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  • Letter
  • Open Access

Fracton models from product codes

Yi Tan1,*, Brenden Roberts1,*, Nathanan Tantivasadakarn2, Beni Yoshida3, and Norman Y. Yao1

  • *These authors contributed equally to this work.
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Phys. Rev. Research 7, L032062 – Published 22 September, 2025

DOI: https://doi.org/10.1103/f48m-rlh3

Abstract

We explore a deep connection between fracton order and product codes. In particular, we propose and analyze conditions on classical seed codes which lead to fracton order in the resulting quantum product codes. Depending on the properties of the input codes, product codes can realize either Type-I or Type-II fracton models, in both nonlocal and local constructions. For the nonlocal case, we show that a recently proposed model of lineons on nonlocal graphs can be obtained as a hypergraph product code. Interestingly, constrained mobility in this model arises only from energy barriers associated with the graph. For the local case, we introduce a novel type of classical LDPC code defined on a planar aperiodic tiling. By considering the specific example of the pinwheel tiling, we demonstrate the systematic construction of local Type-I and Type-II fracton models as product codes. Our work establishes product codes as a natural setting for exploring fracton order.

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References (62)

  1. C. Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotection, Phys. Rev. Lett. 94, 040402 (2005).
  2. J. Haah, Local stabilizer codes in three dimensions without string logical operators, Phys. Rev. A 83, 042330 (2011).
  3. B. Yoshida, Exotic topological order in fractal spin liquids, Phys. Rev. B 88, 125122 (2013).
  4. S. Vijay, J. Haah, and L. Fu, A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations, Phys. Rev. B 92, 235136 (2015).
  5. S. Vijay, J. Haah, and L. Fu, Fracton topological order, generalized lattice gauge theory, and duality, Phys. Rev. B 94, 235157 (2016).
  6. K. Slagle and Y. B. Kim, Quantum field theory of X-cube fracton topological order and robust degeneracy from geometry, Phys. Rev. B 96, 195139 (2017).
  7. M. Pretko, The fracton gauge principle, Phys. Rev. B 98, 115134 (2018).
  8. A. Gromov, Towards classification of fracton phases: The multipole algebra, Phys. Rev. X 9, 031035 (2019).
  9. W. Shirley, K. Slagle, and X. Chen, Foliated fracton order from gauging subsystem symmetries, SciPost Phys. 6, 041 (2019).
  10. A. Dua, I. H. Kim, M. Cheng, and D. J. Williamson, Sorting topological stabilizer models in three dimensions, Phys. Rev. B 100, 155137 (2019).
  11. R. M. Nandkishore and M. Hermele, Fractons, Annu. Rev. Condens. Matter Phys. 10, 295 (2019).
  12. M. Pretko, X. Chen, and Y. You, Fracton phases of matter, Int. J. Mod. Phys. A 35, 2030003 (2020).
  13. N. Manoj and V. B. Shenoy, Arboreal topological and fracton phases, Phys. Rev. B 107, 165136 (2023).
  14. N. P. Breuckmann and J. N. Eberhardt, Balanced product quantum codes, IEEE Trans. Inf. Theory 67, 6653 (2021).
  15. P. Panteleev and G. Kalachev, Asymptotically good quantum and locally testable classical LDPC codes, in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 (Association for Computing Machinery, New York, 2022), pp. 375–388.
  16. N. P. Breuckmann and J. N. Eberhardt, Quantum low-density parity-check codes, PRX Quantum 2, 040101 (2021).
  17. M. B. Hastings, J. Haah, and R. O'Donnell, Fiber bundle codes: Breaking the n1/2polylog(n) barrier for quantum LDPC codes, in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (2021), pp. 1276–1288.
  18. P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Trans. Inf. Theory 68, 213 (2022).
  19. J.-P. Tillich and G. Zémor, Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength, IEEE Trans. Inf. Theory 60, 1193 (2014).
  20. J. Haah, Commuting Pauli Hamiltonians as maps between free modules, Commun. Math. Phys. 324, 351 (2013); Algebraic methods for quantum codes on lattices, Rev. Colomb. Math. 50, 299 (2016).
  21. See Supplemental Material at http://link.aps.org/supplemental/10.1103/f48m-rlh3 for explicit details on these connections.
  22. P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, Gapped lineon and fracton models on graphs, Phys. Rev. B 107, 125121 (2023).
  23. H. Ebisu and B. Han, Anisotropic higher rank ZN topological phases on graphs, SciPost Phys. 14, 106 (2023).
  24. H. Bombín, Single-shot fault-tolerant quantum error correction, Phys. Rev. X 5, 031043 (2015).
  25. A. O. Quintavalle, M. Vasmer, J. Roffe, and E. T. Campbell, Single-shot error correction of three-dimensional homological product codes, PRX Quantum 2, 020340 (2021).
  26. D. J. C. MacKay, Information Theory, Inference, and Learning Algorithms (Cambridge University Press, Cambridge, 2003).
  27. R. Gallager, Low-density parity-check codes, IRE Transactions on Information Theory 8, 21 (1962).
  28. A. R. Calderbank and Peter W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).
  29. A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett. 77, 793 (1996).
  30. R. Tanner, A recursive approach to low complexity codes, IEEE Trans. Inf. Theory 27, 533 (1981).
  31. In the local case, this quantity typically exhibits polynomial scaling (Type I) or follows a polynomial envelope (Type II).
  32. While geometrically local fracton models may support extended immobile excitations [58], the present discussion will show that these are excluded w.h.p.
  33. As in topological order, superselection sectors identify dynamically disconnected excitation patterns; the difference is that rather than supporting finitely many, labeled by elements of the anyon theory, in an HGP code based on a rank-deficient seed code, the number of superselection sectors depends on the system size.
  34. I. H. Kim and J. Haah, Localization from superselection rules in translationally invariant systems, Phys. Rev. Lett. 116, 027202 (2016).
  35. It is interesting that the pointlike nature of excitations precludes a phase transition at finite temperature when coupling the code to a thermal bath [59].
  36. N. Seiberg and S.-H. Shao, Exotic symmetries, duality, and fractons in 2+1-dimensional quantum field theory, SciPost Phys. 10, 027 (2021).
  37. P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, A modified Villain formulation of fractons and other exotic theories, J. Math. Phys. 62, 102301 (2021).
  38. P. Gorantla, H. T. Lam, and S.-H. Shao, Fractons on graphs and complexity, Phys. Rev. B 106, 195139 (2022).
  39. F. R. Chung, Spectral Graph Theory (American Mathematical Society, Providence, RI, 1997), Vol. 92.
  40. N. Delfosse, Tradeoffs for reliable quantum information storage in surface codes and color codes, in 2013 IEEE International Symposium on Information Theory (IEEE, Istanbul, Turkey, 2013), pp. 917–921.
  41. C. Radin and M. Wolff, Space tilings and local isomorphism, Geometriae Dedicata 42, 355 (1992).
  42. C. Radin, The pinwheel tilings of the plane, Ann. Math. 139, 661 (1994).
  43. C. Radin, Aperiodic tilings, ergodic theory, and rotations, NATO ASI Series C Mathematical and Physical Sciences-Advanced Study Institute 489, 499 (1997).
  44. D. Frettlöh, Substitution tilings with statistical circular symmetry, Eur. J. Comb. 29, 1881 (2008).
  45. J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, M. Allman, C. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer et al., Demonstration of the trapped-ion quantum CCD computer architecture, Nature (London) 592, 209 (2021).
  46. D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter et al., Logical quantum processor based on reconfigurable atom arrays, Nature (London) 626, 58 (2024).
  47. We observe that for small values of p short logical operators attached to the boundary can appear, thus we take large enough values to suppress these.
  48. S. Bravyi, D. Poulin, and B. Terhal, Tradeoffs for reliable quantum information storage in 2d systems, Phys. Rev. Lett. 104, 050503 (2010).
  49. If HN is viewed as a classical Hamiltonian, the T=0 residual entropy is determined by the entropy of the code, namely 2k∼2n. Consequently, although the residual entropy density vanishes, the property of confinement ensures that energy barriers produce glassy dynamics.
  50. While we have studied a particular rectangular patch under the substitution rules, more general regions are possible, and due to the lack of spatial symmetry nonidentical codes will not have large coincident subgraphs.
  51. D. J. Williamson and N. Baspin, Layer codes, Nat. Commun. 15, 9528 (2024).
  52. T.-C. Lin, A. Wills, and M.-H. Hsieh, Geometrically local quantum and classical codes from subdivision, arXiv:2309.16104.
  53. G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99, 220405 (2007).
  54. J. Haah, Bifurcation in entanglement renormalization group flow of a gapped spin model, Phys. Rev. B 89, 075119 (2014).
  55. A. Dua, P. Sarkar, D. J. Williamson, and M. Cheng, Bifurcating entanglement-renormalization group flows of fracton stabilizer models, Phys. Rev. Res. 2, 033021 (2020).
  56. J. F. San Miguel, A. Dua, and D. J. Williamson, Bifurcating subsystem symmetric entanglement renormalization in two dimensions, Phys. Rev. B 103, 035148 (2021).
  57. C. T. Aitchison, D. Bulmash, A. Dua, A. C. Doherty, and D. J. Williamson, Boundaries and defects in the cubic code, Phys. Rev. B 109, 205125 (2024).
  58. M.-Y. Li and P. Ye, Fracton physics of spatially extended excitations, Phys. Rev. B 101, 245134 (2020).
  59. Y. Hong, J. Guo, and A. Lucas, Quantum memory at nonzero temperature in a thermodynamically trivial system, Nat. Commuun. 16, 316 (2025).
  60. B. Bollobás and V. Klee, Diameters of random bipartite graphs, Combinatorica 4, 7 (1984).
  61. F. Chung and L. Lu, The diameter of sparse random graphs, Adv. Appl. Math. 26, 257 (2001).
  62. I. Dumer, A. A. Kovalev, and L. P. Pryadko, Thresholds for correcting errors, erasures, and faulty syndrome measurements in degenerate quantum codes, Phys. Rev. Lett. 115, 050502 (2015).

Outline

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Phys. Rev. Research 7, L032062– Published 22 September, 2025
  • Received 9 February 2024
  • Revised 23 July 2024
  • Accepted 10 June 2025
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