics 10.1038/s41567-026-03285-5 (2026)

    work page doi:10.1038/s41567-026-03285-5 2026

  66. [72]

    Féchant, M

    M. Féchant, M. F. Dumas, D. Bénâtre, N. Gosling, P. Lenhard, M. Spiecker, W. Wernsdorfer, B. D’Anjou, A. Blais, and I. M. Pop, Offset charge dependence of measurement-induced transitions in transmons (2025), arXiv:2505.00674 [quant-ph]

    arXiv 2025

  67. [73]

    Connolly, P

    T. Connolly, P. D. Kurilovich, V. D. Kurilovich, C. G. L. Bøttcher, S. Hazra, W. Dai, A. Z. Ding, V. R. Joshi, H. Nho, S. Diamond, D. K. Weiss, V. Fatemi, L. Frunzio, L. I. Glazman, and M. H. Devoret, Full characterization of measurement- induced transitions of a superconducting qubit (2025), arXiv:2506.05306 [quant-ph]

    Pith/arXiv arXiv 2025

  68. [74]

    S. E. Nigg, H. Paik, B. Vlastakis, G. Kirchmair, S. Shankar, L. Frunzio, M. H. Devoret, R. J. Schoelkopf, and S. M. Girvin, Phys. Rev. Lett.108, 240502 (2012)

    2012

  69. [75]

    D. M. Pozar,Microwave Engineering, 4th ed. (John Wiley & Sons, 2012)

    2012

  70. [76]

    C. J. Wood and J. M. Gambetta, Phys. Rev. A97, 032306 (2018)

    2018

  71. [77]

    Marxer, J

    F. Marxer, J. Mrożek, J. Andersson, L. Abdurakhimov, J. Adam, V. Bergholm, R. Beriwal, C. F. Chan, S. Dahl, S. R. Das, F. Deppe, O. Fedorets, Z. Gao, A. Gomez Frieiro, D. Gusenkova, A. Guthrie, T. Hiltunen, H. Hsu, E. Hyyppä, J. Ikonen, S. Inel, S. W. Jolin, A. Karis, S.-G. Kim, W. Kindel, A. Komlev, M. Koistinen, R. Kokkoniemi, S. Kumar, H.-S. Ku, J. Lam...

    2026