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arxiv: 2603.08711 · v2 · submitted 2026-03-09 · 🪐 quant-ph · cond-mat.str-el

Recognition: 2 theorem links

· Lean Theorem

Coupled-Layer Construction of Quantum Product Codes

Shuyu Zhang , Tzu-Chieh Wei , Nathanan Tantivasadakarn
Authors on Pith no claims yet

Pith reviewed 2026-05-15 14:21 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords quantum product codescoupled-layer constructionanyon condensationqLDPC codesstabilizer codestensor productbalanced productCSS codes
0
0 comments X

The pith

Quantum product codes arise by stacking one code and condensing excitations set by the other's checks.

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

The paper establishes that tensor and balanced product codes have a physical construction via coupled layers. It shows this by stacking copies of one code and condensing excitations patterned after the checks of the second code. This unifies algebraic constructions with anyon condensation methods used in topological phases. It also links product codes to gauging in concatenated codes to make them low-density parity-check. This approach works for both classical and quantum CSS codes.

Core claim

Tensor and balanced product codes admit a coupled-layer construction obtained by stacking one constituent code and condensing excitations in the pattern given by the checks of the other code. This construction also connects to concatenated codes, where the tensor product is recovered by gauging large-weight logicals to make the code qLDPC. The method works for both classical and quantum CSS codes and extends known anyon condensation techniques to non-topological codes.

What carries the argument

Coupled-layer construction via condensation of excitations patterned by the checks of the second code.

If this is right

  • The tensor product code is recovered by gauging large-weight logical operators in concatenated codes.
  • The construction unifies with anyon condensation mechanisms for higher-dimensional topological phases.
  • The framework applies equally to classical or quantum CSS input codes.
  • The same layered approach extends to non-topological codes.

Where Pith is reading between the lines

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

  • Different choices of condensation patterns might generate new families of qLDPC codes beyond standard products.
  • Techniques from anyon condensation in topological order could be imported to design stabilizers for error-correcting codes.
  • Numerical checks on small examples would confirm that distance and encoding rate are preserved under the condensation step.

Load-bearing premise

The condensation pattern defined by the checks of the second code produces a valid stabilizer code without introducing new logical operators or violating the CSS structure.

What would settle it

Explicitly compute the stabilizers and logical operators after condensation on small input codes and verify whether the resulting code distance, rate, and generators exactly match the known tensor or balanced product code.

Figures

Figures reproduced from arXiv: 2603.08711 by Nathanan Tantivasadakarn, Shuyu Zhang, Tzu-Chieh Wei.

Figure 2
Figure 2. Figure 2: FIG. 2: The stabilizers of the 2D toric code [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The coupled layer construction from the chain [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (Left) Stabilizers of Shor’s code, obtained by [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1: Left: The stacking configuration of 2D TC [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Left: a unit cell of the RBH cluster state. The gray layers are three stacks of 2D TC labeled by [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Left: a unit cell of the RBH cluster state. The thick lines are copies of the 1D cluster state. The blue and red [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: A stabilizer of CSS [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The term [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Stabilizers after the code switching. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The Double cover [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The left quotient graph defining [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Examples of the code switching terms [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Examples of [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Left: a logical operator on the NM model. Right: fractal logical in NM model which is gauged to obtain the color [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Left two figures are [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Left [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: When CSS [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Schematics of condensation. The red strings are 1D domain walls of the global [PITH_FULL_IMAGE:figures/full_fig_p030_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: The tensor product between the extended complex CSS [PITH_FULL_IMAGE:figures/full_fig_p031_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: The stacked system [PITH_FULL_IMAGE:figures/full_fig_p032_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Stabilizers of the system after the first code switching. After the top-left [PITH_FULL_IMAGE:figures/full_fig_p032_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: From left to right [PITH_FULL_IMAGE:figures/full_fig_p033_20.png] view at source ↗
read the original abstract

Product codes are a class of quantum error correcting codes built from two or more constituent codes. They have recently gained prominence for a breakthrough yielding quantum low-density parity-check (qLDPC) codes with favorable scaling of both code distance and encoding rate. However, despite its powerful algebraic formulation, the physical mechanism for assembling a general product code from its constituents remains unclear. In this letter, we show that the tensor and balanced product codes admit an intuitive coupled-layer construction by taking a stack of one code and condensing a set of excitations in the pattern given by the checks of the other code. We also make a connection to concatenated codes by showing that the tensor product code can be obtained by gauging large-weight logicals in concatenated codes, making them qLDPC. Our framework accommodates both classical or quantum CSS input codes, unifies known physical mechanisms for constructing higher dimensional topological phases via anyon condensation, and naturally extends to non-topological codes.

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 paper claims that tensor and balanced product quantum codes admit a coupled-layer physical construction obtained by stacking one CSS code and condensing excitations whose pattern is dictated by the checks of the second code; it further asserts that the tensor product can be recovered by gauging large-weight logical operators in a concatenated code, thereby producing a qLDPC code, and that the framework unifies anyon condensation in topological phases while applying to both classical and quantum CSS inputs.

Significance. If the central identification between the condensation procedure and the algebraic product construction holds with preserved distance and rate, the work supplies a concrete physical mechanism for a class of qLDPC codes that have recently achieved favorable scaling. The explicit link to gauging in concatenated codes and the extension beyond topological codes are potentially useful for code design and for connecting algebraic and Hamiltonian-based constructions.

major comments (2)
  1. [coupled-layer construction (presumably §2–3)] The central claim that condensation of check-defined excitations exactly reproduces the tensor-product stabilizer group (including logical operators and CSS bipartition) is load-bearing. An explicit algebraic verification is required showing that the selected excitations commute with all original stabilizers and that no extraneous logical operators appear; this should be stated for general CSS inputs, not only topological examples.
  2. [distance and rate analysis] The distance and rate preservation under condensation must be demonstrated. The manuscript should contain a calculation (or reference to one) that the minimum weight of nontrivial logical operators in the condensed code matches the product-code distance formula, at least for a small non-trivial example such as two repetition codes or two surface codes.
minor comments (2)
  1. [introduction and framework] The notation distinguishing 'excitations' in the algebraic (non-topological) case from anyonic excitations should be clarified; a short table comparing the two settings would help readers.
  2. [gauging connection] Figure captions and the description of the gauging step should explicitly state which logical operators are being gauged and why the resulting code remains CSS.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and positive evaluation of our work. We address the major comments below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [coupled-layer construction (presumably §2–3)] The central claim that condensation of check-defined excitations exactly reproduces the tensor-product stabilizer group (including logical operators and CSS bipartition) is load-bearing. An explicit algebraic verification is required showing that the selected excitations commute with all original stabilizers and that no extraneous logical operators appear; this should be stated for general CSS inputs, not only topological examples.

    Authors: We agree that providing an explicit algebraic verification for general CSS inputs is essential for rigor. In the revised manuscript, we will include a new appendix that derives the commutation relations between the condensed excitations and the original stabilizers for arbitrary CSS codes. We will also prove that the resulting stabilizer group coincides exactly with that of the tensor product code, with no additional logical operators introduced. This will cover both the stabilizer generators and the logical operators, as well as the CSS bipartition. revision: yes

  2. Referee: [distance and rate analysis] The distance and rate preservation under condensation must be demonstrated. The manuscript should contain a calculation (or reference to one) that the minimum weight of nontrivial logical operators in the condensed code matches the product-code distance formula, at least for a small non-trivial example such as two repetition codes or two surface codes.

    Authors: We will add an explicit calculation in the revised version demonstrating distance preservation. Specifically, we will work through the example of two repetition codes, computing the logical operators after condensation and verifying that their minimum weights match the product code distance formula. We will also include a brief discussion for two surface codes to illustrate the general case. This addition will confirm that the condensation procedure preserves both distance and rate as required. revision: yes

Circularity Check

0 steps flagged

Coupled-layer construction is presented as a direct physical mechanism without reducing to self-definition or fitted inputs

full rationale

The paper advances a coupled-layer construction in which a stack of one code is condensed according to the checks of the second code, claiming this reproduces the tensor and balanced product codes. This is framed as an intuitive physical mechanism that also connects to gauging in concatenated codes. No equations or derivations in the provided text reduce the claimed equivalence to a tautology by construction, nor do they rely on self-citations for load-bearing uniqueness theorems or ansatzes. The argument remains self-contained against the algebraic product-code definition and does not invoke fitted parameters renamed as predictions. A minor self-citation risk exists in any unification claim, but it is not load-bearing for the central construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated. The construction implicitly relies on standard anyon-condensation rules from topological order.

axioms (1)
  • domain assumption Condensation of excitations according to a set of checks yields a valid stabilizer code
    Invoked in the description of the coupled-layer construction

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Forward citations

Cited by 1 Pith paper

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  1. Topological subsystem bivariate bicycle codes with four-qubit check operators

    quant-ph 2026-05 unverdicted novelty 8.0

    Subsystem bivariate bicycle codes achieve high-rate BB logical qubits with local four-qubit gauge checks, yielding examples such as [[108,12,6]] that outperform surface-code alternatives.

Reference graph

Works this paper leans on

81 extracted references · 81 canonical work pages · cited by 1 Pith paper · 4 internal anchors

  1. [1]

    gauging out

    foliated 1-form symmetry. In comparison to the construction of 3D TC by condensing pairs of anyons in adjacent layers of a stack of 2D TCs, stacking on a 2D layout allows bothmandeanyons to form strings, which commute with each other. Notably, the 2D TC does not have local metachecks since the stabilizers of 2D TC only has global redundancies. This makes ...

    work page

  2. [2]

    P. W. Shor, Scheme for reducing decoherence in quantum com- puter memory, Phys. Rev. A52, R2493 (1995)

    work page 1995

  3. [3]

    S. H. Simon,Topological Quantum(Oxford University Press, 2023)

    work page 2023

  4. [4]

    K. J. Satzinger, Y .-J. Liu, A. Smith, C. Knapp, M. Newman, C. Jones, Z. Chen, C. Quintana, X. Mi, A. Dunsworth, C. Gid- ney, I. Aleiner, F. Arute, K. Arya, J. Atalaya, R. Babbush, J. C. Bardin, R. Barends, J. Basso, A. Bengtsson, A. Bilmes, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bush- nell, B. Chiaro, R. Collins, W. Courtney, S. Demura,...

    work page 2021

  5. [5]

    Krinner, N

    S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann, et al., Realizing repeated quantum error correction in a distance-three surface code, Nature605, 669 (2022)

    work page 2022

  6. [6]

    Het ´enyi and J

    B. Het ´enyi and J. R. Wootton, Creating entangled logical qubits in the heavy-hex lattice with topological codes, PRX Quantum 5, 040334 (2024)

    work page 2024

  7. [7]

    Iqbal, N

    M. Iqbal, N. Tantivasadakarn, T. M. Gatterman, J. A. Ger- ber, K. Gilmore, D. Gresh, A. Hankin, N. Hewitt, C. V . Horst, M. Matheny, T. Mengle, B. Neyenhuis, A. Vishwanath, M. Foss-Feig, R. Verresen, and H. Dreyer, Topological order from measurements and feed-forward on a trapped ion quan- tum computer, Communications Physics7, 205 (2024)

    work page 2024

  8. [8]

    Iqbal, N

    M. Iqbal, N. Tantivasadakarn, R. Verresen, S. L. Campbell, J. M. Dreiling, C. Figgatt, J. P. Gaebler, J. Johansen, M. Mills, S. A. Moses, J. M. Pino, A. Ransford, M. Rowe, P. Siegfried, R. P. Stutz, M. Foss-Feig, A. Vishwanath, and H. Dreyer, Non- abelian topological order and anyons on a trapped-ion proces- sor, Nature626, 505 (2024)

    work page 2024

  9. [9]

    Bluvstein, S

    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, Nature626, 58 (2024)

    work page 2024

  10. [10]

    Iqbal, A

    M. Iqbal, A. Lyons, C. F. B. Lo, N. Tantivasadakarn, J. Dreiling, C. Foltz, T. M. Gatterman, D. Gresh, N. Hewitt, C. A. Holliman, J. Johansen, B. Neyenhuis, Y . Matsuoka, M. Mills, S. A. Moses, P. Siegfried, A. Vishwanath, R. Verresen, and H. Dreyer, Qutrit toric code and parafermions in trapped ions, Nature Communi- cations16, 6301 (2025)

    work page 2025

  11. [11]

    Sales Rodriguez, J

    P. Sales Rodriguez, J. M. Robinson, P. N. Jepsen, Z. He, C. Duckering, C. Zhao, K.-H. Wu, J. Campo, K. Bagnall, M. Kwon,et al., Experimental demonstration of logical magic state distillation, Nature645, 620 (2025)

    work page 2025

  12. [12]

    C. F. B. Lo, A. Lyons, D. Gresh, M. Mills, P. E. Siegfried, M. D. Urmey, N. Tantivasadakarn, H. Dreyer, A. Vishwanath, R. Verresen,et al., Universal topological gates from braid- ing and fusing anyons on quantum hardware, arXiv preprint arXiv:2601.20956 (2026)

    work page arXiv 2026

  13. [13]

    G. Q. AI and Collaborators, Quantum error correction below the surface code threshold, Nature636, 527 (2024)

    work page 2024

  14. [14]

    B. L. Brock, S. Singh, A. Eickbusch, V . V . Sivak, A. Z. Ding, L. Frunzio, S. M. Girvin, and M. H. Devoret, Quantum er- ror correction of qudits beyond break-even, Nature641, 612 (2025)

    work page 2025

  15. [15]

    Bravyi and B

    S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New Journal of Physics11, 043029 (2009)

    work page 2009

  16. [16]

    Bravyi, D

    S. Bravyi, D. Poulin, and B. Terhal, Tradeoffs for reliable quan- tum information storage in 2d systems, Physical review letters 104, 050503 (2010)

    work page 2010

  17. [17]

    S. T. Flammia, J. Haah, M. J. Kastoryano, and I. H. Kim, Limits on the storage of quantum information in a volume of space, Quantum1, 4 (2017)

    work page 2017

  18. [18]

    Ungauging quantum error-correcting codes

    A. Kubica and B. Yoshida, Ungauging quantum error- correcting codes, arXiv preprint arXiv:1805.01836 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  19. [19]

    Rakovszky and V

    T. Rakovszky and V . Khemani, The physics of (good) ldpc codes i. gauging and dualities, arXiv preprint arXiv:2310.16032 (2023)

    work page arXiv 2023

  20. [20]

    Rakovszky and V

    T. Rakovszky and V . Khemani, The physics of (good) ldpc codes ii. product constructions, arXiv preprint arXiv:2402.16831 (2024)

    work page arXiv 2024

  21. [21]

    De Roeck, V

    W. De Roeck, V . Khemani, Y . Li, N. O’Dea, and T. Rakovszky, Ldpc stabilizer codes as gapped quantum phases: stability un- der graph-local perturbations, arXiv preprint arXiv:2411.02384 (2024)

    work page arXiv 2024

  22. [22]

    Placke, T

    B. Placke, T. Rakovszky, N. P. Breuckmann, and V . Khemani, Topological quantum spin glass order and its realization in qldpc codes, arXiv preprint arXiv:2412.13248 (2024)

    work page arXiv 2024

  23. [23]

    Placke, G

    B. Placke, G. M. Sommers, N. P. Breuckmann, T. Rakovszky, and V . Khemani, Expansion creates spin-glass order in finite- connectivity models: a rigorous and intuitive approach from the theory of ldpc codes, arXiv preprint arXiv:2507.13342 (2025)

    work page arXiv 2025

  24. [24]

    Yin and A

    C. Yin and A. Lucas, Low-density parity-check codes as stable phases of quantum matter, PRX Quantum6, 030329 (2025)

    work page 2025

  25. [25]

    Yu and T.-C

    H. Yu and T.-C. Wei, Universal energy-space localization and stable quantum phases against time-dependent perturbations, arXiv preprint arXiv:2510.14160 (2025)

    work page arXiv 2025

  26. [26]

    Quantum LDPC codes with positive rate and minimum distance proportional to n^{1/2}

    J.-P. Tillich and G. Z´emor, Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength, IEEE Trans. Info. Theor.60, 1193 (2014), arXiv:0903.0566 [cs.IT]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  27. [27]

    Bravyi and M

    S. Bravyi and M. B. Hastings, Homological product codes, in Proceedings of the forty-sixth annual ACM symposium on The- ory of computing(2014) pp. 273–282

    work page 2014

  28. [28]

    M. B. Hastings, J. Haah, and R. O’Donnell, Fiber bundle codes: breaking the n 1/2 polylog (n) barrier for quantum ldpc codes, inProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing(2021) pp. 1276–1288

    work page 2021

  29. [29]

    Panteleev and G

    P. Panteleev and G. Kalachev, Quantum ldpc codes with almost linear minimum distance, IEEE Transactions on Information Theory68, 213 (2021)

    work page 2021

  30. [30]

    N. P. Breuckmann and J. N. Eberhardt, Balanced product quan- tum codes, IEEE Transactions on Information Theory67, 6653 (2021)

    work page 2021

  31. [31]

    Panteleev and G

    P. Panteleev and G. Kalachev, Asymptotically good quantum and locally testable classical LDPC codes, inProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Comput- ing, STOC 2022 (Association for Computing Machinery, New York, NY , USA, 2022) p. 375–388

    work page 2022

  32. [32]

    C. L. Kane, R. Mukhopadhyay, and T. C. Lubensky, Fractional 30 quantum hall effect in an array of quantum wires, Phys. Rev. Lett.88, 036401 (2002)

    work page 2002

  33. [33]

    J. C. Y . Teo and C. L. Kane, From luttinger liquid to non-abelian quantum hall states, Phys. Rev. B89, 085101 (2014)

    work page 2014

  34. [34]

    Wang and T

    C. Wang and T. Senthil, Boson topological insulators: A win- dow into highly entangled quantum phases, Phys. Rev. B87, 235122 (2013)

    work page 2013

  35. [35]

    Jian and X.-L

    C.-M. Jian and X.-L. Qi, Layer construction of 3d topologi- cal states and string braiding statistics, Phys. Rev. X4, 041043 (2014)

    work page 2014

  36. [38]

    Prem, S.-J

    A. Prem, S.-J. Huang, H. Song, and M. Hermele, Cage-net frac- ton models, Phys. Rev. X9, 021010 (2019)

    work page 2019

  37. [39]

    A. T. Schmitz, Distilling fractons from layered subsystem- symmetry protected phases, arXiv preprint arXiv:1910.04765 (2019)

    work page arXiv 1910

  38. [40]

    G. B. Hal ´asz, T. H. Hsieh, and L. Balents, Fracton topological phases from strongly coupled spin chains, Phys. Rev. Lett.119, 257202 (2017)

    work page 2017

  39. [41]

    D. J. Williamson and M. Cheng, Designer non-abelian fractons from topological layers, Phys. Rev. B107, 035103 (2023)

    work page 2023

  40. [42]

    Aasen, D

    D. Aasen, D. Bulmash, A. Prem, K. Slagle, and D. J. Williamson, Topological defect networks for fractons of all types, Phys. Rev. Res.2, 043165 (2020)

    work page 2020

  41. [43]

    D. J. Williamson and T. Devakul, Type-ii fractons from coupled spin chains and layers, Phys. Rev. B103, 155140 (2021)

    work page 2021

  42. [44]

    Sullivan, T

    J. Sullivan, T. Iadecola, and D. J. Williamson, Planar p-string condensation: Chiral fracton phases from fractional quantum hall layers and beyond, Phys. Rev. B103, 205301 (2021)

    work page 2021

  43. [45]

    Wen, Systematic construction of gapped nonliquid states, Phys

    X.-G. Wen, Systematic construction of gapped nonliquid states, Phys. Rev. Res.2, 033300 (2020)

    work page 2020

  44. [46]

    Wang, Nonliquid cellular states: Gluing gauge-higher- symmetry-breaking versus gauge-higher-symmetry-extension interfacial defects, Phys

    J. Wang, Nonliquid cellular states: Gluing gauge-higher- symmetry-breaking versus gauge-higher-symmetry-extension interfacial defects, Phys. Rev. Res.4, 023258 (2022)

    work page 2022

  45. [47]

    Z. Song, A. Dua, W. Shirley, and D. J. Williamson, Topological defect network representations of fracton stabilizer codes, PRX Quantum4, 010304 (2023)

    work page 2023

  46. [48]

    Liu and W

    S. Liu and W. Ji, Towards non-invertible anomalies from gener- alized Ising models, SciPost Phys.15, 150 (2023)

    work page 2023

  47. [49]

    See Supplemental Materials ()

    work page

  48. [50]

    Tantivasadakarn, R

    N. Tantivasadakarn, R. Thorngren, A. Vishwanath, and R. Ver- resen, Long-range entanglement from measuring symmetry- protected topological phases, Phys. Rev. X14, 021040 (2024)

    work page 2024

  49. [51]

    D. J. Williamson and T. J. Yoder, Low-overhead fault-tolerant quantum computation by gauging logical operators, Nature Physics22, 598 (2026)

    work page 2026

  50. [52]

    Concatenated Quantum Codes

    E. Knill and R. Laflamme, Concatenated quantum codes, arXiv preprint quant-ph/9608012 (1996)

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  51. [53]

    K. T. Tian, E. Samperton, and Z. Wang, Haah codes on general three-manifolds, Annals of Physics412, 168014 (2020)

    work page 2020

  52. [54]

    The general case and further examples can be found in the sup- plemental materials ()

    work page

  53. [55]

    Haah, Local stabilizer codes in three dimensions without string logical operators, Phys

    J. Haah, Local stabilizer codes in three dimensions without string logical operators, Phys. Rev. A83, 042330 (2011)

    work page 2011

  54. [56]

    Y . Tan, B. Roberts, N. Tantivasadakarn, B. Yoshida, and N. Y . Yao, Fracton models from product codes, Phys. Rev. Res.7, L032062 (2025)

    work page 2025

  55. [57]

    Vijay, J

    S. Vijay, J. Haah, and L. Fu, Fracton topological order, general- ized lattice gauge theory, and duality, Phys. Rev. B94, 235157 (2016)

    work page 2016

  56. [58]

    Leverrier, S

    A. Leverrier, S. Apers, and C. Vuillot, Quantum XYZ Product Codes, Quantum6, 766 (2022), arXiv:2011.09746 [quant-ph]

    work page arXiv 2022

  57. [59]

    Yoshida, S

    S. Yoshida, S. Tamiya, and H. Yamasaki, Concatenate codes, save qubits, npj Quantum Inf.11, 88 (2025), arXiv:2402.09606 [quant-ph]

    work page arXiv 2025

  58. [60]

    Grassl, P

    M. Grassl, P. Shor, G. Smith, J. Smolin, and B. Zeng, Gener- alized concatenated quantum codes, Phys. Rev. A79, 050306 (2009)

    work page 2009

  59. [61]

    Tantivasadakarn, W

    N. Tantivasadakarn, W. Ji, and S. Vijay, Hybrid fracton phases: Parent orders for liquid and nonliquid quantum phases, Phys. Rev. B103, 245136 (2021)

    work page 2021

  60. [62]

    Tantivasadakarn, W

    N. Tantivasadakarn, W. Ji, and S. Vijay, Non-abelian hybrid fracton orders, Phys. Rev. B104, 115117 (2021)

    work page 2021

  61. [63]

    Fuji and A

    Y . Fuji and A. Furusaki, From coupled wires to coupled layers: Model with three-dimensional fractional excitations, Phys. Rev. B99, 241107 (2019)

    work page 2019

  62. [64]

    Fuji and A

    Y . Fuji and A. Furusaki, Bridging three-dimensional coupled- wire models and cellular topological states: Solvable models for topological and fracton orders, Phys. Rev. Res.5, 043108 (2023)

    work page 2023

  63. [65]

    Christos, C

    M. Christos, C. F. B. Lo, V . Khemani, and R. Sahay, Non- abelian quantum low-density parity check codes and non- clifford operations from gauging logical gates via measure- ments, arXiv preprint arXiv:2602.12228 (2026)

    work page arXiv 2026

  64. [66]

    B. T. McDonough, J.-H. Zhang, V . V . Albert, and A. Lucas, Calderbank-shor-steane codes on group-valued qudits, arXiv preprint arXiv:2602.19558 (2026)

    work page arXiv 2026

  65. [67]

    H. Ma, E. Lake, X. Chen, and M. Hermele, Fracton topological order via coupled layers, Phys. Rev. B95, 245126 (2017)

    work page 2017

  66. [68]

    Isotropic Layer Construction and Phase Diagram for Fracton Topological Phases

    S. Vijay, Isotropic Layer Construction and Phase Dia- gram for Fracton Topological Phases, arXiv e-prints 10.48550/arXiv.1701.00762 (2017), arXiv:1701.00762 [cond- mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1701.00762 2017

  67. [69]

    Gorantla, A

    P. Gorantla, A. Prem, N. Tantivasadakarn, and D. J. Williamson, String membrane nets from higher-form gauging: An alter- nate route top-string condensation, Phys. Rev. B112, 125124 (2025)

    work page 2025

  68. [70]

    Gorantla, A

    P. Gorantla, A. Prem, N. Tantivasadakarn, and D. J. Williamson, Gauging nexus between topological and fracton phases, Phys. Rev. B113, 085110 (2026)

    work page 2026

  69. [71]

    Raussendorf, S

    R. Raussendorf, S. Bravyi, and J. Harrington, Long-range quan- tum entanglement in noisy cluster states, Phys. Rev. A71, 062313 (2005)

    work page 2005

  70. [72]

    A. Bolt, G. Duclos-Cianci, D. Poulin, and T. M. Stace, Foliated quantum error-correcting codes, Phys. Rev. Lett.117, 070501 (2016)

    work page 2016

  71. [73]

    Hillmann, G

    T. Hillmann, G. Dauphinais, I. Tzitrin, and M. Vasmer, Single- shot and measurement-based quantum error correction via fault complexes, Phys. Rev. A112, L040401 (2025)

    work page 2025

  72. [74]

    B. J. Brown and S. Roberts, Universal fault-tolerant measurement-based quantum computation, Phys. Rev. Res.2, 033305 (2020)

    work page 2020

  73. [75]

    Tanner, A recursive approach to low complexity codes, IEEE Transactions on Information Theory27, 533 (1981)

    R. Tanner, A recursive approach to low complexity codes, IEEE Transactions on Information Theory27, 533 (1981)

    work page 1981

  74. [76]

    M. E. J. Newman and C. Moore, Glassy dynamics and aging in an exactly solvable spin model, Phys. Rev. E60, 5068 (1999)

    work page 1999

  75. [77]

    Zeng and L

    W. Zeng and L. P. Pryadko, Minimal distances for certain quan- tum product codes and tensor products of chain complexes, Phys. Rev. A102, 062402 (2020)

    work page 2020

  76. [78]

    Li and T

    M. Li and T. J. Yoder, A numerical study of bravyi-bacon-shor and subsystem hypergraph product codes, in2020 IEEE Inter- national Conference on Quantum Computing and Engineering (QCE)(2020) pp. 109–119. 31

    work page 2020

  77. [79]

    Bacon, Operator quantum error-correcting subsystems for self-correcting quantum memories, Phys

    D. Bacon, Operator quantum error-correcting subsystems for self-correcting quantum memories, Phys. Rev. A73, 012340 (2006)

    work page 2006

  78. [80]

    Poulin, Stabilizer formalism for operator quantum error cor- rection, Phys

    D. Poulin, Stabilizer formalism for operator quantum error cor- rection, Phys. Rev. Lett.95, 230504 (2005)

    work page 2005

  79. [81]

    Bacon and A

    D. Bacon and A. Casaccino, Quantum error correcting subsys- tem codes from two classical linear codes, arXiv preprint quant- ph/0610088 (2006)

    work page arXiv 2006

  80. [82]

    Cowtan, Z

    A. Cowtan, Z. He, D. J. Williamson, and T. J. Yoder, Fast and fault-tolerant logical measurements: Auxiliary hypergraphs and transversal surgery, arXiv preprint arXiv:2510.14895 (2025)

    work page arXiv 2025

Showing first 80 references.