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  • Open Access

Multimode rotationally symmetric bosonic codes from group-theoretic construction

Rabsan Galib Ahmed1,2,3, Adithi Udupa2, and Giulia Ferrini2,*

  • *Contact author: ferrini@chalmers.se
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Phys. Rev. Research 8, 023194 – Published 21 May, 2026

DOI: https://doi.org/10.1103/xt48-fxt5

Abstract

We introduce a family of multimode, rotationally symmetric bosonic codes inspired by the group-theoretic framework introduced by Denys and Leverrier [Phys. Rev. Lett. 133, 240603 (2024)]. Such a construction inverts the traditional paradigm of code design by identifying codes from the requirement that a group of chosen logical gates should be implemented by means of physically simple logical operations, such as linear optics. Leveraging previously unexplored degrees of freedom within this framework, our construction results in codes that display rotational symmetry across multiple modes, while enabling linear-optics implementation of the full Pauli group. These codes exhibit improved protection against dephasing noise, outperforming both single-mode analogues and earlier multimode constructions. Notably, they allow exact correction of correlated dephasing and support qudit encoding in arbitrary dimensions. We analytically construct and numerically benchmark two-mode binomial code instances, and demonstrate that, unlike single-mode rotationally symmetric bosonic codes, these exhibit no trade-off between protection against dephasing and photon loss.

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References (45)

  1. V. V. Albert, Bosonic coding: Introduction and use cases, in Quantum Fluids of Light and Matter, edited by A. Bramati, I. Carusotto, and C. Ciuti (IOS Press, Amsterdam, 2025), pp. 79–107.
  2. Z. Ni, S. Li, X. Deng, Y. Cai, L. Zhang, W. Wang, Z.-B. Yang, H. Yu, F. Yan, S. Liu, et al., Beating the break-even point with a discrete-variable-encoded logical qubit, Nature (London) 616, 56 (2023).
  3. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. Brock, A. Ding, L. Frunzio, et al., Real-time quantum error correction beyond break-even, Nature (London) 616, 50 (2023).
  4. D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ultrahigh error threshold for surface codes with biased noise, Phys. Rev. Lett. 120, 050505 (2018).
  5. J. Zhang, Y.-C. Wu, and G.-P. Guo, Concatenation of the Gottesman-Kitaev-Preskill code with the XZZX surface code, Phys. Rev. A 107, 062408 (2023).
  6. K. Noh and C. Chamberland, Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code, Phys. Rev. A 101, 012316 (2020).
  7. K. Noh, C. Chamberland, and F. G. S. L. Brandão, Low-overhead fault-tolerant quantum error correction with the surface-GKP code, PRX Quantum 3, 010315 (2022).
  8. K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, High-threshold fault-tolerant quantum computation with analog quantum error correction, Phys. Rev. X 8, 021054 (2018).
  9. N. Raveendran, N. Rengaswamy, F. Rozpędek, A. Raina, L. Jiang, and B. Vasić, Finite rate QLDPC-GKP coding scheme that surpasses the CSS Hamming bound, Quantum 6, 767 (2022).
  10. L. Berent, T. Hillmann, J. Eisert, R. Wille, and J. Roffe, Analog information decoding of bosonic quantum low-density parity-check codes, PRX Quantum 5, 020349 (2024).
  11. D. Ruiz, J. Guillaud, A. Leverrier, M. Mirrahimi, and C. Vuillot, LDPC-cat codes for low-overhead quantum computing in 2D, Nat. Commun. 16, 1040 (2025).
  12. É. Gouzien, D. Ruiz, F.-M. Le Régent, J. Guillaud, and N. Sangouard, Performance analysis of a repetition cat code architecture: Computing 256-bit elliptic curve logarithm in 9 hours with 126 133 cat qubits, Phys. Rev. Lett. 131, 040602 (2023).
  13. A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quantum computing with rotation-symmetric bosonic codes, Phys. Rev. X 10, 011058 (2020).
  14. A. Denys and A. Leverrier, Quantum error-correcting codes with a covariant encoding, Phys. Rev. Lett. 133, 240603 (2024).
  15. M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, New class of quantum error-correcting codes for a bosonic mode, Phys. Rev. X 6, 031006 (2016).
  16. Even though the physical representation is given using the passive linear operations on the d modes, note its difference from the one given in Ref. [14], which assumes a physical representation that is isomorphic to the Pauli group itself while the one proposed here is more general.
  17. M. Bergmann and P. van Loock, Quantum error correction against photon loss using NOON states, Phys. Rev. A 94, 012311 (2016).
  18. L. García-Álvarez, C. Calcluth, A. Ferraro, and G. Ferrini, Efficient simulatability of continuous-variable circuits with large Wigner negativity, Phys. Rev. Res. 2, 043322 (2020).
  19. Y. Xu, Y. Wang, and V. V. Albert, Multimode rotation-symmetric bosonic codes from homological rotor codes, Phys. Rev. A 110, 022402 (2024).
  20. E. Descamps, A. Keller, and P. Milman, Gottesman-Kitaev-Preskill encoding in continuous modal variables of single photons, Phys. Rev. Lett. 132, 170601 (2024).
  21. We further note that for the code words defined in Eqs. (4) and (5), when N is odd, the implementation of the Z gate remains unchanged and the logical X gate can instead be implemented as ÛBSe−iπĜ12−/2eiπâ1†â1/NÛBS†.
  22. E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature (London) 409, 46 (2001).
  23. J. Guillaud and M. Mirrahimi, Repetition cat qubits for fault-tolerant quantum computation, Phys. Rev. X 9, 041053 (2019).
  24. V. V. Albert, S. O. Mundhada, A. Grimm, S. Touzard, M. H. Devoret, and L. Jiang, Pair-cat codes: Autonomous error-correction with low-order nonlinearity, Quantum Sci. Technol. 4, 035007 (2019).
  25. S. S. Chelluri, S. Sharma, F. Schmidt, S. V. Kusminskiy, and P. van Loock, Bosonic quantum error correction with microwave cavities for quantum repeaters, arXiv:2503.21569.
  26. J. Steinbach and J. Twamley, Motional quantum error correction, J. Mod. Opt. 47, 453 (2000).
  27. Furthermore, our codes are also distinct from the bosonic quantum Fourier codes based on cat states along a similar group-theoretic construction [43].
  28. I. L. Chuang, D. W. Leung, and Y. Yamamoto, Bosonic quantum codes for amplitude damping, Phys. Rev. A 56, 1114 (1997).
  29. A. S. Fletcher, P. W. Shor, and M. Z. Win, Optimum quantum error recovery using semidefinite programming, Phys. Rev. A 75, 012338 (2007).
  30. A. Udupa, T. Hillmann, R. G. Ahmed, A. Smirne, and G. Ferrini, Performance of rotation-symmetric bosonic codes in the presence of non-Markovian effects induced by random telegraph noise, Phys. Rev. Res. 8, 023007 (2026).
  31. U. Leonhardt, J. A. Vaccaro, B. Böhmer, and H. Paul, Canonical and measured phase distributions, Phys. Rev. A 51, 84 (1995).
  32. J. Bergli, Y. M. Galperin, and B. L. Altshuler, Decoherence of a qubit by non-Gaussian noise at an arbitrary working point, Phys. Rev. B 74, 024509 (2006).
  33. J. Bergli, Y. M. Galperin, and B. L. Altshuler, Decoherence in qubits due to low-frequency noise, New J. Phys. 11, 025002 (2009).
  34. J. Schriefl, Y. Makhlin, A. Shnirman, and G. Schön, Decoherence from ensembles of two-level fluctuators, New J. Phys. 8, 1 (2006).
  35. C. Müller, J. H. Cole, and J. Lisenfeld, Towards understanding two-level-systems in amorphous solids: Insights from quantum circuits, Rep. Prog. Phys. 82, 124501 (2019).
  36. C. Benedetti, M. G. A. Paris, and S. Maniscalco, Non-Markovianity of colored noisy channels, Phys. Rev. A 89, 012114 (2014).
  37. R. Lescanne, M. Villiers, T. Peronnin, A. Sarlette, M. Delbecq, B. Huard, T. Kontos, M. Mirrahimi, and Z. Leghtas, Exponential suppression of bit-flips in a qubit encoded in an oscillator, Nat. Phys. 16, 509 (2020).
  38. B. J. Chapman, S. J. de Graaf, S. H. Xue, Y. Zhang, J. Teoh, J. C. Curtis, T. Tsunoda, A. Eickbusch, A. P. Read, A. Koottandavida, S. O. Mundhada, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, High-on-off-ratio beam-splitter interaction for gates on bosonically encoded qubits, PRX Quantum 4, 020355 (2023).
  39. B. Royer, S. Singh, and S. M. Girvin, Encoding qubits in multimode grid states, PRX Quantum 3, 010335 (2022).
  40. Nord Quantique, Suppressing logical errors with multimode quantum error correction, (2025), https://nordquantique.com/articles/suppressing-logical-errors-multimode-quantum-error-correction/.
  41. T. Hillmann, F. Quijandría, G. Johansson, A. Ferraro, S. Gasparinetti, and G. Ferrini, Universal gate set for continuous-variable quantum computation with microwave circuits, Phys. Rev. Lett. 125, 160501 (2020).
  42. A. M. Eriksson, T. Sépulcre, M. Kervinen, T. Hillmann, M. Kudra, S. Dupouy, Y. Lu, M. Khanahmadi, J. Yang, C. Castillo-Moreno, et al., Universal control of a bosonic mode via drive-activated native cubic interactions, Nat. Commun. 15, 2512 (2024).
  43. A. Leverrier, Bosonic quantum Fourier codes, Quantum 10, 2000 (2026).
  44. G. Zheng, W. He, G. Lee, and L. Jiang, Near-optimal performance of quantum error correction codes, Phys. Rev. Lett. 132, 250602 (2024).
  45. T. Hillmann, F. Quijandría, A. L. Grimsmo, and G. Ferrini, Performance of teleportation-based error-correction circuits for bosonic codes with noisy measurements, PRX Quantum 3, 020334 (2022).

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Phys. Rev. Research 8, 023194– Published 21 May, 2026
  • Received 28 October 2025
  • Accepted 6 May 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. Funded by Bibsam.

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