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Generalized Bicycle Codes with Low Connectivity: Minimum Distance Bounds and Hook Errors

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Generalized bicycle codes with four-qubit checks can encode two logical qubits with certified distance d and decoding thresholds near surface-code values.

desk verdict Plausible and potentially solid generalized-bicycle paper, but the supplied full text is unreadable, so the load-bearing uniform-distance claim can't be checked from this version. read the letter →

arxiv 2508.09082 v1 pith:SXOYBEDM submitted 2025-08-12 cs.IT cs.DMmath.ITquant-ph

Reza Dastbasteh , Olatz Sanz Larrarte , Arun John Moncy , Pedro M. Crespo , Josu Etxezarreta Martinez , Ruben M. Otxoa This is my paper
classification cs.ITcs.DMmath.ITquant-ph MSC 81P7094B65
keywords generalizedbicyclecodesquantumLDPCminimumdistanceboundsfault-tolerantlogicalCNOThookerrorsdepolarizingnoisethresholdBP-OSDdecodingMWPM
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's aim is to prove that generalized bicycle (GB) codes—quantum CSS codes built from circulant matrices—can combine surface-code-like connectivity with an extra logical qubit. For two infinite families with parameters $[[d^2+1,2,d]]$ for odd $d\geq 3$ and $[[d^2,2,d]]$ for even $d\geq 4$, it claims new minimum-distance bounds that can certify the true distance $d$, while every check qubit touches exactly four data qubits. If right, these bounds also show a regime where classical bounding techniques fall short, and they guarantee low-connectivity quantum LDPC families with a square-root block-length-to-distance trade-off. The paper further claims a fault-tolerant logical CNOT on the odd family via a simple relabeling of data qubits, a syndrome-extraction pattern that avoids distance reduction from check-to-data fault propagation, and BP-OSD/MWPM decoding thresholds near 14–16%, close to rotated surface codes.

What carries the argument

The construction is a generalized bicycle code: a CSS code whose stabilizer generators are read off from a pair of circulant matrices. The two families are chosen so that every row of the combined check matrix has exactly four non-zero entries, giving each check qubit exactly four data-qubit contacts, a connectivity similar to a surface code. The distance result rests on new upper and lower bounds for the minimum weight of logical operators in these circulant configurations; when the two bounds meet at $d$, the minimum distance is certified. The logical CNOT is carried by a data-qubit relabeling, a permutation of physical qubit labels that maps the stabilizer group to itself and implements t

What would settle it

Take a $d$ larger than any instance whose true distance the paper explicitly computes (for example $d=13$ for the odd family and $d=14$ for the even family), construct the stated circulant code, and compute its rank and exact minimum distance. Finding a logical operator of weight $< d$, or a code dimension other than 2, would falsify the infinite-family parameter claim. For the CNOT claim, apply the proposed relabeling to every stabilizer: if any stabilizer maps outside the stabilizer group, the relabeling is not fault-tolerant.

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Extended reading notes

Core claim

The central discovery is the existence of two infinite families of generalized bicycle codes with parameters $[[d^2+1,2,d]]$ for odd $d\geq 3$ and $[[d^2,2,d]]$ for even $d\geq 4$, together with minimum-distance upper and lower bounds that can be tight enough to determine the true distance for some members. The lower bound rules out every logical Pauli operator of weight below $d$; an explicit weight-$d$ operator reaches the bound, so the stated parameters are achieved. In the odd-distance family, the low-weight logical operators are analyzed and a permutation of the data-qubit labels is shown to implement a logical CNOT between the two logical qubits, commuting with the stabilizer group. Fo

Load-bearing premise

Every member of the two infinite families must have exactly two logical qubits and true minimum distance $d$; if any odd $d\geq 3$ or any even $d\geq 4$ gives a code with different dimension or a logical operator of weight less than $d$, the stated family parameters and the results built on them fail.

Editorial extensions

If this is right

  • The distance bounds give certified parameters for every member of two infinite families, so error correction against up to $\lfloor (d-1)/2\rfloor$ errors is guaranteed by construction rather than by finite-size numerics.
  • With each check qubit touching exactly four data qubits, these codes demand surface-code-grade hardware connectivity while placing two logical qubits in a block of size about $d^2$.
  • The relabeling CNOT gives a fault-tolerant logical gate that requires no extra logical ancilla and no complex gate decomposition, at least for the odd-distance family.
  • The syndrome-extraction pattern means the fault-tolerant measurement circuit itself does not introduce hook errors that shrink the distance from $d$.
  • Code-capacity depolarizing decoding with BP-OSD and MWPM gives thresholds near 14–16%, close to those of rotated surface codes.

Reading between the lines

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

  • Beyond the paper's claims, the same bound technique could be run over other four-connectivity circulant pairs to search for additional $[[n,2,d]]$ codes with exact distances, effectively turning the method into a code-search tool.
  • If relabeling can implement a logical CNOT by mapping stabilizers to stabilizers, then by a similar permutation search the even-distance family may also admit a fault-tolerant CNOT, or other logical Clifford gates may be found by different label permutations—neither is proven in the paper.
  • The threshold claim is made under code-capacity noise; a natural testable extension is circuit-level depolarizing simulation with the proposed syndrome-extraction pattern, which would show whether the 14–16% threshold survives realistic measurement faults.
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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

4 major / 4 minor

Summary. The paper claims new upper and lower bounds on the minimum distance of generalized bicycle (GB) codes, and uses them to analyze two infinite families with parameters [[d^2+1,2,d]] for odd d >= 3 and [[d^2,2,d]] for even d >= 4, where every check qubit has degree four. It further claims a fault-tolerant logical CNOT by a simple relabeling of data qubits, a syndrome extraction pattern that avoids distance reduction from hook errors, and numerical thresholds of approximately 14--16% under code-capacity depolarizing noise using BP-OSD and MWPM decoders. The supplied full text is corruption-garbled: no theorem, lemma, proof, or numeric table can be read, and the body even embeds a line from another arXiv submission. The assessment therefore rests almost entirely on the abstract.

Significance. If correct, the results would be significant: the two families have parameters competitive with surface codes (n = d^2 or d^2+1, k = 2, d, check degree 4) and the decoder thresholds are claimed to be surface-code-like. The distance bounds, if genuine, would be a new contribution to the theory of bicycle codes. However, the manuscript provides no inspectable proof, no explicit polynomial construction, no decoder implementation details, and no reproducible numerical data. The claim that the bounds 'capture the true minimum distance for some cases' is much weaker than the uniform exact-distance claim needed for the infinite families. At present the paper is a plausible abstract with an unverifiable body.

major comments (4)
  1. [Full text (all sections)] The supplied full text is unreadable: it consists of mojibake and garbled figures, and it contains the line 'arXiv:2508.09074v1 [cs.CL] 12 Aug 2025' inside the body. No theorem, lemma, proof, or table can be verified. This is not a minor presentation issue; it prevents any check of the central construction, the distance bounds, and the numerical results. A clean, properly rendered manuscript is a prerequisite for review.
  2. [Abstract and 'Generalized bicycle codes'] The uniform family parameters [[d^2+1,2,d]] and [[d^2,2,d]] require, for every d in the claimed ranges, explicit polynomials A(x), B(x), a proof that the code has exactly k=2 logical qubits, and a proof that the minimum distance is exactly d. The abstract only says the bounds are 'capable of even capturing the true minimum distance for some cases.' That phrase does not establish exact distance for all d. If any family member has distance < d or k != 2, the family-level claim and all downstream conclusions (CNOT, hook-error analysis, thresholds) fail as stated.
  3. [Abstract, logical CNOT by relabeling] The claim that a simple relabeling of data qubits implements a fault-tolerant logical CNOT requires a proof that the relabeling maps the stabilizer group to itself and acts on the two logical qubits as a CNOT. No legible proof or diagram appears in the supplied text. Similarly, the syndrome extraction pattern is claimed to avoid minimum-distance reduction from hook errors, but the circuit, the error-propagation analysis, and the distance-leakage argument are not visible.
  4. [Numerical results / thresholds] The reported thresholds of approximately 14--16% cannot be evaluated without the decoder hyperparameters, code sizes, number of error samples, threshold extraction method, or the noise model details. BP-OSD has many free parameters (e.g., OSD order, search depth), and MWPM requires matching-graph construction; these are not described. The comparison with rotated surface codes is also not backed by any visible simulation curves.
minor comments (4)
  1. [Full text] The body must be re-rendered; the current text is unreadable. The embedded line 'arXiv:2508.09074v1 [cs.CL] 12 Aug 2025' should be removed.
  2. [Title / Abstract] 'Hook errors' is used in the title and abstract but never defined in the legible portion. A precise definition (e.g., errors arising from CNOT hook operations in syndrome extraction) should be given.
  3. [Generalized bicycle codes] The phrase 'each check qubit is connected to exactly four data qubits' should be made precise: does it hold for both X and Z check operators, and is the Tanner graph degree four in each separately or in total? This affects the comparison with surface codes.
  4. [Related work] No legible references or comparison with prior GB code constructions are present. The paper should clarify the novelty of its distance bounds relative to existing bicycle-code literature and cite key prior work.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity identified; family-uniformity gap is a correctness concern, not a circularity

full rationale

No circular step can be exhibited. The paper's central claims are new minimum-distance bounds for generalized bicycle codes, applied to two families with parameters [[d^2+1,2,d]] and [[d^2,2,d]], plus a relabeling CNOT and independent BP-OSD/MWPM threshold simulations. The bounds are presented as results derived from the code structure, not as quantities fitted to the target distances; the abstract's phrase 'capable of even capturing the true minimum distance for some cases' describes validation against computed true distances, which is verification rather than construction-by-fitting. The logical CNOT by relabeling is a constructive claim that must be checked against the stabilizer group, and no visible argument reduces it to an assumption of the theorem. The threshold figures come from standard decoders, i.e., genuinely separate numerical evidence. The supplied full text is corrupted and largely unreadable, so equation-level derivations cannot be inspected; however, the rules require quoting a specific reduction to claim circularity, and none is available. The concern that exact distance d and k=2 are not proven for every member of the infinite families is a uniformity/completeness gap and a possible correctness risk, but it is not a self-definitional, fitted-input, or self-citation circularity. Under the default expectation that most papers are not circular, the appropriate finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claims rest on (i) the generalized bicycle framework itself, (ii) the specific generator polynomials chosen for each family, (iii) the standard depolarizing noise and circuit-fault propagation models, and (iv) unstated decoder calibration for the threshold numbers. The generator choice is the only clearly free object: the abstract does not explain how A and B were selected for the two families. No fitted constants appear in the abstract. Because the body was unreadable, additional parameters may exist in the full derivations.

free parameters (2)
  • Defining polynomial pair (A, B) of the two GB families = not stated in abstract
    The families exist only for a specific circulant pair; the abstract does not say whether the pair was derived via the bound method or found by search. Every family-level claim inherits this choice.
  • BP-OSD and MWPM decoder hyperparameters = not stated in abstract
    Threshold values are sensitive to OSD order, the matching-graph construction, and the number of Monte Carlo samples; none of these are given in the abstract.
assumptions (3)
  • standard math Generalized bicycle CSS framework: the code is defined by circulant matrices A, B from polynomial generators, with the bicycle commutativity condition ensuring a valid quantum code.
    Invoked implicitly throughout; the paper contributes bounds and constructions inside this established framework rather than re-deriving it.
  • domain assumption Standard circuit-fault propagation model in which one check-qubit fault can create at most two data errors (the hook-error model).
    The claim that the syndrome extraction pattern avoids distance reduction quantifies faults against this propagation model; the abstract does not state the circuit-level noise model.
  • domain assumption Code capacity depolarizing noise (independent depolarizing errors on data qubits) as the comparison model for thresholds.
    The similarity to rotated surface codes is only meaningful if both use the same noise model and decoder calibration; the abstract states the noise model but not the decoder calibration.

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Cite this review

Pith. "Pith review of Generalized Bicycle Codes with Low Connectivity: Minimum Distance Bounds and Hook Errors." pith.science (2026). https://pith.science/paper/SXOYBEDM

@misc{pith2026250809082,
  author       = {Pith},
  title        = {Pith review of: Generalized Bicycle Codes with Low Connectivity: Minimum Distance Bounds and Hook Errors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXOYBEDM}},
  note         = {Machine review of arXiv:2508.09082}
}
abstract

We present new upper and lower bounds on the minimum distance of certain generalized bicycle (GB) codes beyond the reach of techniques for classical codes capable of even capturing the true minimum distance for some cases. These bounds are then applied to illustrate the existence and analyze two highly degenerate GB code families with parameters $[[d^2+1,2,d]]$ for odd $d \geq 3$ and $[[d^2,2,d]]$ for even $d \geq 4$, both having the property that each check qubit is connected to exactly four data qubits similar to surface codes. For the odd-distance family, we analyze the structure of low-weight logical Pauli operators and demonstrate the existence of a fault-tolerant logical CNOT gate between the two logical qubits, achievable through a simple relabeling of data qubits. We further construct a syndrome extraction pattern for both families that does not imply minimum distance reduction arising from extraction circuit faults that propagate from the check qubits to the data qubits. Finally, we numerically evaluate their logical error rates under a code capacity depolarizing noise model using the belief propagation ordered statistics decoding (BP-OSD) and minimum-weight perfect-matching (MWPM) decoders, yielding thresholds of approximately $14-16\%$ for the odd and even families, very similar to those of rotated surface codes.

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Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

48 extracted references · 39 canonical work pages · cited by 2 Pith papers

  1. [1]

    Constructions and noise threshold of hyperbolic surface codes

    Nikolas P Breuckmann and Barbara M Terhal. Constructions and noise threshold of hyperbolic surface codes. IEEE transactions on Information Theory , 62(6):3731--3744, 2016

    work page 2016

  2. [2]

    Constructions and performance of hyperbolic and semi-hyperbolic floquet codes

    O Higgott and NP Breuckmann. Constructions and performance of hyperbolic and semi-hyperbolic floquet codes. arXiv preprint arXiv:2308.03750 , 2023

    arXiv 2023

  3. [3]

    Quantum kronecker sum-product low-density parity-check codes with finite rate

    Alexey A Kovalev and Leonid P Pryadko. Quantum kronecker sum-product low-density parity-check codes with finite rate. Physical Review A—Atomic, Molecular, and Optical Physics , 88(1):012311, 2013

    work page 2013

  4. [4]

    Degenerate quantum LDPC codes with good finite length performance

    Pavel Panteleev and Gleb Kalachev. Degenerate quantum LDPC codes with good finite length performance. Quantum , 5:585, 2021

    work page 2021

  5. [5]

    Quantum two-block group algebra codes

    Hsiang-Ku Lin and Leonid P Pryadko. Quantum two-block group algebra codes. Physical Review A , 109(2):022407, 2024

    work page 2024

  6. [6]

    Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength

    Jean-Pierre Tillich and Gilles Z \'e mor. Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength. IEEE Transactions on Information Theory , 60(2):1193--1202, 2013

    work page 2013

  7. [7]

    High-threshold and low-overhead fault-tolerant quantum memory

    Sergey Bravyi, Andrew W Cross, Jay M Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J Yoder. High-threshold and low-overhead fault-tolerant quantum memory. Nature , 627(8005):778--782, 2024

    work page 2024

  8. [8]

    Good quantum LDPC codes with linear time decoders

    Irit Dinur, Min-Hsiu Hsieh, Ting-Chun Lin, and Thomas Vidick. Good quantum LDPC codes with linear time decoders. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing , STOC 2023, page 905–918, New York, NY, USA, 2023. Association for Computing Machinery

    work page 2023

Show all 48 references
  1. [9]

    Quantum LDPC codes of almost linear distance via homological products

    Louis Golowich and Venkatesan Guruswami . Quantum LDPC codes of almost linear distance via homological products. arXiv e-prints , page arXiv:2411.03646, November 2024

    2024 arXiv

  2. [10]

    Quantum Tanner codes

    Anthony Leverrier and Gilles Zémor. Quantum Tanner codes. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) , pages 872--883, 2022

    2022

  3. [11]

    Good quantum LDPC codes with linear time decoder from lossless expanders

    Ting-Chun Lin and Min-Hsiu Hsieh . Good quantum LDPC codes with linear time decoder from lossless expanders. arXiv e-prints , page arXiv:2203.03581, March 2022

    2022 arXiv

  4. [12]

    The Physics of (good) LDPC Codes I

    Tibor Rakovszky and Vedika Khemani . The Physics of (good) LDPC Codes I. Gauging and dualities . arXiv e-prints , page arXiv:2310.16032, October 2023

    2023 arXiv

  5. [13]

    The Physics of (good) LDPC Codes II

    Tibor Rakovszky and Vedika Khemani . The Physics of (good) LDPC Codes II. Product constructions . arXiv e-prints , page arXiv:2402.16831, February 2024

    2024 arXiv

  6. [14]

    Distance bounds for generalized bicycle codes

    Renyu Wang and Leonid P Pryadko. Distance bounds for generalized bicycle codes. Symmetry , 14(7):1348, 2022

    2022

  7. [15]

    Single-shot and two-shot decoding with generalized bicycle codes

    Hsiang-Ku Lin, Xingrui Liu, Pak Kau Lim, and Leonid P Pryadko. Single-shot and two-shot decoding with generalized bicycle codes. arXiv preprint arXiv:2502.19406 , 2025

    2025 arXiv

  8. [16]

    Small quantum codes from algebraic extensions of generalized bicycle codes (2024)

    N Koukoulekidis, F S imkovic IV, M Leib, and FRF Pereira. Small quantum codes from algebraic extensions of generalized bicycle codes (2024). arXiv preprint arXiv:2401.07583 , 2024

    2024 arXiv

  9. [17]

    Abelian and non-abelian quantum two-block codes

    Renyu Wang, Hsiang-Ku Lin, and Leonid P Pryadko. Abelian and non-abelian quantum two-block codes. In 2023 12th International Symposium on Topics in Coding (ISTC) , pages 1--5. IEEE, 2023

    2023

  10. [18]

    Matching generalized-bicycle codes to neutral atoms for low-overhead fault-tolerance

    Joshua Viszlai, Willers Yang, Sophia Fuhui Lin, Junyu Liu, Natalia Nottingham, Jonathan M Baker, and Frederic T Chong. Matching generalized-bicycle codes to neutral atoms for low-overhead fault-tolerance. arXiv preprint arXiv:2311.16980 , 2023

    2023 arXiv

  11. [19]

    Towards early fault tolerance on a 2 n array of qubits equipped with shuttling

    Adam Siegel, Armands Strikis, and Michael Fogarty. Towards early fault tolerance on a 2 n array of qubits equipped with shuttling. PRX Quantum , 5(4):040328, 2024

    2024

  12. [20]

    Design of additive quantum codes via the code-word-stabilized framework

    Alexey A Kovalev, Ilya Dumer, and Leonid P Pryadko. Design of additive quantum codes via the code-word-stabilized framework. Physical Review A—Atomic, Molecular, and Optical Physics , 84(6):062319, 2011

    2011

  13. [21]

    Improved quantum hypergraph-product LDPC codes

    Alexey A Kovalev and Leonid P Pryadko. Improved quantum hypergraph-product LDPC codes. In 2012 IEEE International Symposium on Information Theory Proceedings , pages 348--352. IEEE, 2012

    2012

  14. [22]

    Leveraging automorphisms of quantum codes for fault-tolerant quantum computation

    Markus Grassl and Martin Roetteler. Leveraging automorphisms of quantum codes for fault-tolerant quantum computation. In 2013 IEEE International Symposium on Information Theory , pages 534--538. IEEE, 2013

    2013

  15. [23]

    Fault-tolerant logical clifford gates from code automorphisms

    Hasan Sayginel, Stergios Koutsioumpas, Mark Webster, Abhishek Rajput, and Dan E Browne. Fault-tolerant logical clifford gates from code automorphisms. arXiv preprint arXiv:2409.18175 , 2024

    2024 arXiv

  16. [24]

    (2,2)-gb codes: Classification and comparison with weight-4 surface codes, 2025

    François Arnault, Philippe Gaborit, and Nicolas Saussay. (2,2)-gb codes: Classification and comparison with weight-4 surface codes, 2025

    2025

  17. [25]

    On the generalization of kitaev codes as generalized bicycle codes

    Fran c ois Arnault, Philippe Gaborit, and Nicolas Saussay. On the generalization of kitaev codes as generalized bicycle codes. arXiv preprint arXiv:2504.18360 , 2025

    2025 arXiv

  18. [26]

    Fundamentals of error-correcting codes

    W Cary Huffman and Vera Pless. Fundamentals of error-correcting codes . Cambridge university press, 2010

    2010

  19. [27]

    Quantum error correction via codes over gf (4)

    A Robert Calderbank, Eric M Rains, PM Shor, and Neil JA Sloane. Quantum error correction via codes over gf (4). IEEE Transactions on Information Theory , 44(4):1369--1387, 1998

    1998

  20. [28]

    Class of quantum error-correcting codes saturating the quantum hamming bound

    Daniel Gottesman. Class of quantum error-correcting codes saturating the quantum hamming bound. Physical Review A , 54(3):1862, 1996

    1996

  21. [29]

    Crespo, and Javier Garcia-Frías

    Patricio Fuentes, Josu Etxezarreta Martinez , Pedro M. Crespo, and Javier Garcia-Frías. Degeneracy and its impact on the decoding of sparse quantum codes. IEEE Access , 9:89093--89119, 2021

    2021

  22. [30]

    Asymptotically good quantum and locally testable classical LDPC codes

    Pavel Panteleev and Gleb Kalachev. Asymptotically good quantum and locally testable classical LDPC codes. In Proceedings of the 54th annual ACM SIGACT symposium on theory of computing , pages 375--388, 2022

    2022

  23. [31]

    Good quantum error-correcting codes exist

    A Robert Calderbank and Peter W Shor. Good quantum error-correcting codes exist. Physical Review A , 54(2):1098--1105, 1996

    1996

  24. [32]

    Multiple-particle interference and quantum error correction

    Andrew Steane. Multiple-particle interference and quantum error correction. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences , 452(1954):2551--2577, 1996

    1954

  25. [33]

    Sparse-graph codes for quantum error correction

    David JC MacKay, Graeme Mitchison, and Paul L McFadden. Sparse-graph codes for quantum error correction. IEEE Transactions on Information Theory , 50(10):2315--2330, 2004

    2004

  26. [34]

    Quantum twisted codes

    J \"u rgen Bierbrauer and Yves Edel. Quantum twisted codes. Journal of Combinatorial Designs , 8(3):174--188, 2000

    2000

  27. [35]

    Polynomial representation of additive cyclic codes and new quantum codes

    Reza Dastbasteh and Khalil Shivji. Polynomial representation of additive cyclic codes and new quantum codes. Advances in Mathematics of Communications , 19(1):49--68, 2025

    2025

  28. [36]

    Additive twisted codes: new distance bounds and infinite families of quantum codes

    Reza Dastbasteh and Petr Lison e k. Additive twisted codes: new distance bounds and infinite families of quantum codes. Designs, Codes and Cryptography , pages 1--38, 2025

    2025

  29. [37]

    Surface codes: Towards practical large-scale quantum computation

    Austin G Fowler, Matteo Mariantoni, John M Martinis, and Andrew N Cleland. Surface codes: Towards practical large-scale quantum computation. Physical Review A—Atomic, Molecular, and Optical Physics , 86(3):032324, 2012

    2012

  30. [38]

    Low-distance surface codes under realistic quantum noise

    Yu Tomita and Krysta M Svore. Low-distance surface codes under realistic quantum noise. Physical Review A , 90(6):062320, 2014

    2014

  31. [39]

    Topological quantum memory

    Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics , 43(9):4452--4505, 2002

    2002

  32. [40]

    Error-corrected hadamard gate simulated at the circuit level

    Gy \"o rgy P Geh \'e r, Campbell McLauchlan, Earl T Campbell, Alexandra E Moylett, and Ophelia Crawford. Error-corrected hadamard gate simulated at the circuit level. Quantum , 8:1394, 2024

    2024

  33. [41]

    Distance-preserving stabilizer measurements in hypergraph product codes

    Argyris Giannisis Manes and Jahan Claes. Distance-preserving stabilizer measurements in hypergraph product codes. Quantum , 9:1618, 2025

    2025

  34. [42]

    The magma algebra system i: The user language

    Wieb Bosma, John Cannon, and Catherine Playoust. The magma algebra system i: The user language. Journal of Symbolic Computation , 24(3-4):235--265, 1997

    1997

  35. [43]

    Decoding algorithms for surface codes

    Antonio deMarti iOlius, Patricio Fuentes, Rom \'a n Or \'u s, Pedro M Crespo, and Josu Etxezarreta Martinez. Decoding algorithms for surface codes. Quantum , 8:1498, 2024

    2024

  36. [44]

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

    Oscar Higgott. Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching. https://github.com/oscarhiggott/PyMatching, 2021

    2021

  37. [45]

    Michael A. Perlin. qLDPC . https://github.com/qLDPCOrg/qLDPC, 2023

    2023

  38. [46]

    LDPC: Python tools for low density parity check codes

    Joschka Roffe. LDPC: Python tools for low density parity check codes . https://pypi.org/project/ldpc/, 2022

    2022

  39. [47]

    Towards practical classical processing for the surface code

    Austin G Fowler, Adam C Whiteside, and Lloyd CL Hollenberg. Towards practical classical processing for the surface code. Physical review letters , 108(18):180501, 2012

    2012

  40. [48]

    An almost-linear time decoding algorithm for quantum LDPC codes under circuit-level noise

    Antonio deMarti iOlius , Imanol Etxezarreta Martinez , Joschka Roffe , and Josu Etxezarreta Martinez . An almost-linear time decoding algorithm for quantum LDPC codes under circuit-level noise . arXiv e-prints , page arXiv:2409.01440, September 2024

    2024 arXiv

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