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  • Open Access

Non-Clifford Gates between Stabilizer Codes via Non-Abelian Topological Order

Rohith Sajith1,*, Zijian Song2, Brenden Roberts1, Varun Menon1, and Yabo Li3

  • *Contact author: rohithsajith@g.harvard.edu
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PRX Quantum 7, 010361 – Published 27 March, 2026

DOI: https://doi.org/10.1103/qttm-y8t1

Abstract

We propose protocols to implement non-Clifford logical gates between stabilizer codes by entangling into a non-Abelian topological order as an intermediate step. Generalizing previous approaches, we provide a framework that generates a large class of non-Clifford and nondiagonal logical gates between qudit surface codes by gauging the topological symmetry of symmetry-enriched topological orders. As our main example, we concretely detail a protocol that utilizes the quantum double of S3 to generate a controlled-charge-conjugation (CC) gate between a qubit and qutrit surface code. Both the preparation of non-Abelian states and logical state injection between the Abelian and non-Abelian codes are executed via finite-depth quantum circuits with measurement and feedforward. We discuss aspects of the fault tolerance of our protocol, presenting insights into how to construct a heralded decoder for the quantum double of S3. We also outline how analogous protocols can be used to obtain logical gates between qudit surface codes by entangling into D(G), where G is a semidirect product of Abelian groups. This work serves as a step toward classifying the computational power of non-Abelian quantum phases beyond the paradigm of anyon braiding on near-term quantum devices.

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Popular Summary

Fault-tolerant quantum computing relies on quantum error-correcting codes to protect information from noise. While many logical operations can be performed easily within these codes, achieving universal quantum computation requires implementing the so-called non-Clifford gates, operations that are notoriously difficult to realize in a fault-tolerant way. Our work introduces a new framework for generating such gates by temporarily transforming a quantum system into a more exotic phase of matter.

We explore how non-Abelian topological phases, which can host particles with rich braiding properties, can serve as intermediates for enacting complex logical operations. By leveraging a concept known as symmetry gauging, which promotes a global symmetry of a topological phase to a local one, we construct transitions between distinct quantum codes and use these transitions to realize non-Clifford gates. This generalizes earlier techniques and opens the door to a wide class of nondiagonal, entangling operations. We also propose a heralded decoding scheme for a particular topological phase—the S3 quantum double—to maintain fault tolerance throughout the process.

By fusing ideas from topological quantum matter and quantum error correction, this work offers a powerful new approach to implementing non-Clifford gates, potentially advancing the path toward scalable, universal quantum computing.

Article Text

References (51)

  1. X.-G. Wen, Topological orders in rigid states, Int. J. Mod. Phys. B 4, 239 (1990).
  2. X.-G. Wen and Q. Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces, Phys. Rev. B 41, 9377 (1990).
  3. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  4. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002).
  5. D. Bluvstein et al., Logical quantum processor based on reconfigurable atom arrays, Nature 626, 58 (2024).
  6. S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletić, and M. D. Lukin, High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023).
  7. S. A. Moses et al., A race-track trapped-ion quantum processor, Phys. Rev. X 13, 041052 (2023).
  8. R. Acharya et al., Quantum error correction below the surface code threshold, Nature 638, 920 (2024).
  9. 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, Nat. Commun. 16, 6301 (2025).
  10. 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).
  11. C. Mochon, Anyons from nonsolvable finite groups are sufficient for universal quantum computation, Phys. Rev. A 67, 022315 (2003).
  12. C. Mochon, Anyon computers with smaller groups, Phys. Rev. A 69, 032306 (2004).
  13. S. X. Cui, S.-M. Hong, and Z. Wang, Universal quantum computation with weakly integral anyons, Quantum Inf. Process. 14, 2687 (2015).
  14. M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, Topological quantum computation, Bull. Am. Math. Soc. 40, 31 (2003).
  15. S. B. Bravyi and A. Y. Kitaev, Fermionic quantum computation, Ann. Phys. 298, 210 (2002).
  16. S. Bravyi, I. Kim, A. Kliesch, and R. Koenig, Adaptive constant-depth circuits for manipulating non-Abelian anyons, arXiv:2205.01933.
  17. R. Verresen, N. Tantivasadakarn, and A. Vishwanath, Efficiently preparing Schrödinger’s cat, fractons and non-Abelian topological order in quantum devices, arXiv:2112.03061.
  18. N. Tantivasadakarn, A. Vishwanath, and R. Verresen, A hierarchy of topological order from finite-depth unitaries, measurement and feedforward, PRX Quantum 4, 020339 (2023).
  19. N. Tantivasadakarn, R. Verresen, and A. Vishwanath, The shortest route to non-Abelian topological order on a quantum processor, Phys. Rev. Lett. 131, 060405 (2023).
  20. Y. Ren, N. Tantivasadakarn, and D. J. Williamson, Efficient preparation of solvable anyons with adaptive quantum circuits, Phys. Rev. X 15, 031060 (2025).
  21. Y. Li, H. Sukeno, A. P. Mana, H. P. Nautrup, and T.-C. Wei, Symmetry-enriched topological order from partially gauging symmetry-protected topologically ordered states assisted by measurements, Phys. Rev. B 108, 115144 (2023).
  22. G. Dauphinais and D. Poulin, Fault-tolerant quantum error correction for non-Abelian anyons, Commun. Math. Phys. 355, 519 (2017).
  23. 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).
  24. 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.
  25. A. Bauer and J. C. Magdalena de la Fuente, Planar fault-tolerant circuits for non-Clifford gates on the 2D color code, arXiv:2505.05175.
  26. D. Jing, P. Sala, L. Jiang, and R. Verresen, Intrinsic heralding and optimal decoders for non-Abelian topological order, arXiv:2507.23765.
  27. S.-J. Huang and Y. Chen, Generating logical magic states with the aid of non-Abelian topological order, arXiv:2502.00998.
  28. 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].
  29. Y. Ren and P. Shor, Topological quantum computation assisted by phase transitions, arXiv:2311.00103.
  30. B. Yoshida, Topological phases with generalized global symmetries, Phys. Rev. B 93, 155131 (2016).
  31. A. Kubica and B. Yoshida, Ungauging quantum error-correcting codes, arXiv:1805.01836.
  32. N. Tantivasadakarn, R. Thorngren, A. Vishwanath, and R. Verresen, Long-range entanglement from measuring symmetry-protected topological phases, Phys. Rev. X 14, 021040 (2024).
  33. Y. Li, M. Litvinov, and T.-C. Wei, Measuring topological field theories: Lattice models and field-theoretic description, arXiv:2310.17740 [cond-mat.str-el].
  34. X.-G. Wen, Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders, Phys. Rev. D 88, 045013 (2013).
  35. A. Lyons, C. F. B. Lo, N. Tantivasadakarn, A. Vishwanath, and R. Verresen, Protocols for creating anyons and defects via gauging, Phys. Rev. Lett. 135, 200405 (2025).
  36. Y. Li, Z. Song, A. Kubica, and I. H. Kim, Domain walls from SPT-sewing, arXiv:2411.11967 [quant-ph].
  37. F. H. E. Watson, H. Anwar, and D. E. Browne, Fast fault-tolerant decoder for qubit and qudit surface codes, Phys. Rev. A 92, 032309 (2015).
  38. J. R. Wootton, J. Burri, S. Iblisdir, and D. Loss, Error correction for non-Abelian topological quantum computation, Phys. Rev. X 4, 011051 (2014).
  39. 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).
  40. S.-J. Huang, A. Warman, S. Schafer-Nameki, and Y. Chen, Hybrid lattice surgery: Non-Clifford gates via non-Abelian surface codes, arXiv:2510.20890 [quant-ph].
  41. N. Manjunath, V. Mattei, A. Tiwari, and T. D. Ellison, Universal quantum computation with group surface codes, (2025) (to be published).
  42. M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, Symmetry fractionalization, defects, and gauging of topological phases, Phys. Rev. B 100, 115147 (2019).
  43. D. Bulmash and M. Barkeshli, Gauging fractons: Immobile non-Abelian quasiparticles, fractals, and position-dependent degeneracies, Phys. Rev. B 100, 155146 (2019).
  44. K. Laubscher, D. Loss, and J. R. Wootton, Universal quantum computation in the surface code using non-Abelian islands, Phys. Rev. A 100, 012338 (2019).
  45. L. Chen, Y. Ren, R. Fan, and A. Jaffe, A universal circuit set using the S3 quantum double, npj Quantum Inf. 11, 112 (2025).
  46. L. M. Adleman, J. DeMarrais, and M.-D. A. Huang, Quantum computability, SIAM J. Comput. 26, 1524 (1997).
  47. A. Barenco, A universal two-bit gate for quantum computation, Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci. 449, 679 (1995).
  48. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Phys. Rev. A 52, 3457 (1995).
  49. E. Bernstein and U. Vazirani, in Proceedings of the Twenty-fifth Annual ACM Symposium on Theory of Computing (SIAM Journal on Computing, Philadelphia, PA, 1993), pp. 1411–1473.
  50. D. E. Deutsch, Quantum computational networks, Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 425, 73 (1989).
  51. P. O. Boykin, T. Mor, M. Pulver, V. Roychowdhury, and F. Vatan, On universal and fault-tolerant quantum computing, arXiv:quant-ph/9906054.

Outline

Information

PRX Quantum 7, 010361– Published 27 March, 2026
  • Received 9 August 2025
  • Revised 7 November 2025
  • Accepted 8 January 2026
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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.

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