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arxiv: 2605.24501 · v1 · pith:FY7NJ5CEnew · submitted 2026-05-23 · 🪐 quant-ph

Performance Limits of Fault-Tolerant Quantum Error Correction Schemes

Lorenzo Valentini , Diego Forlivesi , Marco Chiani This is my paper

Pith reviewed 2026-06-30 13:16 UTC · model grok-4.3

classification 🪐 quant-ph
keywords fault-tolerant quantum error correctionShor-style QECfailure probability boundsdepolarizing noiseflag qubitsresidual errorsdecoding errorscircuit-level faults
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The pith

Shor-style fault-tolerant quantum error correction has failure probability bounds derived solely from the number of flag qubits and gates under depolarizing noise.

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

The paper sets out to establish that the failure probability of Shor-style fault-tolerant quantum error correction can be upper-bounded using only basic structural details of the circuit, without requiring exhaustive simulations of every possible error path. It separates the total failure rate into two distinct parts: errors that arise during the decoding step and residual errors that remain after the circuit has acted on faulty gates and measurements. A sympathetic reader would care because this approach gives a practical way to estimate how much circuit imperfections degrade the protection offered by error correction. The bounds therefore quantify fundamental performance ceilings that any such scheme must respect when components are noisy.

Core claim

We derive bounds for the failure probability of Shor-style FT-QEC schemes using limited structural information, such as the number of flag qubits and quantum gates. Our analysis separates and quantifies two key contributors to the failure rate: decoding errors and residual errors arising from circuit-level faults. The derived bounds highlight fundamental limitations in Shor-style FT-QEC performance and quantify how circuit imperfections degrade error correction capabilities, under the assumption of depolarizing noise.

What carries the argument

Bounds on failure probability obtained from limited structural information (number of flag qubits and quantum gates) that separate decoding errors from residual circuit faults.

If this is right

  • Increasing the number of quantum gates raises the contribution of residual errors to the overall failure rate.
  • Decoding errors and residual errors can be quantified and bounded independently from each other.
  • Shor-style schemes reach an inherent performance ceiling set by circuit structure even when the noise model is simple.
  • The separation of error sources shows that adding flag qubits affects the two contributions differently.

Where Pith is reading between the lines

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

  • The same structural bounding approach could be applied to estimate performance of concatenated or surface-code variants before detailed simulation.
  • Experimental data from small Shor-style circuits could be checked directly against these bounds to test whether the two error contributions remain separable in hardware.
  • The method supplies a quick filter for discarding circuit designs whose gate or flag-qubit counts already push the bound above a target logical error rate.
  • Relaxing the depolarizing-noise assumption would require new structural parameters but could still follow the same separation of decoding versus residual terms.

Load-bearing premise

The derivation assumes depolarizing noise and that the failure probability can be bounded from the circuit's flag-qubit count and gate count alone without needing full error-path simulations or other noise models.

What would settle it

A Monte Carlo simulation of any concrete Shor-style circuit whose gate and flag-qubit counts are known, run under depolarizing noise, that yields a failure probability strictly larger than the paper's upper bound would falsify the claimed bound.

Figures

Figures reproduced from arXiv: 2605.24501 by Diego Forlivesi, Lorenzo Valentini, Marco Chiani.

Figure 1
Figure 1. Figure 1: Comparison between upper bounds on the logical error probability. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Examples of FT syndrome extraction of a generator with weight [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: [[7, 1, 3]] Steane code with pFT = p/100. VI. NUMERICAL RESULTS In this section, we present the derived performance bounds and validate them through Monte Carlo simulations. The simulator was implemented in C++ and include: i) the FT￾QEC circuit for syndrome extraction, accounting for faulty gates and measurements; ii) the decoder for the adopted QEC code; and iii) the decoder for the flag qubits. The FT s… view at source ↗
Figure 4
Figure 4. Figure 4: [[13, 1, 3]] Surface code with pFT = p/10. upper bound of the QEC code, as given by (2) with (4) [40], [41]. For the FT-QEC performance limits, we provide both the simple bound in (10) and its enhanced version in (11). The results indicate that the refined bound (11) is tight for small values of pFT, though it becomes loose as pFT increases. Since technological advancements are expected to reduce pFT, thes… view at source ↗
Figure 6
Figure 6. Figure 6: Examples of syndrome extraction faults propagated to data qubits. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
read the original abstract

Quantum error correction (QEC) is essential for realizing scalable quantum computation. However, when evaluating its benefits, most analyses assume idealized components, overlooking the imperfections inherent in realistic fault-tolerant (FT) implementations. In this paper, we investigate the performance of QEC schemes taking into account that quantum gates and measurements are themselves error-prone. We derive bounds for the failure probability of Shor-style FT-QEC schemes using limited structural information, such as the number of flag qubits and quantum gates. Our analysis separates and quantifies two key contributors to the failure rate: decoding errors and residual errors arising from circuit-level faults. The derived bounds highlight fundamental limitations in Shor-style FT-QEC performance and quantify how circuit imperfections degrade error correction capabilities, under the assumption of depolarizing noise.

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

1 major / 0 minor

Summary. The manuscript derives bounds on the failure probability of Shor-style fault-tolerant quantum error correction schemes from limited structural data (gate count and flag-qubit count). It separates the total failure rate into a decoding-error contribution and a residual-error contribution arising from circuit-level faults, under the assumption of depolarizing noise.

Significance. A general, count-based bound that cleanly separates the two error sources would be useful for rapid performance estimates without exhaustive circuit simulation. The result would be strengthened by machine-checked proofs or explicit falsifiable predictions, neither of which is indicated in the provided material.

major comments (1)
  1. [Abstract] Abstract: the central claim that bounds can be obtained from gate and flag counts alone rests on the unverified assertion that these counts uniformly upper-bound all possible error-propagation paths and undetectable residual-error weights. Different circuits realizing the same totals (e.g., distinct transversal vs. non-transversal orderings or flag placements) can possess topologically inequivalent fault sets; without an explicit argument showing that the bound absorbs this variation, both the numerical value and the claimed separation of decoding versus residual errors remain unsubstantiated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting the need for clarity on the generality of our bounds. We respond to the major comment below and are prepared to revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that bounds can be obtained from gate and flag counts alone rests on the unverified assertion that these counts uniformly upper-bound all possible error-propagation paths and undetectable residual-error weights. Different circuits realizing the same totals (e.g., distinct transversal vs. non-transversal orderings or flag placements) can possess topologically inequivalent fault sets; without an explicit argument showing that the bound absorbs this variation, both the numerical value and the claimed separation of decoding versus residual errors remain unsubstantiated.

    Authors: The bounds are constructed as explicit worst-case upper bounds that depend solely on the aggregate gate count (which limits the total number of fault locations) and flag-qubit count (which limits the maximum number of detectable syndromes). Error propagation is bounded by enumerating the maximum number of paths consistent with these counts under depolarizing noise, without assuming any particular ordering or topology. This counting argument is designed to dominate any specific circuit realization, including transversal versus non-transversal implementations, thereby absorbing topological variation. The separation into decoding-error and residual-error contributions follows directly from partitioning the fault locations according to whether they produce a detectable flag or an undetectable weight. We acknowledge that the abstract does not spell out this worst-case reasoning and will add a concise clarifying sentence (and, if space permits, a short lemma in the methods) to make the absorption of circuit variation explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived from structural counts without self-referential reduction

full rationale

The provided abstract and context describe a derivation of failure-probability bounds for Shor-style FT-QEC from limited structural data (gate/flag counts) under depolarizing noise, separating decoding vs. residual errors. No equations, self-citations, or fitted parameters are exhibited that would make any claimed bound equivalent to its inputs by construction. The approach is presented as a general counting argument rather than a fit or renamed ansatz, rendering the derivation self-contained against external benchmarks. No load-bearing self-citation chains or uniqueness theorems imported from the authors appear.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract, the main assumption is the depolarizing noise model; no free parameters or invented entities mentioned.

axioms (1)
  • domain assumption Depolarizing noise model
    The analysis is under the assumption of depolarizing noise as stated in the abstract.

pith-pipeline@v0.9.1-grok · 5655 in / 1104 out tokens · 24443 ms · 2026-06-30T13:16:10.396918+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

66 extracted references · 7 canonical work pages · 3 internal anchors

  1. [1]

    Scheme for reducing decoherence in quantum computer memory,

    P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A, vol. 52, pp. R2493–R2496, Oct 1995

    1995

  2. [2]

    Perfect quantum error correcting code,

    R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, “Perfect quantum error correcting code,” Phys. Rev. Lett., vol. 77, no. 1, p. 198, 1996

    1996

  3. [3]

    Theory of quantum error-correcting codes,

    E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Phys. Rev. A, vol. 55, pp. 900–911, Feb 1997

    1997

  4. [4]

    Structured near-optimal channel-adapted quantum error correction,

    A. S. Fletcher, P. W. Shor, and M. Z. Win, “Structured near-optimal channel-adapted quantum error correction,” Phys. Rev. A , vol. 77, p. 012320, Jan 2008

    2008

  5. [5]

    Entanglement-free parameter estimation of generalized Pauli channels,

    J. ur Rehman and H. Shin, “Entanglement-free parameter estimation of generalized Pauli channels,” Quantum, vol. 5, p. 490, 2021

    2021

  6. [6]

    Surface codes: Towards practical large-scale quantum computation,

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,” Phys. Rev. A, vol. 86, no. 3, sep 2012

    2012

  7. [7]

    Quantum error correction for quantum memories,

    B. M. Terhal, “Quantum error correction for quantum memories,” Rev. Mod. Phys., vol. 87, pp. 307–346, Apr 2015

    2015

  8. [8]

    Optimal architectures for long distance quantum communica- tion,

    S. Muralidharan, L. Li, J. Kim, N. L ¨utkenhaus, M. D. Lukin, and L. Jiang, “Optimal architectures for long distance quantum communica- tion,” Scientific reports, vol. 6, p. 20463, 2016

    2016

  9. [9]

    Duality of quantum and classical error correction codes: Design principles and examples,

    Z. Babar, D. Chandra, H. V . Nguyen, P. Botsinis, D. Alanis, S. X. Ng, and L. Hanzo, “Duality of quantum and classical error correction codes: Design principles and examples,” IEEE Commun. Surveys Tuts., vol. 21, no. 1, pp. 970–1010, Firstquarter 2019. 14

    2019

  10. [10]

    Reliable quantum communications based on asymmetry in distillation and cod- ing,

    L. Valentini, R. B. Christensen, P. Popovski, and M. Chiani, “Reliable quantum communications based on asymmetry in distillation and cod- ing,” IEEE Transactions on Quantum Engineering , vol. 5, pp. 1–13, 2024

    2024

  11. [11]

    On the efficacy of surface codes in compensating for radiation events in superconducting devices,

    M. Vallero, G. Casagranda, F. Vella, and P. Rech, “On the efficacy of surface codes in compensating for radiation events in superconducting devices,” in SC24: International Conference for High Performance Computing, Networking, Storage and Analysis , 2024, pp. 1–15

    2024

  12. [12]

    Towards distributed quantum error correction for distributed quantum computing,

    S. Babaie and C. Qiao, “Towards distributed quantum error correction for distributed quantum computing,” in 2025 International Conference on Quantum Communications, Networking, and Computing (QCNC) , 2025, pp. 66–73

    2025

  13. [13]

    An introduction to quantum error correction and fault- tolerant quantum computation,

    D. Gottesman, “An introduction to quantum error correction and fault- tolerant quantum computation,” in Quantum information science and its contributions to mathematics, PSAM , vol. 68, 2010, pp. 13–58

    2010

  14. [14]

    Good quantum error-correcting codes exist,

    A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,” Phys. Rev. A, vol. 54, no. 2, p. 1098, 1996

    1996

  15. [15]

    Multiple-particle interference and quantum error correction,

    A. Steane, “Multiple-particle interference and quantum error correction,” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences , vol. 452, no. 1954, pp. 2551–2577, 1996

    1954

  16. [16]

    Class of quantum error-correcting codes saturating the quantum Hamming bound,

    D. Gottesman, “Class of quantum error-correcting codes saturating the quantum Hamming bound,” Phys. Rev. A , vol. 54, pp. 1862–1868, Sep 1996

    1996

  17. [17]

    Quantum codes on a lattice with boundary

    S. B. Bravyi and A. Y . Kitaev, “Quantum codes on a lattice with boundary,” arXiv preprint quant-ph/9811052 , 1998

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  18. [18]

    Design of low-density-generator-matrix–based quantum codes for asymmetric quantum channels,

    P. Fuentes, J. E. Martinez, P. M. Crespo, and J. Garcia-Frias, “Design of low-density-generator-matrix–based quantum codes for asymmetric quantum channels,” Phys. Rev. A , vol. 103, p. 022617, Feb 2021. [On- line]. Available: https://link.aps.org/doi/10.1103/PhysRevA.103.022617

    work page doi:10.1103/physreva.103.022617 2021

  19. [19]

    Balanced product quantum codes,

    N. P. Breuckmann and J. N. Eberhardt, “Balanced product quantum codes,” IEEE Trans. Inf. Theory , vol. 67, no. 10, pp. 6653–6674, 2021

    2021

  20. [20]

    Quantum LDPC codes with almost linear minimum distance,

    P. Panteleev and G. Kalachev, “Quantum LDPC codes with almost linear minimum distance,” IEEE Transactions on Information Theory , vol. 68, no. 1, pp. 213–229, 2022

    2022

  21. [21]

    Quantum Tanner codes,

    A. Leverrier and G. Z ´emor, “Quantum Tanner codes,” in 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) , 2022, pp. 872–883

    2022

  22. [22]

    Topological quantum memory,

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,” Journal of Mathematical Physics , vol. 43, no. 9, pp. 4452– 4505, sep 2002

    2002

  23. [23]

    High-threshold universal quantum computation on the surface code,

    A. G. Fowler, A. M. Stephens, and P. Groszkowski, “High-threshold universal quantum computation on the surface code,” Phys. Rev. A , vol. 80, no. 5, nov 2009

    2009

  24. [24]

    Topological quantum distilla- tion,

    H. Bombin and M. A. Martin-Delgado, “Topological quantum distilla- tion,” Phys. Rev. Lett., vol. 97, no. 18, p. 180501, 2006

    2006

  25. [25]

    Surface code quantum computing by lattice surgery,

    C. Horsman, A. G. Fowler, S. Devitt, and R. V . Meter, “Surface code quantum computing by lattice surgery,” New Journal of Physics, vol. 14, no. 12, p. 123011, dec 2012

    2012

  26. [26]

    Cylindrical and M ¨obius quantum codes for asymmetric Pauli errors,

    L. Valentini, D. Forlivesi, and M. Chiani, “Cylindrical and M ¨obius quantum codes for asymmetric Pauli errors,” IEEE Transactions on Information Theory, vol. 71, no. 5, pp. 3766–3778, 2025

    2025

  27. [27]

    Suppressing quantum errors by scaling a surface code logical qubit,

    Google Quantum AI, “Suppressing quantum errors by scaling a surface code logical qubit,” Nature, vol. 614, no. 7949, pp. 676–681, 2023

    2023

  28. [28]

    Realizing repeated quantum error correction in a distance-three surface code,

    S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmannet al., “Realizing repeated quantum error correction in a distance-three surface code,” Nature, vol. 605, no. 7911, pp. 669–674, 2022

    2022

  29. [29]

    Realization of an error-correcting surface code with superconducting qubits,

    Y . Zhao, Y . Ye, H.-L. Huang, Y . Zhang, D. Wu, H. Guan, Q. Zhu, Z. Wei, T. He, S. Cao, F. Chen, T.-H. Chung, H. Deng, D. Fan, M. Gong, C. Guo, S. Guo, L. Han, N. Li, S. Li, Y . Li, F. Liang, J. Lin, H. Qian, H. Rong, H. Su, L. Sun, S. Wang, Y . Wu, Y . Xu, C. Ying, J. Yu, C. Zha, K. Zhang, Y .-H. Huo, C.-Y . Lu, C.-Z. Peng, X. Zhu, and J.-W. Pan, “Reali...

    2022

  30. [30]

    Logical quantum processor based on reconfigurable atom arrays,

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter et al. , “Logical quantum processor based on reconfigurable atom arrays,” Nature, p. 58–65, 2023

    2023

  31. [31]

    Quantum error correction below the surface code threshold,

    R. Acharya, L. Aghababaie-Beni, I. Aleiner, T. I. Andersen, M. Ans- mann, F. Arute, K. Arya, A. Asfaw, N. Astrakhantsev, J. Atalaya et al., “Quantum error correction below the surface code threshold,” arXiv preprint arXiv:2408.13687, 2024

    work page arXiv 2024

  32. [32]

    Almost-linear time decoding algo- rithm for topological codes,

    N. Delfosse and N. H. Nickerson, “Almost-linear time decoding algo- rithm for topological codes,” Quantum, vol. 5, p. 595, 2021

    2021

  33. [33]

    Pymatching: A Python package for decoding quantum codes with minimum-weight perfect matching,

    O. Higgott, “Pymatching: A Python package for decoding quantum codes with minimum-weight perfect matching,” IEEE/ACM Trans. Quant. Comput., vol. 3, no. 3, pp. 1–16, 2022

    2022

  34. [34]

    Spanning tree matching decoder for quantum surface codes,

    D. Forlivesi, L. Valentini, and M. Chiani, “Spanning tree matching decoder for quantum surface codes,”IEEE Commun. Lett., vol. 28, no. 7, pp. 1509–1513, 2024

    2024

  35. [35]

    Conservation laws and quantum error correction: towards a generalised matching decoder,

    B. J. Brown, “Conservation laws and quantum error correction: towards a generalised matching decoder,” IEEE BITS the Information Theory Magazine, 2023

    2023

  36. [36]

    Bubble clustering decoder for quantum topological codes,

    D. Forlivesi, L. Valentini, and M. Chiani, “Bubble clustering decoder for quantum topological codes,” IEEE Transactions on Communications, pp. 1–1, 2025, (Early Access)

    2025

  37. [37]

    Impact of decoding latency in the assessment of quantum surface codes performance,

    L. Valentini, D. Forlivesi, and M. Chiani, “Impact of decoding latency in the assessment of quantum surface codes performance,” in 2025 International Conference on Quantum Communications, Networking, and Computing (QCNC) , 2025, pp. 554–559

    2025

  38. [38]

    Quantum error detection. I. statement of the problem,

    A. E. Ashikhmin, A. M. Barg, E. Knill, and S. N. Litsyn, “Quantum error detection. I. statement of the problem,” IEEE Trans. Inf. Theory , vol. 46, no. 3, pp. 778–788, 2000

    2000

  39. [39]

    Quantum error detection. II. bounds,

    ——, “Quantum error detection. II. bounds,” IEEE Trans. Inf. Theory , vol. 46, no. 3, pp. 789–800, 2000

    2000

  40. [40]

    Logical error rates of XZZX and rotated quantum surface codes,

    D. Forlivesi, L. Valentini, and M. Chiani, “Logical error rates of XZZX and rotated quantum surface codes,” IEEE J. Sel. Areas Commun. , vol. 42, no. 7, pp. 1808–1817, 2024

    2024

  41. [41]

    Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities,

    ——, “Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities,” Quantum, vol. 9, p. 1950, 2025

    1950

  42. [42]

    Performance of CSS quantum codes in asymmetric channels,

    A. Ashikhmin, “Performance of CSS quantum codes in asymmetric channels,” in 2025 International Conference on Quantum Communica- tions, Networking, and Computing (QCNC) , 2025, pp. 187–191

    2025

  43. [43]

    Fault-tolerant quantum computation,

    P. W. Shor, “Fault-tolerant quantum computation,” in Proceedings of 37th conference on foundations of computer science . IEEE, 1996, pp. 56–65

    1996

  44. [44]

    Active stabilization, quantum computation, and quantum state synthesis,

    A. M. Steane, “Active stabilization, quantum computation, and quantum state synthesis,” Physical Review Letters, vol. 78, no. 11, p. 2252–2255, Mar. 1997

    1997

  45. [45]

    Scalable Quantum Computation in the Presence of Large Detected-Error Rates

    E. Knill, “Scalable quantum computation in the presence of large detected-error rates,” arXiv preprint quant-ph/0312190 , 2004

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  46. [46]

    A comparative code study for quantum fault-tolerance,

    A. Cross, D. DiVincenzo, and B. Terhal, “A comparative code study for quantum fault-tolerance,” Quantum Information and Computation , vol. 9, no. 7-8, 2007

    2007

  47. [47]

    Quantum error correction with only two extra qubits,

    R. Chao and B. W. Reichardt, “Quantum error correction with only two extra qubits,” Physical Review Letters , vol. 121, no. 5, Aug. 2018

    2018

  48. [48]

    Flag fault-tolerant error correc- tion with arbitrary distance codes,

    C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correc- tion with arbitrary distance codes,” Quantum, vol. 2, p. 53, Feb. 2018

    2018

  49. [49]

    Flag at origin: a modular fault-tolerant preparation for CSS codes,

    D. Forlivesi and D. Amaro, “Flag at origin: a modular fault-tolerant preparation for CSS codes,” arXiv preprint arXiv:2508.14200 , 2025

    work page arXiv 2025

  50. [50]

    Quantum lego expansion pack: Enumerators from tensor networks,

    C. Cao, M. J. Gullans, B. Lackey, and Z. Wang, “Quantum lego expansion pack: Enumerators from tensor networks,” PRX Quantum , vol. 5, p. 030313, Jul 2024

    2024

  51. [51]

    Quantum circuit tensors and enumerators with applications to quantum fault tolerance,

    A. Kukliansky and B. Lackey, “Quantum circuit tensors and enumerators with applications to quantum fault tolerance,” IEEE Transactions on Information Theory, vol. 71, no. 6, pp. 4406–4427, 2025

    2025

  52. [52]

    Fault-tolerant magic state preparation with flag qubits,

    C. Chamberland and A. W. Cross, “Fault-tolerant magic state preparation with flag qubits,” Quantum, vol. 3, p. 143, 2019

    2019

  53. [53]

    Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code

    A. Paetznick and B. W. Reichardt, “Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit golay code,” arXiv preprint arXiv:1106.2190, 2011

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  54. [54]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, 2010

    2010

  55. [55]

    Quantum weight enumerators and tensor networks,

    C. Cao and B. Lackey, “Quantum weight enumerators and tensor networks,” IEEE Transactions on Information Theory , vol. 70, no. 5, pp. 3512–3528, 2024

    2024

  56. [56]

    Fault-tolerant quantum computation,

    P. Shor, “Fault-tolerant quantum computation,” in Proceedings of 37th Conference on Foundations of Computer Science , 1996, pp. 56–65

    1996

  57. [57]

    Adaptive syndrome measurements for Shor-style error correction,

    T. Tansuwannont, B. Pato, and K. R. Brown, “Adaptive syndrome measurements for Shor-style error correction,” Quantum, vol. 7, p. 1075, Aug. 2023

    2023

  58. [58]

    Flag fault-tolerant error correction for any stabilizer code,

    R. Chao and B. W. Reichardt, “Flag fault-tolerant error correction for any stabilizer code,” PRX Quantum, vol. 1, p. 010302, Sep 2020

    2020

  59. [59]

    Fault-tolerant syndrome extraction and cat state preparation with fewer qubits,

    P. Prabhu and B. W. Reichardt, “Fault-tolerant syndrome extraction and cat state preparation with fewer qubits,” Quantum, vol. 7, p. 1154, Oct. 2023

    2023

  60. [60]

    Quantum error correction: an introductory guide,

    J. Roffe, “Quantum error correction: an introductory guide,” Contempo- rary Physics, vol. 60, no. 3, pp. 226–245, jul 2019

    2019

  61. [61]

    Low-depth flag-style syndrome extraction for small quantum error- 15 correction codes,

    D. Bhatnagar, M. Steinberg, D. Elkouss, C. G. Almudever, and S. Feld, “Low-depth flag-style syndrome extraction for small quantum error- 15 correction codes,” in 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) , vol. 1. IEEE, 2023, pp. 63–69

    2023

  62. [62]

    Parallel syndrome extraction with shared flag qubits for Calderbank-Shor-Steane codes of distance three,

    P.-H. Liou and C.-Y . Lai, “Parallel syndrome extraction with shared flag qubits for Calderbank-Shor-Steane codes of distance three,” Physical Review A, vol. 107, no. 2, p. 022614, 2023

    2023

  63. [63]

    Ultra low overhead syndrome extraction for the Steane code,

    B. Po ´or, B. Rodatz, and A. Kissinger, “Ultra low overhead syndrome extraction for the Steane code,” arXiv preprint arXiv:2511.13700, 2025

    work page arXiv 2025

  64. [64]

    Distributions of numbers of failures and successes until the first consecutive k successes,

    S. Aki and K. Hirano, “Distributions of numbers of failures and successes until the first consecutive k successes,” Annals of the Institute of Statistical Mathematics , vol. 46, pp. 193–202, 1994

    1994

  65. [65]

    Feller, An Introduction to Probability Theory and Its Applications: Volume I

    W. Feller, An Introduction to Probability Theory and Its Applications: Volume I. John Wiley & sons., 1968

    1968

  66. [66]

    Ryan and S

    W. Ryan and S. Lin, Channel codes: classical and modern . Cambridge university press, 2009

    2009