- 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

Measurement-Free Code-Switching Protocol for Low-Overhead Quantum Computation Using Permutation-Invariant Codes

Yingkai Ouyang1,*, Yumang Jing2, and Gavin K. Brennen2,3

  • 1School of Mathematical and Physical Sciences, University of Sheffield, Sheffield S3 7RH, United Kingdom
  • 2Center for Engineered Quantum Systems, Dept. of Physics and Astronomy, Macquarie University, 2109 NSW, Australia
  • 3BTQ Technologies, 16-104 555 Burrard Street, Vancouver, British Columbia, Canada V7X 1M8

  • *Contact author: y.ouyang@sheffield.ac.uk
PDF

PRX Quantum 6, 040341 – Published 20 November, 2025

DOI: https://doi.org/10.1103/p28m-hb6x

Abstract

Transversal gates on quantum error correction codes have been a promising approach for fault-tolerant quantum computing, but are limited by the Eastin-Knill no-go theorem. Existing solutions such as gate teleportation and magic state distillation are resource-intensive. We present a measurement-free code-switching protocol for universal quantum computation, switching between a stabilizer code for transversal Clifford gates and a permutation-invariant (PI) code for transversal non-Clifford gates that are logical Z rotations for any rational multiple of π. The novel non-Clifford gates enabled by this code-switching protocol provide for a lower gate count implementation of a universal gate set relative to the Clifford + T gate set. To achieve this, we present a protocol for performing controlled-not gates between the codes using near-term quantum control operations that employ a catalytic bosonic mode. We also present a new class of PI codes with tunable code distance, supporting transversal non-Clifford gates, and demonstrate their reduced gate count overhead relative to a comparable stabilizer code to stabilizer code-switching scheme.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Fault-tolerant quantum computation incurs substantial overhead when performing non-Clifford gates. Our protocol implements measurement-free switching between conventional stabilizer codes (used for transversal logical Clifford gates) and new permutation-invariant (PI) codes of tunable code-distance that allow direct operations of transversal non-Clifford gates that are logical Z rotations of rational multiples of π.

At the core of our proposed code-switching algorithm is (1) a measurement-free state-teleportation protocol that uses only logical controlled-not (cnot) gates between a stabilizer code and a PI code and (2) our compilation of cnot gates that act between a stabilizer code and a PI code in terms of near-term quantum control. The requirements of our protocol are modest. First, we need strong, linear coupling between the spins and a catalytic bosonic mode. In the case of coupling to a quantized cavity mode, this translates into the requirement for high cooperativity. Second, we require that the stabilizer code admits transversal implementations of logical Clifford gates. Third, the stabilizer code should be an even-odd quantum code meaning its logical zero (one) codeword can be written as a superposition of even (odd)-weight computational basis states.

The novel transversal non-Clifford gates that we introduce allow the implementation of gates in the binary icosahedral group and also an approximate supergolden gate to allow the implementation of the most efficient known single-qubit universal gate set. We then demonstrate the reduced gate count overhead of our protocol in terms of quantum control complexity relative to a comparable stabilizer code to stabilizer code-switching scheme.

Article Text

References (74)

  1. B. Eastin, and E. Knill, Restrictions on transversal encoded quantum gate sets, Phys. Rev. Lett. 102, 110502 (2009).
  2. E. T. Campbell, B. M. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature 549, 172 (2017).
  3. D. Gottesman, and I. L. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations, Nature 402, 390 (1999).
  4. X. Zhou, D. W. Leung, and I. L. Chuang, Methodology for quantum logic gate construction, Phys. Rev. A 62, 052316 (2000).
  5. S. Bravyi, and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005).
  6. S. Bravyi, and J. Haah, Magic-state distillation with low overhead, Phys. Rev. A 86, 052329 (2012).
  7. A. Krishna, and J.-P. Tillich, Towards low overhead magic state distillation, Phys. Rev. Lett. 123, 070507 (2019).
  8. E. T. Campbell, and M. Howard, Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost, Phys. Rev. A 95, 022316 (2017).
  9. E. T. Campbell, and M. Howard, Unifying gate synthesis and magic state distillation, Phys. Rev. Lett. 118, 060501 (2017).
  10. B. Eastin, Distilling one-qubit magic states into Toffoli states, Phys. Rev. A 87, 032321 (2013).
  11. M. B. Hastings, and J. Haah, Distillation with sublogarithmic overhead, Phys. Rev. Lett. 120, 050504 (2018).
  12. J. O’Gorman, and E. T. Campbell, Quantum computation with realistic magic-state factories, Phys. Rev. A 95, 032338 (2017).
  13. M. E. Beverland, A. Kubica, and K. M. Svore, Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes, PRX Quantum 2, 020341 (2021).
  14. D. Litinski, Magic state distillation: Not as costly as you think, Quantum 3, 205 (2019).
  15. 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).
  16. H. Bombín, Dimensional jump in quantum error correction, New J. Phys. 18, 043038 (2016).
  17. A. Kubica, and M. E. Beverland, Universal transversal gates with color codes: A simplified approach, Phys. Rev. A 91, 032330 (2015).
  18. H. Poulsen Nautrup, N. Friis, and H. J. Briegel, Fault-tolerant interface between quantum memories and quantum processors, Nat. Commun. 8, 1321 (2017).
  19. F. Butt, S. Heußen, M. Rispler, and M. Müller, Fault-tolerant code-switching protocols for near-term quantum processors, PRX Quantum 5, 020345 (2024).
  20. I. Pogorelov, F. Butt, L. Postler, C. D. Marciniak, P. Schindler, M. Müller, and T. Monz, Experimental fault-tolerant code switching, ArXiv:2403.13732.
  21. J. Huang, S. M. Li, L. Yeh, A. Kissinger, M. Mosca, and M. Vasmer, Graphical CSS code transformation using ZX calculus, ArXiv:2307.02437.
  22. L. Daguerre, and I. H. Kim, Code switching revisited: Low-overhead magic state preparation using color codes, ArXiv:2410.07327.
  23. S. Heußen, and J. Hilder, Efficient fault-tolerant code switching via one-way transversal cnot gates, ArXiv:2409.13465.
  24. B. J. Brown, A fault-tolerant non-Clifford gate for the surface code in two dimensions, Sci. Adv. 6, eaay4929 (2020).
  25. S. Heußen, D. F. Locher, and M. Müller, Measurement-free fault-tolerant quantum error correction in near-term devices, PRX Quantum 5, 010333 (2024).
  26. E. Kubischta, and I. Teixeira, Family of quantum codes with exotic transversal gates, Phys. Rev. Lett. 131, 240601 (2023).
  27. O. Parzanchevski, and P. Sarnak, Super-golden-gates for PU(2), Adv. Math. 327, 869 (2018), special volume honoring David Kazhdan.
  28. H. Bombín, Gauge color codes: Optimal transversal gates and gauge fixing in topological stabilizer codes, New J. Phys. 17, 083002 (2015).
  29. A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett. 77, 793 (1996).
  30. P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995).
  31. D. Bacon, Operator quantum error-correcting subsystems for self-correcting quantum memories, Phys. Rev. A 73, 012340 (2006).
  32. P. Aliferis, and A. W. Cross, Subsystem fault tolerance with the Bacon-Shor code, Phys. Rev. Lett. 98, 220502 (2007).
  33. M. B. Ruskai, Pauli exchange errors in quantum computation, Phys. Rev. Lett. 85, 194 (2000).
  34. H. Pollatsek, and M. B. Ruskai, Permutationally invariant codes for quantum error correction, Linear Algebra Appl. 392, 255 (2004).
  35. Y. Ouyang, Permutation-invariant quantum codes, Phys. Rev. A 90, 062317 (2014).
  36. Y. Ouyang, and J. Fitzsimons, Permutation-invariant codes encoding more than one qubit, Phys. Rev. A 93, 042340 (2016).
  37. Y. Ouyang, Permutation-invariant qudit codes from polynomials, Linear Algebra Appl. 532, 43 (2017).
  38. R. Movassagh, and Y. Ouyang, Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians, Quantum 8, 1541 (2024).
  39. M. T. Johnsson, N. R. Mukty, D. Burgarth, T. Volz, and G. K. Brennen, Geometric pathway to scalable quantum sensing, Phys. Rev. Lett. 125, 190403 (2020).
  40. Y. Ouyang, Permutation-invariant quantum coding for quantum deletion channels, in 2021 IEEE International Symposium on Information Theory (ISIT) (IEEE, Melbourne, 2021), p. 1499.
  41. Y. Ouyang, and G. K. Brennen, Finite round quantum error correction on symmetric quantum sensors, ArXiv:2212.06285.
  42. M. Hagiwara, The four qubits deletion code is the first quantum insertion code, IEICE Commun. Express 10, 243 (2021).
  43. T. Shibayama, and Y. Ouyang, The equivalence between correctability of deletions and insertions of separable states in quantum codes, in 2021 IEEE Information Theory Workshop (ITW) (IEEE, Kanazawa, 2021), p. 1.
  44. T. Shibayama, and M. Hagiwara, Equivalence of quantum single insertion and single deletion error-correctabilities, and construction of codes and decoders, in 2022 IEEE International Symposium on Information Theory (ISIT) (IEEE, Espoo, Finland, 2022), p. 2957.
  45. Y. Ouyang, N. Shettell, and D. Markham, Robust quantum metrology with explicit symmetric states, IEEE Trans. Inf. Theory 68, 1809 (2022).
  46. Y. Ouyang, Quantum storage in quantum ferromagnets, Phys. Rev. B 103, 144417 (2021).
  47. E. Kubischta, and I. Teixeira, The not-so-secret fourth parameter of quantum codes, ArXiv:2310.17652.
  48. A. Aydin, M. A. Alekseyev, and A. Barg, A family of permutationally invariant quantum codes, Quantum 8, 1321 (2024).
  49. J. A. Gross, Designing codes around interactions: The case of a spin, Phys. Rev. Lett. 127, 010504 (2021).
  50. E. Kubischta, and I. Teixeira, Family of quantum codes with exotic transversal gates, Phys. Rev. Lett. 131, 240601 (2023).
  51. P. O. Boykin, T. Mor, M. Pulver, V. Roychowdhury, and F. Vatan, On universal and fault-tolerant quantum computing: A novel basis and a new constructive proof of universality for Shor’s basis, in 40th Annual Symposium on Foundations of Computer Science, 1999 (IEEE Computer Society, New York City, NY, 1999), p. 486.
  52. K. Mølmer, and A. Sørensen, Multiparticle Entanglement of Hot Trapped Ions, Phys. Rev. Lett. 82, 1835 (1999).
  53. A. Luis, Quantum mechanics as a geometric phase: phase-space interferometers, J. Phys. A: Math. Gen. 34, 7677 (2001).
  54. D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate, Nature 422, 412 (2003).
  55. S. Jandura, V. Srivastava, L. Pecorari, G. K. Brennen, and G. Pupillo, Nonlocal multiqubit quantum gates via a driven cavity, Phys. Rev. A 110, 062610 (2024).
  56. A. S. Sørensen, and K. Mølmer, Measurement induced entanglement and quantum computation with atoms in optical cavities, Phys. Rev. Lett. 91, 097905 (2003).
  57. L. B. Nguyen, Y.-H. Lin, A. Somoroff, R. Mencia, N. Grabon, and V. E. Manucharyan, High-coherence fluxonium qubit, Phys. Rev. X 9, 041041 (2019).
  58. A. André, D. DeMille, J. M. Doyle, M. D. Lukin, S. E. Maxwell, P. Rabl, R. J. Schoelkopf, and P. Zoller, A coherent all-electrical interface between polar molecules and mesoscopic superconducting resonators, Nat. Phys. 2, 636 (2006).
  59. X. Wang, and P. Zanardi, Simulation of many-body interactions by conditional geometric phases, Phys. Rev. A 65, 032327 (2002).
  60. N. Gutman, A. Gorlach, O. Tziperman, R. Ruimy, and I. Kaminer, Universal control of symmetric states using spin squeezing, Phys. Rev. Lett. 132, 153601 (2024).
  61. L. J. Bond, M. J. Davis, J. c. v. Minář, R. Gerritsma, G. K. Brennen, and A. Safavi-Naini, Global variational quantum circuits for arbitrary symmetric state preparation, Phys. Rev. Res. 7, L022072 (2025).
  62. V. Srivastava, S. Jandura, G. K. Brennen, and G. Pupillo, Entanglement-enhanced quantum sensing via optimal global control, ArXiv:2409.12932.
  63. G. K. Brennen, D. P. O’Leary, and S. S. Bullock, Criteria for exact qudit universality, Phys. Rev. A 71, 052318 (2005).
  64. A. Gilchrist, N. K. Langford, and M. A. Nielsen, Distance measures to compare real and ideal quantum processes, Phys. Rev. A–At., Mol. Opt. Phys. 71, 062310 (2005).
  65. P. Aliferis, D. Gottesman, and J. Preskill, Quantum accuracy threshold for concatenated distance-3 codes, Quantum. Inf. Comput. 6, 97 (2006).
  66. Y. Ouyang, Avoiding coherent errors with rotated concatenated stabilizer codes, Npj Quantum Inf. 7 (2021).
  67. T. Zhang, M. Wu, S. R. Cohen, L. Xin, D. Das, K. K. S. Multani, N. Peard, A.-M. Valente-Feliciano, P. B. Welander, A. H. Safavi-Naeini, E. A. Nanni, and M. Schleier-Smith, Optically accessible high-finesse millimeter-wave resonator for cavity quantum electrodynamics with atom arrays, ArXiv:2506.05804.
  68. S. Kuhr, S. Gleyzes, C. Guerlin, J. Bernu, U. B. Hoff, S. Deléglise, S. Osnaghi, M. Brune, J.-M. Raimond, S. Haroche, E. Jacques, P. Bosland, and B. Visentin, Ultrahigh finesse fabry-pérot superconducting resonator, Appl. Phys. Lett. 90, 164101 (2007).
  69. C. D. Marciniak, T. Feldker, I. Pogorelov, R. Kaubruegger, D. V. Vasilyev, R. van Bijnen, P. Schindler, P. Zoller, R. Blatt, and T. Monz, Optimal metrology with programmable quantum sensors, Nature 603, 604 (2022).
  70. S. P. Jain, and V. V. Albert, Transversal Clifford and T-gate codes of short length and high distance, IEEE J. Sel. Areas Inf. Theory 6, 127 (2025).
  71. D. Gottesman, Fault-tolerant quantum computation with constant overhead, Quantum Inf. Comput. 14, 1338 (2014).
  72. R. Chao, and B. W. Reichardt, Quantum error correction with only two extra qubits, Phys. Rev. Lett. 121, 050502 (2018).
  73. R. Chao, and B. W. Reichardt, Flag fault-tolerant error correction for any stabilizer code, PRX Quantum 1, 010302 (2020).
  74. Y. Ouyang, and R. Chao, Permutation-invariant constant-excitation quantum codes for amplitude damping, IEEE Trans. Inf. Theory 66, 2921 (2019).

Outline

Information

PRX Quantum 6, 040341– Published 20 November, 2025
  • Received 15 January 2025
  • Revised 10 June 2025
  • Accepted 17 October 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