- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Non-Clifford and Parallelizable Fault-Tolerant Logical Gates on Constant and Almost-Constant Rate Homological Quantum Low-Density Parity-Check Codes via Higher Symmetries

Guanyu Zhu1,*, Shehryar Sikander2, Elia Portnoy3, Andrew W. Cross1, and Benjamin J. Brown1

  • *Contact author: guanyu.zhu@ibm.com
PDF

PRX Quantum 6, 040361 – Published 11 December, 2025

DOI: https://doi.org/10.1103/wcxs-w69t

Abstract

We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive a generic formula for a transversal T gate on color codes defined on general 3-manifolds, which acts as collective non-Clifford logical ccz gates on any triplet of logical qubits with their logical-X membranes having a Z2 triple intersection at a single point. The triple-intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory (TQFT): the Z23 gauge theory. Moreover, the transversal S gate of the color code corresponds to a higher-form symmetry in TQFT supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical cz gates. A construction of constant-depth circuits of the above logical gates via cup-product cohomology operation is also presented for three copies of identical toric codes on arbitrary 3-manifolds. We have developed a generic formalism to compute the triple-intersection invariants for 3-manifolds, with the structure encoded into an interaction hypergraph which determines the logical gate property and also corresponds to the hypergraph magic state that can be injected into the code without distillation (“magic-state fountain”). We also study the scaling of the Betti number and systoles with volume for various 3-manifolds, which translates to the encoding rate and distance. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate k/n=O(1/log(n)) and O(log(n)) distance; (2) A homological fiber-bundle code from twisting the product by an isometry of the surface based on the construction by Freedman-Meyer-Luo, with O(1/log12(n)) rate and O(log12(n)) distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping-torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical-X measurements.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Quantum low-density parity-check (LDPC) codes are one of the most prominent candidates for near-term realization of fault-tolerant quantum computation due to their very low qubit overhead. While significant progress has been made in terms of the information storage properties of LDPC codes, little has been known about how to perform logical operations on such codes, especially for the individual addressability of the logical gates.

This work has made progress on this problem. In particular, non-Clifford and individually addressable logical gates have been constructed for a particular family of homological quantum LDPC codes defined on 3-manifolds, leading to a parallelizable universal logical gate set. The crucial discovery is that the logical gate structure is completely determined by the topology of the manifolds, while the geometry decides the code parameters such as encoding rate and code distance. This work introduces cup products to construct logical non-Clifford gates in high-rate qLDPC codes. This work further proposes the “magic-state fountain” scheme which uses transversal non-Clifford gates to fault tolerantly prepare magic states in qLDPC codes in a single shot as an alternative to the more costly magic-state distillation scheme. The geometric and topological insights obtained in this work also shed light on the future design of logical gates for a wide range of quantum LDPC codes.

Article Text

References (88)

  1. M. Hastings, J. Haah, and R. O’Donnell, in Proceedings of the ACM STOC (Association for Computing Machinery, New York, NY, USA, 2021), p. 1276.
  2. P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Trans. Inf. Theory 68, 213 (2022).
  3. N. P. Breuckmann and J. N. Eberhardt, Balanced product quantum codes, IEEE Trans. Inf. Theory 67, 6653 (2021).
  4. M. Hastings, On quantum weight reduction, arXiv:2102.10030.
  5. P. Panteleev and G. Kalachev, in Proceedings of the ACM STOC (Association for Computing Machinery, New York, NY, USA, 2022), p. 375.
  6. A. Leverrier and G. Zemor, in Proceedings of the IEEE FOCS (IEEE Computer Society, Los Alamitos, CA, USA, 2022), p. 872.
  7. T. Lin and M. Hsieh, Good quantum LDPC codes with linear time decoder from lossless expanders, arXiv:2203.03581.
  8. S. Gu, C. Pattison, and E. Tang, An efficient decoder for a linear distance quantum LDPC code, arXiv:2206.06557.
  9. I. Dinur, M. Hsieh, T. Lin, and T. Vidick, in Proceedings of the ACM STOC (Association for Computing Machinery, New York, NY, USA, 2023), p. 905.
  10. A. Leverrier and G. Zémor, Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes, in Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2023), p. 1216.
  11. S. Gu, C. A. Pattison, and E. Tang, An Efficient Decoder for a Linear Distance Quantum LDPC Code, in STOC 2023: Proceedings of the 55th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, USA, 2023), pp. 919–932.
  12. D. Gottesman, Fault-tolerant quantum computation with constant overhead, Quantum Inform. Comput. 14, 1339 (2014).
  13. A. Kubica, B. Yoshida, and F. Pastawski, Unfolding the color code, New J. Phys. 17, 083026 (2015).
  14. J. E. Moussa, Transversal Clifford gates on folded surface codes, Phys. Rev. A 94, 042316 (2016).
  15. G. Zhu, M. Hafezi, and M. Barkeshli, Quantum origami: Transversal gates for quantum computation and measurement of topological order, Phys. Rev. Res. 2, 013285 (2020).
  16. N. P. Breuckmann and S. Burton, Fold-transversal Clifford gates for quantum codes, Quantum 8, 1372 (2024).
  17. A. O. Quintavalle, P. Webster, and M. Vasmer, Partitioning qubits in hypergraph product codes to implement logical gates, Quantum 7, 1153 (2023).
  18. 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).
  19. H. Bombin, Gauge color codes: Optimal transversal gates and gauge fixing in topological stabilizer codes, New J. Phys. 17, 083002 (2015).
  20. H. Bombin, Transversal gates and error propagation in 3D topological codes, arXiv:1810.09575.
  21. L. Cohen, I. Kim, S. Bartlett, and B. Brown, Low-overhead fault-tolerant quantum computing using long-range connectivity, Sci. Adv. 8, eabn1717 (2022).
  22. H. Yamasaki and M. Koashi, Time-efficient constant-space-overhead fault-tolerant quantum computation, Nat. Phys. 20, 247 (2024).
  23. S. Burton and D. Browne, Limitations on transversal gates for hypergraph product codes, IEEE Trans. Inf. Theory 68, 1772 (2021).
  24. S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005).
  25. A. Lavasani, G. Zhu, and M. Barkeshli, Universal logical gates with constant overhead: Instantaneous Dehn twists for hyperbolic quantum codes, Quantum 3, 180 (2019).
  26. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  27. R. Koenig, G. Kuperberg, and B. W. Reichardt, Quantum computation with Turaev-Viro codes, Ann. Phys. 325, 2707 (2010).
  28. 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).
  29. A. Schotte, L. Burgelman, and G. Zhu, Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature, arXiv:2301.00054.
  30. H. Bombin and M. A. Martin-Delgado, Exact topological quantum order in D=3 and beyond: Branyons and brane-net condensates, Phys. Rev. B 75, 075103 (2007).
  31. C. Vuillot and N. P. Breuckmann, Quantum pin codes, IEEE Trans. Inf. Theory 68, 5955 (2022).
  32. 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).
  33. D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, arXiv:1412.5148.
  34. F. Benini, C. Córdova, and P.-S. Hsin, On 2-group global symmetries and their anomalies, J. High Energy Phys. 2019, 118 (2019).
  35. C. Cordova, T. T. Dumitrescu, K. Intriligator, and S.-H. Shao, Snowmass white paper: Generalized symmetries in quantum field theory and beyond, arXiv:2205.09545.
  36. J. McGreevy, Generalized symmetries in condensed matter, Annu. Rev. Condens. Matter Phys. 14, 57 (2023).
  37. L. Bhardwaj, S. Schäfer-Nameki, and J. Wu, Universal non-invertible symmetries, Fortschr. Phys. 70, 2200143 (2022).
  38. S. Bravyi and R. König, Classification of topologically protected gates for local stabilizer codes, Phys. Rev. Lett. 110, 170503 (2013).
  39. H. Bombín, Gauge color codes: Optimal transversal gates and gauge fixing in topological stabilizer codes, New J. Phys. 17, 083002 (2015).
  40. B. Yoshida, Topological phases with generalized global symmetries, Phys. Rev. B 93, 155131 (2016).
  41. B. Yoshida, Gapped boundaries, group cohomology and fault-tolerant logical gates, Ann. Phys. 377, 387 (2017).
  42. 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).
  43. M. Vasmer and A. Kubica, Morphing quantum codes, PRX Quantum 3, 030319 (2022).
  44. P.-S. Hsin and A. Turzillo, Symmetry-enriched quantum spin liquids in (3+1)D, JHEP 2020, 22 (2020).
  45. A. Kapustin and R. Thorngren, Fermionic SPT phases in higher dimensions and bosonization, J. High Energy Phys. 2017, 080 (2017).
  46. H. Bombin, 2d quantum computation with 3D topological codes, arXiv:1810.09571.
  47. M. Vasmer and D. E. Browne, Three-dimensional surface codes: Transversal gates and fault-tolerant architectures, Phys. Rev. A 100, 012312 (2019).
  48. B. Yoshida, Topological color code and symmetry-protected topological phases, Phys. Rev. B 91, 245131 (2015).
  49. P. Webster and S. D. Bartlett, Locality-preserving logical operators in topological stabilizer codes, Phys. Rev. A 97, 012330 (2018).
  50. G. Zhu, T. Jochym-O’Connor, and A. Dua, Topological order, quantum codes, and quantum computation on fractal geometries, PRX Quantum 3, 030338 (2022).
  51. G. Zhu, A topological theory for qLDPC: Non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the n1/3 distance barrier, arXiv:2501.19375.
  52. P.-S. Hsin, R. Kobayashi, and G. Zhu, Classifying logical gates in quantum codes via cohomology operations and symmetry, arXiv:2411.15848.
  53. M. Gromov, Volume and bounded cohomology, Publ. Math. l’IHÉS 56, 5 (1982).
  54. M. Rossi, M. Huber, D. Bruß, and C. Macchiavello, Quantum hypergraph states, New J. Phys. 15, 113022 (2013).
  55. J. Chen, Y. Yan, and Y. Zhou, Magic of quantum hypergraph states, Quantum 8, 1351 (2024).
  56. Y. Takeuchi, T. Morimae, and M. Hayashi, Quantum computational universality of hypergraph states with Pauli-X and Z basis measurements, Sci. Rep. 9, 13585 (2019).
  57. S. Bravyi and A. Y. Kitaev, Quantum codes on a lattice with boundary, arXiv:quant-ph/9811052.
  58. L. Guth and A. Lubotzky, Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds, J. Math. Phys. 55, 082202 (2014).
  59. M. H. Freedman, D. A. Meyer, and F. Luo, in Mathematics of Quantum Computation (Chapman and Hall/CRC, 2002), p. 303, https://www.taylorfrancis.com/chapters/edit/10.1201/9781420035377-13/z2-systolic-freedom-quantum-codes-michael-freedman-david-meyer-feng-luo.
  60. It is based on the 3-manifold they constructed before doing the Dehn surgery to kill most of the homology cycles.
  61. I. Agol, C. J. Leininger, and D. Margalit, Pseudo-Anosov stretch factors and homology of mapping tori, J. Lond. Math. Soc. 93, 664 (2016).
  62. S. Kojima and G. McShane, Normalized entropy versus volume for pseudo-Anosovs, Geom. Topol. 22, 2403 (2018).
  63. A. Lavasani and M. Barkeshli, Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes, Phys. Rev. A 98, 052319 (2018).
  64. D. Sullivan, On the intersection ring of compact three manifolds, Topology 14, 275 (1975).
  65. H. Bombin and M. A. Martin-Delgado, Topological computation without braiding, Phys. Rev. Lett. 98, 160502 (2007).
  66. A. Kubica and M. E. Beverland, Universal transversal gates with color codes: A simplified approach, Phys. Rev. A 91, 032330 (2015).
  67. For conciseness, we drop Z2 in the followed texts and when we say Betti number we will always refer to the Z2 Betti number.
  68. A more general quantity to be considered is the algebraic intersection number |α1∩β2|Z∈Z, which has a ‘±’ sign for each intersection point depending the orientation of the cycles and is a topological invariant. In contrast, geometric intersection number only associated a ‘+’ sign for each intersection point, and is hence not topological but depending the choice of the representative of each cycle. In the Z2 case, ‘+′ and ‘−′ signs are identified and it turns out the algebraic and geometric intersection numbers mod 2 are the same. For more general case like the ZN toric code, one should consider algebraic intersection number mod N.
  69. Strictly speaking Poincaré dual is between the cycle α2 and the cocycle α1. In this paper, we sometimes relax the use of this terminology by calling α1 the Poincaré dual cycle of α2.
  70. P.-S. Hsin, R. Kobayashi, and G. Zhu, Non-Abelian self-correcting quantum memory, arXiv:2405.11719.
  71. A. Hatcher, Algebraic Topology (CUP, Cambridge, 2001).
  72. Y.-A. Chen and S. Tata, Higher cup products on hypercubic lattices: Application to lattice models of topological phases, J. Math. Phys. 64, 091902 (2023).
  73. T. Lan, C. Zhu, and X.-G. Wen, Fermion decoration construction of symmetry-protected trivial order for fermion systems with any symmetry and in any dimension, Phys. Rev. B 100, 235141 (2019).
  74. X. Chen, A. Dua, P.-S. Hsin, C.-M. Jian, W. Shirley, and C. Xu, Loops in 4 + 1D topological phases, SciPost Phys. 15, 001 (2023).
  75. S. Tata, R. Kobayashi, D. Bulmash, and M. Barkeshli, Anomalies in (2 + 1)D fermionic topological phases and (3 + 1)D path integral state sums for fermionic SPTs, Commun. Math. Phys. 397, 199 (2023).
  76. This is in analogy to the famous gauge transformation a→→a→+∇χ in Maxwell’s theory or the corresponding gauge theory in the continuum, where a→ is the quantum vector potential. The gauge invariance of the magnetic field B=∇×a→=0 is satisfied due to the Stokes theorem: ∇×∇χ=0.
  77. For the definition and construction of the D-dimensional arithmetic manifold, see Ref. [58] for details.
  78. (2,2) means both the logical-X and logical-Z operators are supported on 2-cycles.
  79. P. S. Schaller, Extremal Riemann surfaces with a large number of systoles (1995).
  80. J. Stallings, in Topology of 3-Manifolds and Related Topics; Proceedings of the University of Georgia Institute, 1961 (Prentice-Hall, Englewood Cliffs, NJ, 1962), p. 95, https://books.google.com/books/about/Topology_of_3_manifolds.html?id=Sss-AAAAIAAJ.
  81. I. Agol, Criteria for virtual fibering, J. Topol. 1, 269 (2008).
  82. B. Farb and D. Margalit, A Primer on Mapping Class Groups (PMS-49) (Princeton University Press, 2012).
  83. J. Brock and K. Bromberg, Inflexibility, Weil-Petersson distance, and volumes of fibered 3-manifolds, arXiv:1412.0733.
  84. B. Eastin and E. Knill, Restrictions on transversal encoded quantum gate sets, Phys. Rev. Lett. 102, 81 (2009).
  85. C. Chamberland, A. Kubica, T. J. Yoder, and G. Zhu, Triangular color codes on trivalent graphs with flag qubits, New J. Phys. 22, 023019 (2020).
  86. K. Sahay and B. J. Brown, Decoder for the triangular color code by matching on a Möbius strip, PRX Quantum 3, 010310 (2022).
  87. 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).
  88. P. Sarnak, Prime geodesic theorems, thesis, Stanford University, 1980.

Outline

Information

PRX Quantum 6, 040361– Published 11 December, 2025
  • Received 7 April 2024
  • Revised 17 June 2025
  • Accepted 6 November 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation