- 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

Fault-Tolerant Operations in Majorana-Based Quantum Codes: Gates, Measurements, and High-Rate Constructions

Maryam Mudassar1,*, Alexander Schuckert1,2, and Daniel Gottesman1

  • *Contact author: maryammu@umd.edu
PDF

PRX Quantum 7, 020315 – Published 24 April, 2026

DOI: https://doi.org/10.1103/bglx-ssjg

Abstract

Majorana-based quantum computation in nanowires and neutral atoms has gained prominence as a promising platform to encode qubits and protect them against noise. In order to run computations reliably on such devices, a fully fault-tolerant scheme is needed for state preparation, gates, and measurements. However, current fault-tolerant schemes have either been limited to specific code families or have not been developed fully. In this work, we extend the tools of qubit fault tolerance to Majorana codes. We emphasize the division between even and odd Majorana codes and how it manifests when constructing fault-tolerant gadgets for these families. We provide transversal constructions and supplement them with measurements to obtain several examples of fault-tolerant Clifford gadgets. For the case of odd codes, we give a construction for gadgets using quantum reference frames that allows one to implement operations that are forbidden due to parity superselection. We also provide a fault-tolerant measurement scheme for Majorana codes inspired by Steane error correction, enabling state preparation, measurement of logical operations, and error correction. We also point out a construction for odd Majorana codes with transversal T gates. Finally, we construct a high-rate quantum low density parity check codes (LDPC) Majorana code with logical qubits. Our work shows that all necessary elements of fault-tolerant quantum computation can be consistently implemented in fermionic hardware such as Majorana nanowires and fermionic neutral atoms.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Building large-scale quantum computers is a formidable challenge due to the fragility of quantum states. While most architectures focus on qubits, fermionic platforms such as Majorana nanowires and neutral atom arrays are emerging as competitive alternatives. However, these systems face unique constraints, such as the conservation of fermion parity, which forbids certain essential operations. In this work, we provide a theoretical foundation for fault-tolerant computing using Majorana-based codes. By utilizing quantum reference frames, we demonstrate how to bypass parity restrictions to implement logical gates previously thought impossible. We successfully adapt established techniques, including Steane error correction and magic state distillation, to the native Majorana architecture. Finally, we propose high-rate Majorana codes capable of encoding O(N) logical qubits on N physical fermions. This research provides a dictionary for translating fault-tolerant protocols into the language of fermions, paving the way for the next generation of quantum hardware.

Article Text

References (63)

  1. M. Aghaee, et al., Interferometric single-shot parity measurement in InAs–Al hybrid devices, Nature 638, 651 (2025).
  2. T. Liu, C. Y. Wan, H. Yang, Y. Zhao, B. Xie, W. Zheng, Z. Yi, D. Guan, S. Wang, H. Zheng, C. Liu, L. Fu, J. Liu, Y. Li, and J. Jia, Signatures of hybridization of multiple Majorana zero modes in a vortex, Nature 633, 71 (2024).
  3. J. Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Rep. Prog. Phys. 75, 076501 (2012).
  4. D. Aasen et al., Roadmap to fault tolerant quantum computation using topological qubit arrays, arXiv:2502.12252 [quant-ph].
  5. A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. 44, 131 (2001).
  6. J. K. Pachos, Introduction to Topological Quantum Computation (Cambridge University Press, Cambridge, 2012).
  7. P. Bonderson, M. Freedman, and C. Nayak, Measurement-only topological quantum computation, Phys. Rev. Lett. 101, 010501 (2008).
  8. A. Alase, M. C. Goffage, M. C. Cassidy, and S. N. Coppersmith, Decoherence of Majorana qubits by 1/f noise, arXiv:2506.22394 [cond-mat.mes-hall].
  9. S. Bravyi, B. Leemhuis, and B. M. Terhal, Majorana fermion codes, New J. Phys. 12, 083039 (2010).
  10. M. B. Hastings, Small Majorana fermion codes, arXiv:1703.00612 [quant-ph].
  11. D. Litinski and F. von Oppen, Quantum computing with Majorana fermion codes, Phys. Rev. B 97, 205404 (2018).
  12. C. McLauchlan and B. Béri, A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation, Quantum 8, 1400 (2024).
  13. Y. Li, Fault-tolerant fermionic quantum computation based on color code, Phys. Rev. A 98, 012336 (2018).
  14. A. Tran, A. Bocharov, B. Bauer, and P. Bonderson, Optimizing Clifford gate generation for measurement-only topological quantum computation with Majorana zero modes, SciPost Phys. 8, 091 (2020).
  15. A. Paetznick, C. Knapp, N. Delfosse, B. Bauer, J. Haah, M. B. Hastings, and M. P. da Silva, Performance of planar floquet codes with Majorana-based qubits, PRX Quantum 4, 010310 (2023).
  16. Y. Li, Noise threshold and resource cost of fault-tolerant quantum computing with Majorana fermions in hybrid systems, Phys. Rev. Lett. 117, 120403 (2016).
  17. R. Chao, M. E. Beverland, N. Delfosse, and J. Haah, Optimization of the surface code design for Majorana-based qubits, Quantum 4, 352 (2020).
  18. S. Kundu and B. Reichardt, Majorana subsystem qubit codes that also correct odd-weight errors, New J. Phys. 26, 073029 (2024).
  19. V. Bettaque and B. Swingle, The structure of the Majorana Clifford group, arXiv:2407.11319 [quant-ph].
  20. M. Mudassar, R. W. Chien, and D. Gottesman, Encoding Majorana codes, Phys. Rev. A 110, 032430 (2024).
  21. C. K. McLauchlan and B. Béri, Fermion-parity-based computation and its Majorana-zero-mode implementation, Phys. Rev. Lett. 128, 180504 (2022).
  22. D. Litinski and F. von Oppen, Braiding by Majorana tracking and long-range CNOT gates with color codes, Phys. Rev. B 96, 205413 (2017).
  23. T. E. O’Brien, P. Rożek, and A. R. Akhmerov, Majorana-based fermionic quantum computation, Phys. Rev. Lett. 120, 220504 (2018).
  24. Z. Z. Yan, B. M. Spar, M. L. Prichard, S. Chi, H.-T. Wei, E. Ibarra-García-Padilla, K. R. Hazzard, and W. S. Bakr, Two-dimensional programmable tweezer arrays of fermions, Phys. Rev. Lett. 129, 123201 (2022).
  25. D. González-Cuadra, D. Bluvstein, M. Kalinowski, R. Kaubruegger, N. Maskara, P. Naldesi, T. V. Zache, A. M. Kaufman, M. D. Lukin, H. Pichler, B. Vermersch, J. Ye, and P. Zoller, Fermionic quantum processing with programmable neutral atom arrays, Proc. Natl. Acad. Sci. 120, e2304294120 (2023).
  26. P. Bojović, T. Hilker, S. Wang, J. Obermeyer, M. Barendregt, D. Tell, T. Chalopin, P. M. Preiss, I. Bloch, and T. Franz, High-fidelity collisional quantum gates with fermionic atoms, arXiv:2506.14711 [cond-mat.quant-gas].
  27. A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, Fermion-qubit fault-tolerant quantum computing, arXiv:2411.08955 [quant-ph].
  28. R. Ott, D. González-Cuadra, T. V. Zache, P. Zoller, A. M. Kaufman, and H. Pichler, Error-corrected fermionic quantum processors with neutral atoms, arXiv:2412.16081 [quant-ph].
  29. A. J. Landahl and B. C. A. Morrison, Logical fermions for fault-tolerant quantum simulation, arXiv:2110.10280 [quant-ph].
  30. C.-Y. Xu, Z.-C. Liu, and Y. Xu, Fermion-to-fermion low-density parity-check codes, arXiv:2508.15323 [quant-ph].
  31. M. Chiew, B. Harrison, and S. Strelchuk, Ternary tree transformations are equivalent to linear encodings of the fock basis, arXiv:2412.07578 [quant-ph].
  32. F. Šimkovic IV, M. Leib, and F. R. F. Pereira, Low-weight high-distance error-correcting fermionic encodings, Phys. Rev. Res. 6, 043123 (2024).
  33. N. Maskara, M. Kalinowski, D. Gonzalez-Cuadra, and M. D. Lukin, Fast simulation of fermions with reconfigurable qubits, arXiv:2509.08898.
  34. N. Constantinides, J. Yu, D. Devulapalli, A. Fahimniya, L. Schaeffer, A. M. Childs, M. J. Gullans, A. Schuckert, and A. V. Gorshkov, Low-depth fermion routing without ancillas, arXiv:2510.05099 [quant-ph].
  35. D. Zhang and T. Cubitt, Quantum error transmutation, arXiv:2310.10278 [quant-ph].
  36. X. Wang, E. Khatami, F. Fei, J. Wyrick, P. Namboodiri, R. Kashid, A. F. Rigosi, G. Bryant, and R. Silver, Experimental realization of an extended Fermi-Hubbard model using a 2D lattice of dopant-based quantum dots, Nat. Commun. 13, 1 (2022).
  37. A. Rad, A. Schuckert, E. Crane, G. Nambiar, F. Fei, J. Wyrick, R. M. Silver, M. Hafezi, Z. Davoudi, and M. J. Gullans, Analog quantum simulator of a quantum field theory with fermion-spin systems in silicon, arXiv:2407.03419.
  38. D. Gottesman, Surviving as a Quantum Computer in a Classical World (University of Maryland, 2024), draft textbook for CMSC858G: Quantum Error Correction, https://www.cs.umd.edu/∼dgottesm/QECCbook-2024.pdf.
  39. B. Eastin and E. Knill, Restrictions on transversal encoded quantum gate sets, Phys. Rev. Lett. 102, 110502 (2009).
  40. C. Knapp, M. Beverland, D. I. Pikulin, and T. Karzig, Modeling noise and error correction for Majorana-based quantum computing, Quantum 2, 88 (2018).
  41. R. W. Chien and J. Klassen, Optimizing fermionic encodings for both hamiltonian and hardware, arXiv:2210.05652 [quant-ph].
  42. N. T. Vidal, M. L. Bera, A. Riera, M. Lewenstein, and M. N. Bera, Quantum operations in an information theory for fermions, Phys. Rev. A 104, 032411 (2021).
  43. S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reference frames, superselection rules, and quantum information, Rev. Mod. Phys. 79, 555 (2007).
  44. S. Vijay and L. Fu, Quantum error correction for complex and Majorana fermion qubits, arXiv:1703.00459 [cond-mat.mes-hall].
  45. A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett. 77, 793 (1996).
  46. L. H. Kauffman, Majorana fermions and representations of the braid group, Int. J. Mod. Phys. A 33, 18300235 (2018).
  47. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett. 85, 2208 (2000).
  48. C. Moore and M. Nilsson, Parallel quantum computation and quantum codes, arXiv:quant-ph/9808027 [quant-ph].
  49. G. Gour and R. W. Spekkens, The resource theory of quantum reference frames: Manipulations and monotones, New J. Phys. 10, 033023 (2008).
  50. E. Kubischta and I. Teixeira, Family of quantum codes with exotic transversal gates, Phys. Rev. Lett. 131, 240601 (2023).
  51. A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction via codes over gf(4), arXiv:quant-ph/9608006 [quant-ph].
  52. H. Sayginel, S. Koutsioumpas, M. Webster, A. Rajput, and D. E. Browne, Fault-tolerant logical Clifford gates from code automorphisms, arXiv:2409.18175 [quant-ph].
  53. N. Berthusen, M. J. Gullans, Y. Hong, M. Mudassar, and S. J. S. Tan, Automorphism gadgets in homological product codes, arXiv:2508.04794 [quant-ph].
  54. D. Gottesman, Theory of fault-tolerant quantum computation, Phys. Rev. A 57, 127 (1998).
  55. S. B. Bravyi and A. Y. Kitaev, Fermionic quantum computation, Ann. Phys. 298, 210 (2002).
  56. S. Bravyi and J. Haah, Magic-state distillation with low overhead, Phys. Rev. A 86, 052329 (2012).
  57. J. T. Anderson, G. Duclos-Cianci, and D. Poulin, Fault-tolerant conversion between the steane and reed-muller quantum codes, Phys. Rev. Lett. 113, 080501 (2014).
  58. F. Butt, S. Heußen, M. Rispler, and M. Müller, Fault-tolerant code-switching protocols for near-term quantum processors, PRX Quantum 5, 020345 (2024).
  59. J. Farinholt, Quantum LDPC codes constructed from point-line subsets of the finite projective plane, arXiv:1207.0732 [quant-ph].
  60. D. J. C. MacKay, G. Mitchison, and P. L. McFadden, Sparse-graph codes for quantum error correction, IEEE Trans. Inf. Theory 50, 2315 (2004).
  61. A. Couvreur, N. Delfosse, and G. Zémor, A construction of quantum LDPC codes from Cayley graphs, IEEE Trans. Inf. Theory 59, 6087 (2013).
  62. 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).
  63. L. Spagnoli, A. Roggero, and N. Wiebe, Fault-tolerant simulation of lattice gauge theories with gauge covariant codes, Quantum 10, 1968 (2026).

Outline

Information

PRX Quantum 7, 020315– Published 24 April, 2026
  • Received 16 September 2025
  • Revised 9 January 2026
  • Accepted 25 March 2026
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