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Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate Beyond the n1/3 Distance Barrier

Po-Shen Hsin1,2,*, Ryohei Kobayashi3,4,†, and Guanyu Zhu5,‡

  • *Contact author: po-shen.hsin@kcl.ac.uk
  • †Contact author: ryok@ias.edu
  • ‡Contact author: guanyu.zhu@ibm.com
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PRX Quantum 6, 040360 – Published 10 December, 2025

DOI: https://doi.org/10.1103/hkgn-hhqx

Abstract

We construct a family of infinitely many candidate non-Abelian self-correcting topological quantum memories in D≥5+1 spacetime dimensions without particle excitations using local commuting non-Pauli stabilizer lattice models and field theories of Z23 higher-form gauge fields with nontrivial topological action. We call such non-Pauli stabilizer models magic stabilizer codes. The family of topological orders have Abelian electric excitations and non-Abelian magnetic excitations that obey Ising-like fusion rules and non-Abelian braiding, including Borromean ring type braiding which is a signature of non-Abelian topological order, generalizing the dihedral group D8 gauge theory in (2+1)D. The simplest example includes a non-Abelian self-correcting memory in (5+1)D with Abelian loop excitations and non-Abelian membrane excitations. We prove the self-correction property and the thermal stability and devise a probabilistic local cellular-automaton decoder. We also construct fault-tolerant non-Clifford CCZ logical gate using constant-depth circuits from higher-cup products in the 5D non-Abelian code. The use of higher-cup products and non-Pauli stabilizers allows us to get an O(n2/5) distance overcoming the O(n1/3) distance barrier in conventional topological stabilizer codes, including the 3D color code and the 6D self-correcting color code.

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Popular Summary

Anyons in topological orders are quasiparticles that are neither bosons nor fermions. Among them, non-Abelian anyons—whose fusion can yield multiple possible outcomes—play a central role in universal quantum computation. However, at any temperature above absolute zero, anyons become thermally unstable, leading to continuous errors that must be corrected in quantum computers based on them.

In this work, the authors introduce thermally stable variants of non-Abelian anyons, which are extended excitations such as loops or membranes instead of particles. These new excitations enable quantum memories that remain robust without the need for constant error correction. Furthermore, the authors demonstrate that, like conventional non-Abelian anyons, these quantum memories support a non-Clifford logical gate that is crucial for universal quantum computation. The quantum memories discovered by the authors have high-dimensional structures and can be realized on quantum computing architectures such as trapped ions, superconducting qubits, or neutral atoms with long-range couplings.

The results obtained by the authors deepen the understanding of thermally stable quantum memories and topological phases of matter. They also open a promising route toward practical architectures for universal quantum computing that are both fault-tolerant and thermally robust.

Article Text

References (109)

  1. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002).
  2. B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. R. Wootton, Quantum memories at finite temperature, Rev. Mod. Phys. 88, 045005 (2016).
  3. H. Bombin, Single-shot fault-tolerant quantum error correction, Phys. Rev. X 5, 031043 (2015).
  4. R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki, On thermal stability of topological qubit in Kitaev’s 4D model, Open Syst. Inf. Dyn. 17, 1 (2010).
  5. H. Bombin, R. W. Chhajlany, M. Horodecki, and M. A. Martin-Delgado, Self-correcting quantum computers, New J. Phys. 15, 055023 (2013).
  6. N. Berthusen, J. Dreiling, C. Foltz, J. P. Gaebler, T. M. Gatterman, D. Gresh, N. Hewitt, M. Mills, S. A. Moses, B. Neyenhuis, P. Siegfried, and D. Hayes, Experiments with the four-dimensional surface code on a quantum charge-coupled device quantum computer, Phys. Rev. A 110, 062413 (2024).
  7. Higher dimensional topological orders above the physical dimension can be realized using long-range connections and are relevant for practical fault-tolerant quantum computation [87, 88, 89], and they are ubiquitous in quantum low-density parity-check (LDPC) codes [67, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101].
  8. M. H. Freedman, M. Larsen, and Z. H. Wang, A modular functor which is universal for quantum computation, Commun. Math. Phys. 227, 605 (2002).
  9. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  10. R. Koenig, G. Kuperberg, and B. W. Reichardt, Quantum computation with Turaev-Viro codes, Ann. Phys. (N.Y.) 325, 2707 (2010).
  11. A. Schotte, G. Zhu, L. Burgelman, and F. Verstraete, Quantum error correction thresholds for the Universal Fibonacci Turaev-Viro Code, Phys. Rev. X 12, 021012 (2022).
  12. Z. K. Minev, K. Najafi, S. Majumder, J. Wang, A. Stern, E.-A. Kim, C.-M. Jian, and G. Zhu, arXiv:2406.12820.
  13. Z. Nussinov and G. Ortiz, Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems, Phys. Rev. B 77, 064302 (2008).
  14. S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New J. Phys. 11, 043029 (2009).
  15. M. B. Hastings, Topological order at nonzero temperature, Phys. Rev. Lett. 107, 210501 (2011).
  16. B. Yoshida, Feasibility of self-correcting quantum memory and thermal stability of topological order, Ann. Phys. (N.Y.) 326, 2566 (2011).
  17. T. Johnson-Freyd and M. Yu, Fusion 2-categories with no line operators are grouplike, Bull. Aust. Math. Soc. 104, 434 (2021).
  18. T. Johnson-Freyd and M. Yu, Topological orders in (4+1)-dimensions, SciPost Phys. 13, 068 (2022).
  19. C. Cordova, P.-S. Hsin, and C. Zhang, Anomalies of non-invertible symmetries in (3+1)d, arXiv:2308.11706 [hep-th].
  20. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  21. Higher-form gauge theories have been studied in e.g., Refs. [57, 102, 103, 104, 105, 106, 107, 108, 109], especially in high energy theory context. However, the properties of higher-form gauge theories with general gauge groups and general topological actions of the gauge fields in general dimensions remain unknown. In this work we study the properties of a family of particular higher-form gauge theories whose properties have not been explored.
  22. H. Bombin, Gauge color codes: Optimal transversal gates and gauge fixing in topological stabilizer codes, New J. Phys. 17, 083002 (2015).
  23. A. Kubica and M. E. Beverland, Universal transversal gates with color codes: A simplified approach, Phys. Rev. A 91, 032330 (2015).
  24. M. Vasmer and D. E. Browne, Three-dimensional surface codes: Transversal gates and fault-tolerant architectures, Phys. Rev. A 100, 012312 (2019).
  25. S. Bravyi and R. König, Classification of topologically protected gates for local stabilizer codes, Phys. Rev. Lett. 110, 170503 (2013).
  26. G. Zhu, arXiv:2501.19375.
  27. 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 processor, Nature 626, 505 (2024).
  28. H. Bombin, arXiv:1810.09571v1.
  29. B. J. Brown, A fault-tolerant non-Clifford gate for the surface code in two dimensions, Sci. Adv. 6, eaay4929 (2020).
  30. M. Davydova, A. Bauer, J. C. M. de la Fuente, M. Webster, D. J. Williamson, and B. J. Brown, Universal fault tolerant quantum computation in 2D without getting tied in knots, arXiv:2503.15751 [quant-ph].
  31. E. T. Campbell, B. M. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature 549, 172 (2017).
  32. G. Zhu, A. Lavasani, and M. Barkeshli, Universal logical gates on topologically encoded qubits via constant-depth unitary circuits, Phys. Rev. Lett. 125, 050502 (2020).
  33. G. Zhu, A. Lavasani, and M. Barkeshli, Instantaneous braids and Dehn twists in topologically ordered states, Phys. Rev. B 102, 075105 (2020).
  34. A. Lavasani, G. Zhu, and M. Barkeshli, Universal logical gates with constant overhead: Instantaneous Dehn twists for hyperbolic quantum codes, Quantum 3, 180 (2019).
  35. X. Chen, A. Dua, P.-S. Hsin, C.-M. Jian, W. Shirley, and C. Xu, Loops in 4+1d topological phases, arXiv:2112.02137 [cond-mat.str-el].
  36. T. Lan, L. Kong, and X.-G. Wen, Classification of (3+1)D bosonic topological orders: The case when pointlike excitations are all bosons, Phys. Rev. X 8, 021074 (2018).
  37. Despite the naming similarity, the theories are not related to Haah’s cubic code.
  38. B. Yoshida, Topological phases with generalized global symmetries, Phys. Rev. B 93, 155131 (2016).
  39. N. Madras and G. Slade, The Self-Avoiding Walk, Modern Birkhäuser Classics (Springer, New York, 2012).
  40. H. Bombin, R. W. Chhajlany, M. Horodecki, and M. A. Martin-Delgado, Self-correcting quantum computers, New J. Phys. 15, 055023 (2013).
  41. D. Gaiotto and T. Johnson-Freyd, Condensations in higher categories, arXiv:1905.09566 [math.CT].
  42. X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013).
  43. M. D. F. de Wild Propitius, Topological interactions in broken gauge theories, Ph.D. thesis, Amsterdam University, 1995. arXiv:hep-th/9511195.
  44. B. Yoshida, Topological phases with generalized global symmetries, Phys. Rev. B 93, 155131 (2016).
  45. L. Tsui and X.-G. Wen, Lattice models that realize Zn−1 symmetry-protected topological states for even n, Phys. Rev. B 101, 035101 (2020).
  46. Y.-A. Chen and P.-S. Hsin, Exactly solvable lattice Hamiltonians and gravitational anomalies, SciPost Phys. 14, 089 (2023).
  47. J. W. Milnor and J. D. Stasheff, Characteristic Classes, Annals of Mathematics Studies (Princeton University Press, Princeton, 1974), http://www.jstor.org/stable/j.ctt1b7x751.
  48. F. Benini, C. Córdova, and P.-S. Hsin, On 2-group global symmetries and their anomalies, J. High Energy Phys. 2019, 118 (2019).
  49. Y.-A. Chen and S. Tata, Higher cup products on hypercubic lattices: Application to lattice models of topological phases, arXiv:2106.05274 [cond-mat.str-el].
  50. M. Levin and Z.-C. Gu, Braiding statistics approach to symmetry-protected topological phases, Phys. Rev. B 86, 115109 (2012).
  51. L. Bhardwaj, D. Gaiotto, and A. Kapustin, State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter, J. High Energy Phys. 2017, 096 (2017).
  52. W. Shirley, K. Slagle, and X. Chen, Foliated fracton order from gauging subsystem symmetries, SciPost Phys. 6, 041 (2019).
  53. M. de Wild Propitius, Confinement in partially broken abelian Chern-Simons theories, Phys. Lett. B 410, 188 (1997).
  54. A. Coste, T. Gannon, and P. Ruelle, Finite group modular data, Nucl. Phys. B 581, 679 (2000).
  55. P.-S. Hsin and A. Turzillo, Symmetry-enriched quantum spin liquids in (3+1)d, J. High Energy Phys. 2020, 022 (2020).
  56. M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and R. Kobayashi, Higher-group symmetry in finite gauge theory and stabilizer codes, SciPost Phys. 16, 089 (2024).
  57. E. Witten, AdS/CFT correspondence and topological field theory, J. High Energy Phys. 1998, 012 (1998).
  58. E. Witten, Five-brane effective action in M theory, J. Geom. Phys. 22, 103 (1997).
  59. E. Witten, Geometric Langlands from six dimensions, arXiv:0905.2720 [hep-th].
  60. D. S. Freed and C. Teleman, Relative quantum field theory, Commun. Math. Phys. 326, 459 (2014).
  61. C. Cordova, D. B. Costa, and P.-S. Hsin, Non-invertible symmetries in finite-group gauge theory, SciPost Phys. 18, 019 (2025).
  62. B. Yoshida, Topological color code and symmetry-protected topological phases, Phys. Rev. B 91, 245131 (2015).
  63. B. Yoshida, Gapped boundaries, group cohomology and fault-tolerant logical gates, Ann. Phys. 377, 387 (2017).
  64. M. Barkeshli, Y.-A. Chen, S.-J. Huang, R. Kobayashi, N. Tantivasadakarn, and G. Zhu, Codimension-2 defects and higher symmetries in (3+1)D topological phases, SciPost Phys. 14, 065 (2023).
  65. A. Kubica, B. Yoshida, and F. Pastawski, Unfolding the color code, New J. Phys. 17, 083026 (2015).
  66. H. Bombin and M. A. Martin-Delgado, Topological quantum distillation, Phys. Rev. Lett. 97, 180501 (2006).
  67. G. Zhu, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, arXiv:2310.16982.
  68. M. Barkeshli, P.-S. Hsin, and R. Kobayashi, Higher-group symmetry of (3+1)D fermionic Z2 gauge theory: Logical CCZ, CS, and T gates from higher symmetry, SciPost Phys. 16, 122 (2024).
  69. The winding modes will give additional (l−1)-form, (m−1)-form, and (n−1)-form gauge fields ∫al,∫bm,∫cn.
  70. M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, Symmetry fractionalization, defects, and gauging of topological phases, Phys. Rev. B 100, 115147 (2019).
  71. J. C. Teo, T. L. Hughes, and E. Fradkin, Theory of twist liquids: Gauging an anyonic symmetry, Ann. Phys. (N. Y) 360, 349 (2015).
  72. N. Tarantino, N. H. Lindner, and L. Fidkowski, Symmetry fractionalization and twist defects, New J. Phys. 18, 035006 (2016).
  73. P. Putrov, J. Wang, and S.-T. Yau, Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions, Ann. Phys. 384, 254 (2017).
  74. R. Kobayashi, G. Zhu, and P.-S. Hsin, Clifford hierarchy stabilizer codes: Transversal non-Clifford gates and magic, arXiv:2511.02900.
  75. P.-S. Hsin and R. Kobayashi, Automorphism in Gauge theories: Higher symmetries and transversal non-Clifford logical gates, arXiv:2511.15783.
  76. P.-S. Hsin, R. Kobayashi, and G. Zhu, arXiv:2411.15848.
  77. B. Eastin and E. Knill, Restrictions on transversal encoded quantum gate sets, Phys. Rev. Lett. 102, 81 (2009).
  78. L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, Low-overhead fault-tolerant quantum computing using long-range connectivity, Sci. Adv. 8, eabn1717 (2022).
  79. D. J. Williamson and T. J. Yoder, Low-overhead fault-tolerant quantum computation by gauging logical operators, arXiv:2410.02213 [quant-ph].
  80. J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Phys. Rev. A 94, 052325 (2016).
  81. B. Durhuus, J. Fröhlich, and T. Jonsson, Self-avoiding and planar random surfaces on the lattice, Nucl. Phys. B 225, 185 (1983).
  82. A. Dua, T. Jochym-O’Connor, and G. Zhu, Quantum error correction with fractal topological codes, Quantum 7, 1122 (2023).
  83. S.-J. Huang and Y. Chen, Generating logical magic states with the aid of non-Abelian topological order, arXiv:2502.00998 [quant-ph].
  84. N. E. Steenrod, Products of cocycles and extensions of mappings, Ann. Math. 48, 290 (1947).
  85. D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, J. High Energy Phys. 2015, 172 (2015).
  86. P.-S. Hsin, R. Kobayashi, and G. Zhu, Classifying logical gates in quantum codes via cohomology operations and symmetry, arXiv:2411.15848 [quant-ph].
  87. L. Guth and A. Lubotzky, Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds, J. Math. Phys. 55, 082202 (2014).
  88. N. P. Breuckmann and V. Londe, arXiv:2001.03568.
  89. M. Freedman and M. B. Hastings, arXiv:2012.02249.
  90. S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low-overhead fault-tolerant quantum memory, Nature 627, 778 (2024).
  91. N. P. Breuckmann, C. Vuillot, E. Campbell, A. Krishna, and B. M. Terhal, Hyperbolic and semi-hyperbolic surface codes for quantum storage, Quantum Sci. Technol. 2, 035007 (2017).
  92. M. Hastings, J. Haah, and R. O’Donnell, in Proceedings of ACM STOC (Association for Computing Machinery, New York, NY, USA, 2021), pp. 1276–1288.
  93. P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Trans. Inf. Theory 68, 213 (2022).
  94. N. P. Breuckmann and J. N. Eberhardt, Balanced product quantum codes, IEEE Trans. Inf. Theory 67, 6653 (2021).
  95. M. Hastings, arXiv:2102.10030.
  96. P. Panteleev and G. Kalachev, in Proceedings of ACM STOC (Association for Computing Machinery, New York, NY, USA, 2022), pp. 375–388.
  97. T. Lin and M. Hsieh, arXiv:2203.03581.
  98. S. Gu, C. Pattison, and E. Tang, arXiv:2206.06557.
  99. I. Dinur, M. Hsieh, T. Lin, and T. Vidick, in Proceedings of ACM STOC (Association for Computing Machinery, New York, NY, USA, 2023), pp. 905–918.
  100. A. Leverrier and G. Zémor, in Proceedings of ACM SODA (Association for Computing Machinery, New York, NY, USA, 2023), pp. 1216–1244.
  101. S. Gu, E. Tang, L. Caha, S. Choe, Z. He, and A. Kubica, arXiv:2306.12470.
  102. G. T. Horowitz, Exactly soluble diffeomorphism invariant theories, Commun. Math. Phys. 125, 417 (1989).
  103. N. J. Hitchin, The geometry of three-forms in six dimensions, J. Differ. Geom. 55, 547 (2000).
  104. J. M. Maldacena, G. W. Moore, and N. Seiberg, D-brane charges in five-brane backgrounds, J. High Energy Phys. 2001, 005 (2001).
  105. R. Dijkgraaf, S. Gukov, A. Neitzke, and C. Vafa, Topological M-theory as unification of form theories of gravity, Adv. Theor. Math. Phys. 9, 603 (2005).
  106. K. Walker and Z. Wang, (3+1)-TQFTs and topological insulators, arXiv:1104.2632 [cond-mat.str-el].
  107. A. Kapustin and R. Thorngren, Higher symmetry and gapped phases of gauge theories, in Algebra, Geometry, and Physics in the 21st Century: Kontsevich Festschrift, edited by D. Auroux, L. Katzarkov, T. Pantev, Y. Soibelman, and Y. Tschinkel (Springer International Publishing, Cham, 2017).
  108. A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, J. High Energy Phys. 2014, 001 (2014).
  109. P.-S. Hsin, H. T. Lam, and N. Seiberg, Comments on one-form global symmetries and their gauging in 3d and 4d, SciPost Phys. 6, 039 (2019).

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PRX Quantum 6, 040360– Published 10 December, 2025
  • Received 30 August 2024
  • Revised 9 May 2025
  • Accepted 30 October 2025
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