pith. sign in

arxiv: 2409.11312 · v2 · pith:RVGHPPQ7new · submitted 2024-09-17 · 🪐 quant-ph · cs.IT· math.IT

Synchronizable hybrid subsystem codes

Theerapat Tansuwannont , Andrew Nemec This is my paper

Pith reviewed 2026-05-23 20:29 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords synchronizable codeshybrid codessubsystem codesquantum error correctionCSS constructioncyclic codessynchronization errorsgauge errors
0
0 comments X

The pith

From pairs of classical cyclic codes one can build synchronizable hybrid subsystem codes that correct both Pauli and synchronization errors while carrying classical and quantum information.

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

The paper shows how to extend the CSS construction from a pair of classical cyclic codes C and D satisfying Cperp subset C subset D to produce a single code that is simultaneously synchronizable, hybrid, and subsystem. This code corrects Pauli errors on the qubits and block synchronization errors, tolerates gauge errors by design, and transmits both classical bits and quantum qubits at once. Explicit trade-off relations are given among the number of correctable synchronization errors, the number of gauge qubits, and the number of logical classical bits. Separate general constructions for hybrid and hybrid-subsystem CSS codes from arbitrary classical codes are also supplied.

Core claim

Starting from classical cyclic codes C and D with Cperp subset C subset D, a CSS-type construction produces a synchronizable hybrid subsystem code whose stabilizer and gauge operators allow simultaneous correction of Pauli and synchronization errors, resilience to gauge errors, and transmission of both classical and quantum information, together with trade-off formulas relating the synchronization-correction distance, gauge-qubit count, and logical classical-bit count.

What carries the argument

The CSS-type extension of the classical cyclic code pair (C, D) with Cperp subset C subset D to a synchronizable hybrid subsystem code.

If this is right

  • The code corrects a number of synchronization errors determined by the minimum distances of C and D.
  • Gauge errors are automatically tolerated because of the subsystem structure.
  • The number of logical classical bits can be traded against the number of gauge qubits and the synchronization-correction capability.
  • General CSS constructions for hybrid and hybrid-subsystem codes follow from the same classical-code starting point.

Where Pith is reading between the lines

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

  • The same classical-code pairs might be reusable to add synchronization correction to existing hybrid or subsystem codes without rebuilding the entire code.
  • The trade-off formulas could be used to optimize code parameters for quantum communication links that must also maintain block alignment.
  • If the classical-code inclusion condition can be relaxed or replaced by other algebraic conditions, the construction might extend to non-cyclic base codes.

Load-bearing premise

Suitable pairs of classical cyclic codes C and D with the required inclusion chain must exist, and the CSS construction must extend to the hybrid-subsystem-synchronizable setting while keeping the claimed error-correction and transmission properties intact.

What would settle it

Exhibit a concrete pair of classical cyclic codes satisfying Cperp subset C subset D for which the resulting hybrid subsystem code either fails to correct the claimed number of synchronization errors or loses the ability to transmit both classical and quantum information simultaneously.

Figures

Figures reproduced from arXiv: 2409.11312 by Andrew Nemec, Theerapat Tansuwannont.

Figure 1
Figure 1. Figure 1: A circuit diagram describing the entire encoding procedure of a QSC. [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A Venn diagram displaying properties of the codes in a family of the synchronizable hybrid subsystem [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relationship between each of the codes from Theorem [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
read the original abstract

Quantum synchronizable codes are quantum error correcting codes that can correct not only Pauli errors but also errors in block synchronization. The code can be constructed from two classical cyclic codes $\mathcal{C}$, $\mathcal{D}$ satisfying $\mathcal{C}^{\perp} \subset \mathcal{C} \subset \mathcal{D}$ through the Calderbank-Shor-Steane (CSS) code construction. In this work, we establish connections between quantum synchronizable codes, subsystem codes, and hybrid codes constructed from the same pair of classical cyclic codes. We also propose a method to construct a synchronizable hybrid subsystem code which can correct both Pauli and synchronization errors, is resilient to gauge errors by virtue of the subsystem structure, and can transmit both classical and quantum information, all at the same time. The trade-offs between the number of synchronization errors that the code can correct, the number of gauge qubits, and the number of logical classical bits of the code are also established. In addition, we propose general methods to construct hybrid and hybrid subsystem codes of CSS type from classical codes, which cover relevant codes from our main construction.

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 / 2 minor

Summary. The manuscript proposes a construction of synchronizable hybrid subsystem codes starting from classical cyclic codes C and D satisfying C^perp ⊂ C ⊂ D, lifted via a CSS-type construction. The resulting codes are claimed to correct both Pauli and synchronization errors, remain resilient to gauge errors due to the subsystem structure, and simultaneously transmit classical and quantum information. Trade-offs are stated between the number of correctable synchronization errors, the number of gauge qubits, and the number of logical classical bits. General methods for constructing hybrid and hybrid-subsystem CSS codes from classical codes are also proposed.

Significance. If the construction and trade-offs hold, the work would provide a unified approach combining synchronization correction, gauge freedom, and hybrid classical-quantum transmission within a single code family derived from standard cyclic codes. This could be useful for designing quantum communication systems that must handle frame synchronization alongside mixed information types. The explicit connection between synchronizable, subsystem, and hybrid codes from the same classical pair is a positive feature.

major comments (2)
  1. [main construction (following the abstract description)] The central construction begins from the CSS lift of C^perp ⊂ C ⊂ D but the manuscript does not supply an explicit verification that the synchronization coset representatives remain compatible with the introduced gauge group and logical classical bits while preserving the claimed distances; this is load-bearing for the simultaneous correction claim.
  2. [trade-off statements] No explicit families of classical cyclic codes satisfying the inclusion relation, no resulting quantum code parameters, and no numerical checks of the trade-offs are provided, making it impossible to confirm that the gauge resilience and hybrid transmission do not reduce the synchronization or Pauli correction capability.
minor comments (2)
  1. Notation for the gauge group and the hybrid logical operators should be introduced with explicit definitions rather than relying on the classical-code inclusions alone.
  2. The general methods for hybrid and hybrid-subsystem CSS codes in the final section would benefit from a short comparison table relating them to the main synchronizable construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the identification of points where additional explicit verification and examples would improve the manuscript. We address each major comment below and will incorporate the suggested clarifications and concrete illustrations in a revised version.

read point-by-point responses

  1. Referee: The central construction begins from the CSS lift of C^perp ⊂ C ⊂ D but the manuscript does not supply an explicit verification that the synchronization coset representatives remain compatible with the introduced gauge group and logical classical bits while preserving the claimed distances; this is load-bearing for the simultaneous correction claim.

    Authors: We agree that an explicit verification of compatibility is necessary to fully substantiate the simultaneous correction claim. In the revised manuscript we will insert a new lemma (or subsection) that derives the conditions under which the synchronization coset representatives commute appropriately with the gauge operators and logical classical operators, and that confirms the minimum distances for Pauli and synchronization errors are unchanged by the hybrid-subsystem structure. revision: yes

  2. Referee: No explicit families of classical cyclic codes satisfying the inclusion relation, no resulting quantum code parameters, and no numerical checks of the trade-offs are provided, making it impossible to confirm that the gauge resilience and hybrid transmission do not reduce the synchronization or Pauli correction capability.

    Authors: We accept that concrete families and parameter tables are required to make the trade-off statements verifiable. The revision will add a dedicated section containing at least two explicit families of cyclic codes C and D satisfying C^perp ⊂ C ⊂ D, the resulting hybrid-subsystem code parameters (n, k_q, k_c, g, t_s, d), and a small numerical table or example demonstrating that the gauge and classical-logical dimensions can be varied without degrading the synchronization or Pauli distances below the claimed values. revision: yes

Circularity Check

0 steps flagged

No circularity: direct CSS construction from classical codes with stated inclusion

full rationale

The paper's derivation begins from the standard CSS construction applied to any classical cyclic codes C, D satisfying C^perp ⊂ C ⊂ D (a condition external to the quantum construction). It then extends this to hybrid subsystem synchronizable codes and derives trade-offs directly from the parameters of C and D. No step reduces a claimed prediction or property to a fitted input, self-definition, or load-bearing self-citation; the central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard CSS construction (a domain assumption) and on the existence of classical cyclic codes satisfying the stated inclusion relation; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Existence of classical cyclic codes C, D satisfying C^perp subset C subset D
    This nesting condition is the explicit starting point for the CSS construction of the quantum, hybrid, and subsystem codes.

pith-pipeline@v0.9.0 · 5716 in / 1398 out tokens · 39409 ms · 2026-05-23T20:29:04.462032+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

80 extracted references · 80 canonical work pages · 3 internal anchors

  1. [1]

    Balanced product quantum codes

    Nikolas P. Breuckmann and Jens N. Eberhardt. “Balanced product quantum codes”. IEEE Transactions on Information Theory67, 6653–6674 (2021)

    work page 2021

  2. [2]

    Fiber bundle codes: break- ing the n1/2polylog(n) barrier for quantum LDPC codes

    Matthew B Hastings, Jeongwan Haah, and Ryan O’Donnell. “Fiber bundle codes: break- ing the n1/2polylog(n) barrier for quantum LDPC codes”. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. Pages 1276–1288. (2021)

    work page 2021

  3. [3]

    Decodable quantum LDPC codes beyond the√n distance barrier using high-dimensional expanders

    Shai Evra, Tali Kaufman, and Gilles Zémor. “Decodable quantum LDPC codes beyond the√n distance barrier using high-dimensional expanders”. SIAM Journal on Computing 0, FOCS20–276 (2022)

    work page 2022

  4. [4]

    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)

    work page 2022

  5. [5]

    Quantum LDPC codes with almost linear minimum distance

    Pavel Panteleev and Gleb Kalachev. “Quantum LDPC codes with almost linear minimum distance”. IEEE Transactions on Information Theory68, 213–229 (2022)

    work page 2022

  6. [6]

    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)

    work page 2022

  7. [7]

    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. Pages 905–918. (2023)

    work page 2023

  8. [8]

    Constant overhead quantum fault tolerance with quantum expander codes

    Omar Fawzi, Antoine Grospellier, and Anthony Leverrier. “Constant overhead quantum fault tolerance with quantum expander codes”. Commun. ACM64, 106–114 (2020)

    work page 2020

  9. [9]

    Low- overhead fault-tolerant quantum computing using long-range connectivity

    Lawrence Z. Cohen, Isaac H. Kim, Stephen D. Bartlett, and Benjamin J. Brown. “Low- overhead fault-tolerant quantum computing using long-range connectivity”. Science Ad- vances 8, eabn1717 (2022). 47

    work page 2022

  10. [10]

    Constant-overhead quantum error correction with thin planar connectivity

    Maxime A. Tremblay, Nicolas Delfosse, and Michael E. Beverland. “Constant-overhead quantum error correction with thin planar connectivity”. Phys. Rev. Lett. 129, 050504 (2022)

    work page 2022

  11. [11]

    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, 778–782 (2024)

    work page 2024

  12. [12]

    Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays

    Qian Xu, J Pablo Bonilla Ataides, Christopher A Pattison, Nithin Raveendran, Dolev Bluvstein, Jonathan Wurtz, Bane Vasić, Mikhail D Lukin, Liang Jiang, and Hengyun Zhou. “Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays”. Nature Physics20, 1084–1090 (2024)

    work page 2024

  13. [13]

    Time-efficient constant-space-overhead fault- tolerant quantum computation

    Hayata Yamasaki and Masato Koashi. “Time-efficient constant-space-overhead fault- tolerant quantum computation”. Nature Physics20, 247–253 (2024)

    work page 2024

  14. [14]

    Algorithmic fault tolerance for fast quantum computing

    Hengyun Zhou, Chen Zhao, Madelyn Cain, Dolev Bluvstein, Casey Duckering, Hong-Ye Hu, Sheng-Tao Wang, Aleksander Kubica, and Mikhail D. Lukin. “Algorithmic fault tolerance for fast quantum computing” (2024). arXiv:2406.17653

    work page arXiv 2024

  15. [15]

    Fault- tolerant control of an error-corrected qubit

    Laird Egan, Dripto M Debroy, Crystal Noel, Andrew Risinger, Daiwei Zhu, Debopriyo Biswas, Michael Newman, Muyuan Li, Kenneth R Brown, Marko Cetina, et al. “Fault- tolerant control of an error-corrected qubit”. Nature598, 281–286 (2021)

    work page 2021

  16. [16]

    Entanglinglogicalqubitswithlatticesurgery

    Alexander Erhard, Hendrik Poulsen Nautrup, Michael Meth, Lukas Postler, Roman Stricker, Martin Stadler, Vlad Negnevitsky, Martin Ringbauer, Philipp Schindler, Hans J Briegel, etal. “Entanglinglogicalqubitswithlatticesurgery”. Nature589, 220–224(2021)

    work page 2021

  17. [17]

    Realization of real-time fault- tolerant quantum error correction

    C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz. “Realization of real-time fault- tolerant quantum error correction”. Phys. Rev. X11, 041058 (2021)

    work page 2021

  18. [18]

    Demonstration of fault-tolerant universal quantum gate operations

    Lukas Postler, Sascha Heuβen, Ivan Pogorelov, Manuel Rispler, Thomas Feldker, Michael Meth, Christian D Marciniak, Roman Stricker, Martin Ringbauer, Rainer Blatt, et al. “Demonstration of fault-tolerant universal quantum gate operations”. Nature605, 675– 680 (2022)

    work page 2022

  19. [19]

    Ryan-Anderson, N

    C. Ryan-Anderson, N. C. Brown, M. S. Allman, B. Arkin, G. Asa-Attuah, C. Baldwin, J. Berg, J. G. Bohnet, S. Braxton, N. Burdick, et al. “Implementing fault-tolerant entan- gling gates on the five-qubit code and the color code” (2022). arXiv:2208.01863

    work page arXiv 2022

  20. [20]

    Suppressing quantum errors by scaling a surface code logical qubit

    Rajeev Acharya, Igor Aleiner, Richard Allen, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, et al. “Suppressing quantum errors by scaling a surface code logical qubit”. Nature614, 676–681 (2023)

    work page 2023

  21. [21]

    Real-time quantum error correction beyond break-even

    VV Sivak, Alec Eickbusch, Baptiste Royer, Shraddha Singh, Ioannis Tsioutsios, Suhas Ganjam, Alessandro Miano, BL Brock, AZ Ding, Luigi Frunzio, et al. “Real-time quantum error correction beyond break-even”. Nature616, 50–55 (2023). 48

    work page 2023

  22. [22]

    Logical quantum processor based on reconfigurable atom arrays

    Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, et al. “Logical quantum processor based on reconfigurable atom arrays”. Nature626, 58–65 (2024)

    work page 2024

  23. [23]

    Demonstration of logical qubits and repeated error correction with better-than-physical error rates

    M. P. da Silva, C. Ryan-Anderson, J. M. Bello-Rivas, A. Chernoguzov, J. M. Dreiling, C. Foltz, F. Frachon, J. P. Gaebler, T. M. Gatterman, L. Grans-Samuelsson, D. Hayes, N. Hewitt, et al. “Demonstration of logical qubits and repeated error correction with better-than-physical error rates” (2024). arXiv:2404.02280

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  24. [24]

    Acharya, L

    Rajeev Acharya, Laleh Aghababaie-Beni, Igor Aleiner, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Nikita Astrakhantsev, Juan Atalaya, et al. “Quantum error correction below the surface code threshold” (2024). arXiv:2408.13687

    work page arXiv 2024

  25. [25]

    Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects

    C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim. “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects”. Phys. Rev. A89, 022317 (2014)

    work page 2014

  26. [26]

    Modular entanglement of atomic qubits using photons and phonons

    David Hucul, Ismail V Inlek, Grahame Vittorini, Clayton Crocker, Shantanu Debnath, Susan M Clark, and Christopher Monroe. “Modular entanglement of atomic qubits using photons and phonons”. Nature Physics11, 37–42 (2015)

    work page 2015

  27. [27]

    Co-designing a scalable quantum computer with trapped atomic ions

    Kenneth R Brown, Jungsang Kim, and Christopher Monroe. “Co-designing a scalable quantum computer with trapped atomic ions”. npj Quantum Information2, 1–10 (2016)

    work page 2016

  28. [28]

    Network architecture for a topological quantum computer in silicon

    Brandon Buonacorsi, Zhenyu Cai, Eduardo B Ramirez, Kyle S Willick, Sean M Walker, Jiahao Li, Benjamin D Shaw, Xiaosi Xu, Simon C Benjamin, and Jonathan Baugh. “Network architecture for a topological quantum computer in silicon”. Quantum Science and Technology4, 025003 (2019)

    work page 2019

  29. [29]

    Interleaving: Modular architectures for fault-tolerant photonic quantum computing

    Hector Bombin, Isaac H. Kim, Daniel Litinski, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Sam Roberts, and Terry Rudolph. “Interleaving: Modular architectures for fault-tolerant photonic quantum computing” (2021). arXiv:2103.08612

    work page arXiv 2021

  30. [30]

    En- tanglement across separate silicon dies in a modular superconducting qubit device

    Alysson Gold, JP Paquette, Anna Stockklauser, Matthew J Reagor, M Sohaib Alam, Andrew Bestwick, Nicolas Didier, Ani Nersisyan, Feyza Oruc, Armin Razavi, et al. “En- tanglement across separate silicon dies in a modular superconducting qubit device”. npj Quantum Information 7, 142 (2021)

    work page 2021

  31. [31]

    The future of quantum computing with superconducting qubits

    Sergey Bravyi, Oliver Dial, Jay M. Gambetta, Darío Gil, and Zaira Nazario. “The future of quantum computing with superconducting qubits”. Journal of Applied Physics132, 160902 (2022)

    work page 2022

  32. [32]

    Fusion-based quantum computation

    Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, et al. “Fusion-based quantum computation”. Nature Communications14, 912 (2023). 49

    work page 2023

  33. [33]

    Low-loss interconnects for modular super- conducting quantum processors

    Jingjing Niu, Libo Zhang, Yang Liu, Jiawei Qiu, Wenhui Huang, Jiaxiang Huang, Hao Jia, Jiawei Liu, Ziyu Tao, Weiwei Wei, et al. “Low-loss interconnects for modular super- conducting quantum processors”. Nature Electronics6, 235–241 (2023)

    work page 2023

  34. [34]

    Quantum low-density parity-check codes for modular architectures

    Armands Strikis and Lucas Berent. “Quantum low-density parity-check codes for modular architectures”. PRX Quantum4, 020321 (2023)

    work page 2023

  35. [35]

    Fault- tolerant connection of error-corrected qubits with noisy links

    Joshua Ramette, Josiah Sinclair, Nikolas P Breuckmann, and Vladan Vuletić. “Fault- tolerant connection of error-corrected qubits with noisy links”. npj Quantum Information 10, 58 (2024)

    work page 2024

  36. [36]

    Digital communications: fundamentals and applications

    Bernard Sklar. “Digital communications: fundamentals and applications”. Pearson. (2021)

    work page 2021

  37. [37]

    Synchronization of digital telecommunications networks

    Stefano Bregni. “Synchronization of digital telecommunications networks”. John Wiley & Sons, Inc. USA (2002)

    work page 2002

  38. [38]

    Block synchronization for quantum information

    Yuichiro Fujiwara. “Block synchronization for quantum information”. Phys. Rev. A87, 022344 (2013)

    work page 2013

  39. [39]

    Good quantum error-correcting codes exist

    A. R. Calderbank and Peter W. Shor. “Good quantum error-correcting codes exist”. Phys. Rev. A 54, 1098–1105 (1996)

    work page 1996

  40. [40]

    Simple quantum error-correcting codes

    A. M. Steane. “Simple quantum error-correcting codes”. Phys. Rev. A 54, 4741– 4751 (1996)

    work page 1996

  41. [41]

    Algebraic techniques in designing quantum synchronizable codes

    Yuichiro Fujiwara, Vladimir D. Tonchev, and Tony W. H. Wong. “Algebraic techniques in designing quantum synchronizable codes”. Phys. Rev. A88, 012318 (2013)

    work page 2013

  42. [42]

    Quantum synchronizable codes from finite geometries

    Yuichiro Fujiwara and Peter Vandendriessche. “Quantum synchronizable codes from finite geometries”. IEEE Transactions on Information Theory60, 7345–7354 (2014)

    work page 2014

  43. [43]

    Quantum synchronizable codes from quadratic residue codes and their supercodes

    Yixuan Xie, Jinhong Yuan, and Yuichiro Fujiwara. “Quantum synchronizable codes from quadratic residue codes and their supercodes”. In 2014 IEEE Information Theory Work- shop (ITW 2014). Pages 172–176. (2014)

    work page 2014

  44. [44]

    q-ary chain-containing quantum synchroniz- able codes

    Yixuan Xie, Lei Yang, and Jinhong Yuan. “q-ary chain-containing quantum synchroniz- able codes”. IEEE Communications Letters20, 414–417 (2016)

    work page 2016

  45. [45]

    Algebraic quantum synchronizable codes

    K. Guenda, G. G. La Guardia, and T. A. Gulliver. “Algebraic quantum synchronizable codes”. J. Appl. Math. Comput.55, 393–407 (2017)

    work page 2017

  46. [46]

    Non-binary quantum synchronizable codes from repeated-root cyclic codes

    Lan Luo and Zhi Ma. “Non-binary quantum synchronizable codes from repeated-root cyclic codes”. IEEE Transactions on Information Theory64, 1461–1470 (2018)

    work page 2018

  47. [47]

    Unified and generalized approach to quantum error correction

    David Kribs, Raymond Laflamme, and David Poulin. “Unified and generalized approach to quantum error correction”. Phys. Rev. Lett.94, 180501 (2005)

    work page 2005

  48. [48]

    Stabilizer formalism for operator quantum error correction

    David Poulin. “Stabilizer formalism for operator quantum error correction”. Phys. Rev. Lett. 95, 230504 (2005)

    work page 2005

  49. [49]

    Operator quantum error-correcting subsystems for self-correcting quantum memories

    Dave Bacon. “Operator quantum error-correcting subsystems for self-correcting quantum memories”. Phys. Rev. A73, 012340 (2006). 50

    work page 2006

  50. [50]

    Classical enhancement of quantum- error-correcting codes

    Isaac Kremsky, Min-Hsiu Hsieh, and Todd A. Brun. “Classical enhancement of quantum- error-correcting codes”. Phys. Rev. A78, 012341 (2008)

    work page 2008

  51. [51]

    Codes for simultaneous transmission of quantum and classical information

    Markus Grassl, Sirui Lu, and Bei Zeng. “Codes for simultaneous transmission of quantum and classical information”. In 2017 IEEE International Symposium on Information Theory (ISIT). Pages 1718–1722. (2017)

    work page 2017

  52. [52]

    Hybrid codes

    Andrew Nemec and Andreas Klappenecker. “Hybrid codes”. In 2018 IEEE International Symposium on Information Theory (ISIT). Pages 796–800. (2018)

    work page 2018

  53. [53]

    Infinite families of quantum-classical hybrid codes

    Andrew Nemec and Andreas Klappenecker. “Infinite families of quantum-classical hybrid codes”. IEEE Transactions on Information Theory67, 2847–2856 (2021)

    work page 2021

  54. [54]

    Encoding classical information in gauge subsystems of quantum codes

    Andrew Nemec and Andreas Klappenecker. “Encoding classical information in gauge subsystems of quantum codes”. International Journal of Quantum Information 20, 2150041 (2022)

    work page 2022

  55. [55]

    Stabilizer formalism for operator algebra quantum error correction

    Guillaume Dauphinais, David W. Kribs, and Michael Vasmer. “Stabilizer formalism for operator algebra quantum error correction”. Quantum8, 1261 (2024)

    work page 2024

  56. [56]

    Stabilizer Codes and Quantum Error Correction

    Daniel Gottesman. “Stabilizer Codes and Quantum Error Correction”. PhD thesis. Cali- fornia Institute of Technology. (1997)

    work page 1997

  57. [57]

    Good quantum error-correcting codes exist

    A Robert Calderbank and Peter W Shor. “Good quantum error-correcting codes exist”. Physical Review A54, 1098 (1996)

    work page 1996

  58. [58]

    Multiple-particle interference and quantum error correction

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

    work page 1996

  59. [59]

    Subsystem codes

    Salah A. Aly, Andreas Klappenecker, and Pradeep Kiran Sarvepalli. “Subsystem codes”. In Proceedings of the 44th Annual Allerton Conference on Communication, Control, and Computing (Allerton). Pages 528–535. (2006)

    work page 2006

  60. [60]

    Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat’s lemma

    Michael Liaofan Liu, Nathanan Tantivasadakarn, and Victor V. Albert. “Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat’s lemma”. Quantum8, 1403 (2024)

    work page 2024

  61. [61]

    The Heisenberg Representation of Quantum Computers

    Daniel Gottesman. “The Heisenberg representation of quantum computers” (1998). arXiv:quant-ph/9807006

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  62. [62]

    Cyclic quantum error–correcting codes and quantum shift registers

    Markus Grassl and Thomas Beth. “Cyclic quantum error–correcting codes and quantum shift registers”. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences456, 2689–2706 (2000)

    work page 2000

  63. [63]

    Quantum-shift-register circuits

    Mark M. Wilde. “Quantum-shift-register circuits”. Phys. Rev. A79, 062325 (2009)

    work page 2009

  64. [64]

    Universal fault-tolerant quantum computation with only transversal gates and error correction

    Adam Paetznick and Ben W. Reichardt. “Universal fault-tolerant quantum computation with only transversal gates and error correction”. Phys. Rev. Lett.111, 090505 (2013)

    work page 2013

  65. [65]

    Fault-tolerant con- version between the Steane and Reed-Muller quantum codes

    Jonas T. Anderson, Guillaume Duclos-Cianci, and David Poulin. “Fault-tolerant con- version between the Steane and Reed-Muller quantum codes”. Phys. Rev. Lett.113, 080501 (2014). 51

    work page 2014

  66. [66]

    Fault-tolerant quantum computation

    P.W. Shor. “Fault-tolerant quantum computation”. In Proceedings of 37th Conference on Foundations of Computer Science. Pages 56–65. (1996)

    work page 1996

  67. [67]

    Effective fault-tolerant quantum computation with slow measurements

    David P DiVincenzo and Panos Aliferis. “Effective fault-tolerant quantum computation with slow measurements”. Physical Review Letters98, 020501 (2007)

    work page 2007

  68. [68]

    Adaptive syndrome measurements for Shor-style error correction

    Theerapat Tansuwannont, Balint Pato, and Kenneth R. Brown. “Adaptive syndrome measurements for Shor-style error correction”. Quantum7, 1075 (2023)

    work page 2023

  69. [69]

    Scalable quantum computing in the presence of large detected-error rates

    Emanuel Knill. “Scalable quantum computing in the presence of large detected-error rates”. Physical Review A71, 042322 (2005)

    work page 2005

  70. [70]

    Quantum error correction with only two extra qubits

    Rui Chao and Ben W Reichardt. “Quantum error correction with only two extra qubits”. Physical Review Letters121, 050502 (2018)

    work page 2018

  71. [71]

    Flag fault-tolerant error correction for any stabilizer code

    Rui Chao and Ben W Reichardt. “Flag fault-tolerant error correction for any stabilizer code”. PRX Quantum1, 010302 (2020)

    work page 2020

  72. [72]

    Flag fault- tolerant error correction, measurement, and quantum computation for cyclic Calderbank- Shor-Steane codes

    Theerapat Tansuwannont, Christopher Chamberland, and Debbie Leung. “Flag fault- tolerant error correction, measurement, and quantum computation for cyclic Calderbank- Shor-Steane codes”. Physical Review A101, 012342 (2020)

    work page 2020

  73. [73]

    Low-depth flagged syndrome extraction for Calderbank-Shor-Steane codes of distance 3

    Chao Du, Zhi Ma, Yiting Liu, Hong Wang, and Qianheng Duan. “Low-depth flagged syndrome extraction for Calderbank-Shor-Steane codes of distance 3”. IEEE Transactions on Information Theory70, 6326–6349 (2024)

    work page 2024

  74. [74]

    Flag gadgets based on classical codes, 2024

    Benjamin Anker and Milad Marvian. “Flag gadgets based on classical codes” (2024). arXiv:2212.10738

    work page arXiv 2024

  75. [75]

    Fault-tolerant quantum computation by anyons

    A.Yu. Kitaev. “Fault-tolerant quantum computation by anyons”. Annals of Physics303, 2–30 (2003)

    work page 2003

  76. [76]

    Quantum codes on a lattice with boundary

    S. B. Bravyi and A. Yu. Kitaev. “Quantum codes on a lattice with boundary” (1998). arXiv:quant-ph/9811052

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  77. [77]

    Topological quantum memory

    Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. “Topological quantum memory”. Journal of Mathematical Physics43, 4452–4505 (2002)

    work page 2002

  78. [78]

    Topological quantum distillation

    Hector Bombin and Miguel Angel Martin-Delgado. “Topological quantum distillation”. Physical Review Letters97, 180501 (2006)

    work page 2006

  79. [79]

    Correcting quantum errors with entan- glement

    Todd Brun, Igor Devetak, and Min-Hsiu Hsieh. “Correcting quantum errors with entan- glement”. Science314, 436–439 (2006)

    work page 2006

  80. [80]

    Logical operators of quantum codes

    Mark M. Wilde. “Logical operators of quantum codes”. Phys. Rev. A79, 062322 (2009). 52

    work page 2009