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arxiv: 2605.03187 · v1 · submitted 2026-05-04 · 🪐 quant-ph · cond-mat.mes-hall

Recognition: 3 theorem links

· Lean Theorem

Operating a bistable qubit

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:55 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall
keywords bistable qubittwo-level systemsTLS defects1-bit feedbacksuperconducting qubitdephasing mitigationFPGA controlgate fidelity
0
0 comments X

The pith

A 1-bit feedback protocol estimates a superconducting qubit's bistable frequency from one single-shot measurement, reaching the qubit's intrinsic entropy limit.

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

The paper shows how parasitic two-level system defects make a qubit's frequency jump between two stable values, causing dephasing that limits gate performance. The authors introduce a simple adaptive protocol that uses an FPGA controller to read the current frequency state from a single projective measurement and immediately retunes the drive. This approach extracts the maximum possible information allowed by the qubit's entropy and is tested on a real superconducting device. It removes the characteristic Ramsey beating pattern and keeps gate error rates low over long times.

Core claim

We experimentally demonstrate an adaptive protocol for operating a bistable qubit with high fidelity using a classical controller powered by a field-programmable gate array (FPGA). Our 1-bit feedback protocol estimates the qubit's bistable frequency from only one single-shot measurement, reaching the information limit set by the qubit's intrinsic entropy. We validate the protocol in a superconducting qubit by suppressing TLS-induced Ramsey beating, and deploy it to stabilize gate fidelities over time with approximately 136 kHz estimation bandwidth and a 77% error reduction.

What carries the argument

The 1-bit feedback protocol, which infers the current TLS state (and thus the qubit frequency) from a single projective measurement and updates the drive frequency in real time.

If this is right

  • TLS-induced Ramsey beating is suppressed in the qubit's coherence measurements.
  • Gate fidelities remain stable over extended operation periods.
  • Estimation bandwidth reaches approximately 136 kHz.
  • Gate error rates are reduced by 77% compared with the uncontrolled case.
  • The method supplies a practical route to mitigate dephasing from strongly coupled TLS defects.

Where Pith is reading between the lines

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

  • The same single-shot inference idea could be applied to other discrete noise sources that produce telegraph-like frequency jumps in quantum devices.
  • Scaling the protocol to arrays with several independent TLS per qubit would require only modest additional classical logic.
  • The bandwidth and error-reduction numbers set a concrete target for controller hardware in future large-scale processors.

Load-bearing premise

The bistability is produced by one strongly coupled TLS whose state can be correctly read out from a single measurement and whose switching is slower than the controller's update speed.

What would settle it

Running the protocol on a device where the TLS switching rate exceeds the FPGA update time or where multiple TLS contribute comparable frequency shifts, and observing that Ramsey beating is not suppressed or that the 77% error reduction disappears.

Figures

Figures reproduced from arXiv: 2605.03187 by Fabrizio Berritta, Ferdinand Kuemmeth, Jan A. Krzywda, Jeroen Danon, Paul Buttles, Stanislav Eilhart, Tom Dvir.

Figure 1
Figure 1. Figure 1: FIG. 1. Dephasing and control infidelity of a bistable qubit. (a) Illus view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Mitigating dephasing of a bistable qubit by a feedback view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Quantifying improvement of gate fidelities by randomized view at source ↗
read the original abstract

Parasitic two-level-system (TLS) defects limit the stability and performance of solid-state quantum processors. Their interaction with a qubit can cause discrete, stochastic shifts of the qubit frequency, making the qubit bistable. We experimentally demonstrate an adaptive protocol for operating a bistable qubit with high fidelity using a classical controller powered by a field-programmable gate array (FPGA). Our "1-bit feedback" protocol estimates the qubit's bistable frequency from only one single-shot measurement, reaching the information limit set by the qubit's intrinsic entropy. We validate the protocol in a superconducting qubit by suppressing TLS-induced Ramsey beating, and deploy it to stabilize gate fidelities over time with approximately 136 kHz estimation bandwidth and a 77% error reduction. Our approach provides a simple, yet fundamentally efficient strategy for mitigating dephasing errors induced by strongly coupled TLS defects, and may enable the operation of large future qubit arrays suffering from few remaining, discrete instabilities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript experimentally demonstrates a '1-bit feedback' protocol for operating a superconducting qubit made bistable by interaction with a strongly coupled TLS defect. Using an FPGA-based classical controller, a single projective measurement infers the TLS state to estimate and correct the qubit frequency in real time, claimed to reach the information limit set by the qubit's intrinsic entropy. Validation includes suppression of TLS-induced Ramsey beating and stabilization of gate fidelities with ~136 kHz bandwidth and 77% error reduction.

Significance. If the central claims hold, the work provides a simple, low-overhead method for mitigating discrete dephasing from TLS defects that is information-theoretically efficient. This could be impactful for scaling solid-state quantum processors by handling remaining instabilities without complex multi-shot protocols. The experimental results in a superconducting platform add practical relevance, and the high estimation bandwidth is a notable strength.

major comments (2)
  1. [Abstract] Abstract and experimental results: The central claim that the protocol reaches the qubit's intrinsic entropy limit via one single-shot measurement requires that the projective readout reliably infers the TLS state (near-unit fidelity) and that feedback latency is shorter than the TLS switching time. The manuscript reports suppression of Ramsey beating and 77% error reduction but provides no per-shot inference error rate, readout fidelity, or measured TLS switching time versus loop latency, leaving the information-limit assertion unverified.
  2. [Experimental results] Experimental validation section: The reported outcomes (Ramsey beating suppressed, 77% error reduction, 136 kHz bandwidth) lack raw data, error bars, or statistical details on the number of trials and variability, as noted in the abstract's presentation of clear experimental outcomes without supporting quantification. This undermines assessment of the quantitative improvements.
minor comments (1)
  1. [Abstract] The abstract uses 'bistable frequency' without a brief inline definition or reference to the underlying TLS-qubit coupling model for readers unfamiliar with the context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of the work's significance and for the constructive comments on verification and quantification. We address each major point below and have revised the manuscript accordingly.

read point-by-point responses
  1. Referee: [Abstract] Abstract and experimental results: The central claim that the protocol reaches the qubit's intrinsic entropy limit via one single-shot measurement requires that the projective readout reliably infers the TLS state (near-unit fidelity) and that feedback latency is shorter than the TLS switching time. The manuscript reports suppression of Ramsey beating and 77% error reduction but provides no per-shot inference error rate, readout fidelity, or measured TLS switching time versus loop latency, leaving the information-limit assertion unverified.

    Authors: We agree that explicit per-shot metrics would strengthen the information-limit claim. The manuscript's design uses a single projective measurement whose outcome directly selects the estimated frequency; the observed suppression of Ramsey beating to the intrinsic coherence limit and the 77% gate-error reduction are consistent with high inference fidelity. In the revised manuscript we add the calibrated single-shot readout fidelity (extracted from the measurement contrast), the independently measured TLS switching time, and the FPGA loop latency (well below the switching time), allowing direct comparison to the entropy bound. revision: yes

  2. Referee: [Experimental results] Experimental validation section: The reported outcomes (Ramsey beating suppressed, 77% error reduction, 136 kHz bandwidth) lack raw data, error bars, or statistical details on the number of trials and variability, as noted in the abstract's presentation of clear experimental outcomes without supporting quantification. This undermines assessment of the quantitative improvements.

    Authors: We acknowledge that the original presentation omitted error bars and trial counts. The revised manuscript now includes error bars on all quantitative plots, states the number of experimental repetitions for each dataset (e.g., Ramsey traces and fidelity measurements), and reports the standard deviation across repeated runs. Raw time traces and histograms are added to the supplementary material. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental protocol validated directly against measured performance

full rationale

The paper describes an FPGA-based 1-bit feedback protocol for a superconducting qubit experiencing TLS-induced bistability. The core claim that the protocol reaches the information limit set by the qubit's intrinsic entropy is presented as a design goal justified by the single-shot projective measurement providing one bit, with experimental validation through observed suppression of Ramsey beating and 77% gate-error reduction at 136 kHz bandwidth. No equations reduce a fitted parameter to a renamed prediction, no self-citation chain supports a uniqueness theorem, and the protocol steps are stated explicitly without self-definition. The work is self-contained as an experimental demonstration; the information-theoretic bound is invoked as an external limit rather than derived from the paper's own data fits.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The protocol rests on the domain assumption that TLS defects produce discrete, long-lived frequency jumps that can be treated as a classical two-state Markov process observable in a single projective measurement.

axioms (2)
  • domain assumption TLS defects cause discrete stochastic shifts of the qubit frequency making the qubit bistable
    Stated directly in the abstract as the source of instability.
  • domain assumption A single projective measurement extracts enough information to estimate the current TLS state at the information limit set by qubit entropy
    Central to the 1-bit feedback claim.

pith-pipeline@v0.9.0 · 5478 in / 1301 out tokens · 58468 ms · 2026-05-08T17:55:51.207580+00:00 · methodology

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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matches
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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
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contradicts
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unclear
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Reference graph

Works this paper leans on

51 extracted references · 39 canonical work pages

  1. [1]

    and Cole, J

    author author C. M \"u ller , author J. H. \ Cole , \ and\ author J. Lisenfeld ,\ title title Towards understanding two-level-systems in amorphous solids: insights from quantum circuits , \ 10.1088/1361-6633/ab3a7e journal journal Reports on Progress in Physics \ volume 82 ,\ pages 124501 ( year 2019 ) NoStop

  2. [2]

    author author I. Siddiqi ,\ title title Engineering high-coherence superconducting qubits , \ 10.1038/s41578-021-00370-4 journal journal Nature Reviews Materials \ volume 6 ,\ pages 875--891 ( year 2021 ) NoStop

  3. [3]

    author author C. E. \ Murray ,\ title title Material matters in superconducting qubits , \ 10.1016/j.mser.2021.100646 journal journal Materials Science and Engineering: R: Reports \ volume 146 ,\ pages 100646 ( year 2021 ) NoStop

  4. [4]

    author author J. S. \ Rojas-Arias , author A. Noiri , author P. Stano , author T. Nakajima , author J. Yoneda , author K. Takeda , author T. Kobayashi , author A. Sammak , author G. Scappucci , author D. Loss , et al. ,\ title title Spatial noise correlations beyond nearest neighbors in 28 Si/Si-Ge spin qubits , \ @noop journal journal Physical Review App...

  5. [5]

    Ye , author A

    author author F. Ye , author A. Ellaboudy , author D. Albrecht , author R. Vudatha , author N. T. \ Jacobson , \ and\ author J. M. \ Nichol ,\ title title Characterization of individual charge fluctuators in Si/SiGe quantum dots , \ 10.1103/PhysRevB.110.235305 journal journal Physical Review B \ volume 110 ,\ pages 235305 ( year 2024 ) NoStop

  6. [6]

    Donnelly , author J

    author author M. Donnelly , author J. Rowlands , author L. Kranz , author Y. Hsueh , author Y. Chung , author A. Timofeev , author H. Geng , author P. Singh-Gregory , author S. Gorman , author J. Keizer , et al. ,\ title title Noise correlations in an atom-based quantum dot array , \ @noop journal journal Physical Review Applied \ volume 23 ,\ pages 06405...

  7. [7]

    author author J. S. \ Rojas-Arias , author A. Noiri , author J. Yoneda , author P. Stano , author T. Nakajima , author K. Takeda , author T. Kobayashi , author G. Scappucci , author S. Tarucha , \ and\ author D. Loss ,\ title title Inferring charge-noise source locations from correlations in spin qubits , \ 10.1103/glp5-bb5b journal journal Physical Revie...

  8. [8]

    Hakoshima , author Y

    author author H. Hakoshima , author Y. Matsuzaki , \ and\ author S. Endo ,\ title title Relationship between costs for quantum error mitigation and non-Markovian measures , \ 10.1103/PhysRevA.103.012611 journal journal Physical Review A \ volume 103 ,\ pages 012611 ( year 2021 ) NoStop

  9. [9]

    author author J. F. \ Kam , author S. Gicev , author K. Modi , author A. Southwell , \ and\ author M. Usman ,\ title title Detrimental non-Markovian errors for surface code memory , \ 10.48550/arXiv.2410.23779 \ ( year 2024 ),\ 10.48550/arXiv.2410.23779 NoStop

  10. [10]

    Gustavsson , author F

    author author S. Gustavsson , author F. Yan , author J. Bylander , author F. Yoshihara , author Y. Nakamura , author T. P. \ Orlando , \ and\ author W. D. \ Oliver ,\ title title Dynamical decoupling and dephasing in interacting two-level systems , \ @noop journal journal Physical Review Letters \ volume 109 ,\ pages 010502 ( year 2012 ) NoStop

  11. [11]

    Sza \'n kowski , author G

    author author P. Sza \'n kowski , author G. Ramon , author J. Krzywda , author D. Kwiatkowski , et al. ,\ title title Environmental noise spectroscopy with qubits subjected to dynamical decoupling , \ @noop journal journal Journal of Physics: Condensed Matter \ volume 29 ,\ pages 333001 ( year 2017 ) NoStop

  12. [12]

    Pataki , author \'A

    author author D. Pataki , author \'A . M \'a rton , author J. K. \ Asb \'o th , \ and\ author A. P \'a lyi ,\ title title Coherent errors in stabilizer codes caused by quasistatic phase damping , \ https://doi.org/10.1103/PhysRevA.110.012417 journal journal Physical Review A \ volume 110 ,\ pages 012417 ( year 2024 ) NoStop

  13. [13]

    author author C. R. H. \ McRae , author G. M. \ Stiehl , author H. Wang , author S.-X. \ Lin , author S. A. \ Caldwell , author D. P. \ Pappas , author J. Mutus , \ and\ author J. Combes ,\ title title Reproducible coherence characterization of superconducting quantum devices , \ 10.1063/5.0060370 journal journal Applied Physics Letters \ volume 119 ,\ pa...

  14. [14]

    Weeden , author D

    author author S. Weeden , author D. C. \ Harrison , author S. Patel , author M. Snyder , author E. J. \ Blackwell , author G. Spahn , author S. Abdullah , author Y. Takeda , author B. L. T. \ Plourde , author J. M. \ Martinis , \ and\ author R. McDermott ,\ title title Statistics of strongly coupled defects in superconducting qubits , \ 10.48550/arXiv.250...

  15. [15]

    author author O. F. \ Wolff , author H. Mantry , author R. Raja , author W.-H. \ Peng , author K. Singirikonda , author S. Lee , author S. Sudhaman , author R. Goncalves , author P. Y. \ Huang , author A. Kou , et al. ,\ title title Structural control of two-level defect density revealed by high-throughput correlative measurements of josephson junctions ,...

  16. [16]

    author author G. J. \ Grabovskij , author T. Peichl , author J. Lisenfeld , author G. Weiss , \ and\ author A. V. \ Ustinov ,\ title title Strain tuning of individual atomic tunneling systems detected by a superconducting qubit , \ 10.1126/science.1226487 journal journal Science \ volume 338 ,\ pages 232--234 ( year 2012 ) NoStop

  17. [17]

    Lisenfeld , author A

    author author J. Lisenfeld , author A. Bilmes , author A. Megrant , author R. Barends , author J. Kelly , author P. Klimov , author G. Weiss , author J. M. \ Martinis , \ and\ author A. V. \ Ustinov ,\ title title Electric field spectroscopy of material defects in transmon qubits , \ 10.1038/s41534-019-0224-1 journal journal npj Quantum Information \ volu...

  18. [18]

    Bilmes , author A

    author author A. Bilmes , author A. Megrant , author P. Klimov , author G. Weiss , author J. M. \ Martinis , author A. V. \ Ustinov , \ and\ author J. Lisenfeld ,\ title title Resolving the positions of defects in superconducting quantum bits , \ 10.1038/s41598-020-59749-y journal journal Scientific reports \ volume 10 ,\ pages 3090 ( year 2020 ) NoStop

  19. [19]

    Kim , author L

    author author Y. Kim , author L. C. \ Govia , author A. Dane , author E. v. d. \ Berg , author D. M. \ Zajac , author B. Mitchell , author Y. Liu , author K. Balakrishnan , author G. Keefe , author A. Stabile , et al. ,\ title title Error mitigation with stabilized noise in superconducting quantum processors , \ 10.48550/arXiv.2407.02467 \ ( year 2024 ),\...

  20. [20]

    Chen , author K.-H

    author author L. Chen , author K.-H. \ Lee , author C.-H. \ Liu , author B. Marinelli , author R. K. \ Naik , author Z. Kang , author N. Goss , author H. Kim , author D. I. \ Santiago , \ and\ author I. Siddiqi ,\ title title Scalable and site-specific frequency tuning of two-level system defects in superconducting qubit arrays , \ 10.48550/arXiv.2503.047...

  21. [21]

    Dane , author K

    author author A. Dane , author K. Balakrishnan , author B. Wacaser , author L.-W. \ Hung , author H. J. \ Mamin , author D. Rugar , author R. M. \ Shelby , author C. Murray , author K. Rodbell , \ and\ author J. Sleight ,\ title title Performance stabilization of high-coherence superconducting qubits , \ 10.48550/arXiv.2503.12514 \ ( year 2025 ),\ 10.4855...

  22. [22]

    Ye , author A

    author author F. Ye , author A. Ellaboudy , \ and\ author J. M. \ Nichol ,\ title title Stabilizing an individual charge fluctuator in a Si/Si-Ge quantum dot , \ @noop journal journal Physical Review Applied \ volume 23 ,\ pages 044063 ( year 2025 ) NoStop

  23. [23]

    Degnan , author C.-C

    author author Z. Degnan , author C.-C. \ Chiu , author Y.-H. \ Chen , author D. Sommers , author L. Abdurakhimov , author L. Zhu , author A. Fedorov , \ and\ author P. Jacobson ,\ title title Reducing TLS loss in tantalum CPW resonators using titanium sacrificial layers , \ 10.48550/arXiv.2601.16369 \ ( year 2026 ),\ 10.48550/arXiv.2601.16369 NoStop

  24. [24]

    Roy , author X

    author author T. Roy , author X. You , author D. van Zanten , author F. Crisa , author S. Garattoni , author S. Zhu , author A. Grassellino , \ and\ author A. Romanenko ,\ title title Two-level system spectroscopy from correlated multilevel relaxation in superconducting qubits , \ 10.48550/arXiv.2602.11127 \ ( year 2026 ),\ 10.48550/arXiv.2602.11127 NoStop

  25. [25]

    Gebhart , author R

    author author V. Gebhart , author R. Santagati , author A. A. \ Gentile , author E. M. \ Gauger , author D. Craig , author N. Ares , author L. Banchi , author F. Marquardt , author L. Pezz \`e , \ and\ author C. Bonato ,\ title title Learning quantum systems , \ https://doi.org/10.1038/s42254-022-00552-1 journal journal Nature Reviews Physics \ volume 5 ,...

  26. [26]

    author author M. J. \ Arshad , author C. Bekker , author B. Haylock , author K. Skrzypczak , author D. White , author B. Griffiths , author J. Gore , author G. W. \ Morley , author P. Salter , author J. Smith , author I. Zohar , author A. Finkler , author Y. Altmann , author E. M. \ Gauger , \ and\ author C. Bonato ,\ title title Real-time adaptive estima...

  27. [27]

    Berritta , author T

    author author F. Berritta , author T. Rasmussen , author J. A. \ Krzywda , author J. van der Heijden , author F. Fedele , author S. Fallahi , author G. C. \ Gardner , author M. J. \ Manfra , author E. van Nieuwenburg , author J. Danon , author A. Chatterjee , \ and\ author F. Kuemmeth ,\ title title Real-time two-axis control of a spin qubit , \ 10.1038/s...

  28. [28]

    Dumoulin Stuyck , author A

    author author N. Dumoulin Stuyck , author A. E. \ Seedhouse , author S. Serrano , author T. Tanttu , author W. Gilbert , author J. Y. \ Huang , author F. Hudson , author K. M. \ Itoh , author A. Laucht , author W. H. \ Lim , author C. H. \ Yang , author A. Saraiva , \ and\ author A. S. \ Dzurak ,\ title title Silicon spin qubit noise characterization usin...

  29. [29]

    Berritta , author J

    author author F. Berritta , author J. Benestad , author L. Pahl , author M. Mathews , author J. A. \ Krzywda , author R. Assouly , author Y. Sung , author D. K. \ Kim , author B. M. \ Niedzielski , author K. Serniak , author M. E. \ Schwartz , author J. L. \ Yoder , author A. Chatterjee , author J. A. \ Grover , author J. Danon , author W. D. \ Oliver , \...

  30. [30]

    author author M. A. \ Marciniak , author R. T. \ Birke , author J. B. \ Severin , author F. Berritta , author D. Kjær , author F. Nilsson , author S. N. \ Themadath , author S. Kallatt , author J. L. \ Webb , author K. Bentsen , author T. Madsen , author Z. Sun , author S. Krøjer , author C. W. \ Warren , author J. Hastrup , \ and\ author M. Kjaergaard ,\...

  31. [31]

    author author P. J. J. \ O’Malley , author J. Kelly , author R. Barends , author B. Campbell , author Y. Chen , author Z. Chen , author B. Chiaro , author A. Dunsworth , author A. G. \ Fowler , author I.-C. \ Hoi , author E. Jeffrey , author A. Megrant , author J. Mutus , author C. Neill , author C. Quintana , et al. ,\ title title Qubit metrology of ultr...

  32. [32]

    Berritta , author J

    author author F. Berritta , author J. Benestad , author J. A. \ Krzywda , author O. Krause , author M. A. \ Marciniak , author S. Kr jer , author C. W. \ Warren , author E. Hogedal , author A. Nylander , author I. Ahmad , author A. Osman , author J. Bizn\'arov\'a , author M. Rommel , author A. F. \ Roudsari , author J. Bylander , author G. Tancredi , auth...

  33. [33]

    A quantum engineer’s guide to superconducting qubits,

    author author P. Krantz , author M. Kjaergaard , author F. Yan , author T. P. \ Orlando , author S. Gustavsson , \ and\ author W. D. \ Oliver ,\ title title A quantum engineer's guide to superconducting qubits , \ 10.1063/1.5089550 journal journal Applied Physics Reviews \ volume 6 ,\ pages 021318 ( year 2019 ) NoStop

  34. [34]

    L., Girvin, S

    author author A. Blais , author A. L. \ Grimsmo , author S. M. \ Girvin , \ and\ author A. Wallraff ,\ title title Circuit quantum electrodynamics , \ https://doi.org/10.1103/RevModPhys.93.025005 journal journal Reviews of Modern Physics \ volume 93 ,\ pages 025005 ( year 2021 ) NoStop

  35. [35]

    Schlör , author J

    author author S. Schlör , author J. Lisenfeld , author C. Müller , author A. Bilmes , author A. Schneider , author D. P. \ Pappas , author A. V. \ Ustinov , \ and\ author M. Weides ,\ title title Correlating decoherence in transmon qubits: Low frequency noise by single fluctuators , \ 10.1103/physrevlett.123.190502 journal journal Physical Review Letters ...

  36. [36]

    author author J. M. \ Gertler , author B. Baker , author J. Li , author S. Shirol , author J. Koch , \ and\ author C. Wang ,\ title title Protecting a bosonic qubit with autonomous quantum error correction , \ @noop journal journal Nature \ volume 590 ,\ pages 243--248 ( year 2021 ) NoStop

  37. [37]

    Levine , author A

    author author H. Levine , author A. Haim , author J. S. \ Hung , author N. Alidoust , author M. Kalaee , author L. DeLorenzo , author E. A. \ Wollack , author P. Arrangoiz-Arriola , author A. Khalajhedayati , author R. Sanil , et al. ,\ title title Demonstrating a long-coherence dual-rail erasure qubit using tunable transmons , \ @noop journal journal Phy...

  38. [38]

    ( eq:avg_Rabi_prob ) of Appendix app:infidelity using P(f_ H )=P(f_ L ) = 1/2 NoStop

    note Computed from Eq. ( eq:avg_Rabi_prob ) of Appendix app:infidelity using P(f_ H )=P(f_ L ) = 1/2 NoStop

  39. [39]

    Luthi , author T

    author author F. Luthi , author T. Stavenga , author O. Enzing , author A. Bruno , author C. Dickel , author N. Langford , author M. A. \ Rol , author T. S. \ Jespersen , author J. Nyg rd , author P. Krogstrup , et al. ,\ title title Evolution of nanowire transmon qubits and their coherence in a magnetic field , \ @noop journal journal Physical review let...

  40. [40]

    Berritta , author J

    author author F. Berritta , author J. A. \ Krzywda , author J. Benestad , author J. van der Heijden , author F. Fedele , author S. Fallahi , author G. C. \ Gardner , author M. J. \ Manfra , author E. van Nieuwenburg , author J. Danon , author A. Chatterjee , \ and\ author F. Kuemmeth ,\ title title Physics-informed tracking of qubit fluctuations , \ 10.11...

  41. [41]

    \ Liu , author Y.-Y

    author author B.-J. \ Liu , author Y.-Y. \ Wang , author T. Sheffer , \ and\ author C. Wang ,\ title title Observation of discrete charge states of a coherent two-level system in a superconducting qubit , \ https://doi.org/10.1103/PhysRevLett.133.160602 journal journal Physical Review Letters \ volume 133 ,\ pages 160602 ( year 2024 ) NoStop

  42. [42]

    Hutin , author P

    author author H. Hutin , author P. Bilous , author C. Ye , author S. Abdollahi , author L. Cros , author T. Dvir , author T. Shah , author Y. Cohen , author A. Bienfait , author F. Marquardt , et al. ,\ title title Preparing Schr \"o dinger cat states in a microwave cavity using a neural network , \ @noop journal journal PRX Quantum \ volume 6 ,\ pages 01...

  43. [43]

    author author D. C. \ McKay , author C. J. \ Wood , author S. Sheldon , author J. M. \ Chow , \ and\ author J. M. \ Gambetta ,\ title title Efficient Z gates for quantum computing , \ 10.1103/physreva.96.022330 journal journal Physical Review A \ volume 96 ,\ pages 022330 ( year 2017 ) NoStop

  44. [44]

    note For the virtual detunings, we choose f = 2 for Ramsey cycles without mode estimation and f = 2.33 for Ramsey cycles with mode estimation. Stop

  45. [45]

    Park , author H

    author author J. Park , author H. Jang , author H. Sohn , author J. Yun , author Y. Song , author B. Kang , author L. E. A. \ Stehouwer , author D. D. \ Esposti , author G. Scappucci , \ and\ author D. Kim ,\ title title Passive and active suppression of transduced noise in silicon spin qubits , \ 10.1038/s41467-024-55338-z journal journal Nature Communic...

  46. [46]

    Knill , author D

    author author E. Knill , author D. Leibfried , author R. Reichle , author J. Britton , author R. B. \ Blakestad , author J. D. \ Jost , author C. Langer , author R. Ozeri , author S. Seidelin , \ and\ author D. J. \ Wineland ,\ title title Randomized benchmarking of quantum gates , \ 10.1103/physreva.77.012307 journal journal Physical Review A \ volume 77...

  47. [47]

    fig:3 (b), as the corresponding randomized-benchmarking fit failed NoStop

    note One data point in the no-feedback trace (laboratory time 575 ) was excluded from Fig. fig:3 (b), as the corresponding randomized-benchmarking fit failed NoStop

  48. [48]

    Shalibo , author Y

    author author Y. Shalibo , author Y. Rofe , author D. Shwa , author F. Zeides , author M. Neeley , author J. M. \ Martinis , \ and\ author N. Katz ,\ title title Lifetime and coherence of two-level defects in a Josephson junction , \ 10.1103/PhysRevLett.105.177001 journal journal Physical Review Letters \ volume 105 ,\ pages 177001 ( year 2010 ) NoStop

  49. [49]

    Colao Zanuz, Q

    author author D. Colao Zanuz , author Q. Ficheux , author L. Michaud , author A. Orekhov , author K. Hanke , author A. Flasby , author M. Bahrami Panah , author G. J. \ Norris , author M. Kerschbaum , author A. Remm , author F. Swiadek , author C. Hellings , author S. Laz a a r , author C. Scarato , author N. Lacroix , et al. ,\ title title Mitigating los...

  50. [50]

    author author P. W. \ Anderson ,\ title title A mathematical model for the narrowing of spectral lines by exchange or motion , \ 10.1143/JPSJ.9.316 journal journal Journal of the Physical Society of Japan \ volume 9 ,\ pages 316--339 ( year 1954 ) NoStop

  51. [51]

    author author R. Kubo ,\ title title Note on the stochastic theory of resonance absorption , \ 10.1143/JPSJ.9.935 journal journal Journal of the Physical Society of Japan \ volume 9 ,\ pages 935--944 ( year 1954 ) NoStop