- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Concatenated dual displacement code for continuous-variable quantum error correction

Fucheng Guo* and Frank Mueller†

Yuan Liu‡

  • Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, North Carolina 27695, USA; Department of Computer Science, North Carolina State University, Raleigh, North Carolina 27695, USA; and Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA

  • *Contact author: fguo22@ncsu.edu
  • †Contact author: fmuelle@ncsu.edu
  • ‡Contact author: q_yuanliu@ncsu.edu
PDF

Phys. Rev. Research 8, 023158 – Published 12 May, 2026

DOI: https://doi.org/10.1103/d1zs-cy4t

Abstract

The continuous-variable (CV) Gaussian no-go theorem fundamentally limits the suppression of Gaussian displacement errors using only Gaussian gates and states. Prior studies have employed Gottesman-Kitaev-Preskill (GKP) states as ancillary qumodes to suppress small Gaussian displacement errors. However, when the displacement magnitude becomes large, inevitable lattice-crossing errors arise beyond the correctable range of the GKP state. To address this issue, we concatenate the Gaussian-noise-suppression circuit with an outer analog Steane code that corrects such occasional lattice-crossing events as well as other abrupt displacement errors. Contrary to conventional concatenation, which primarily aims to reduce logical error rates, the Steane-GKP duality in encoding provides complementary protection against displacement errors at different scales: The inner GKP layer employs non-Gaussian resources to suppress continuous Gaussian noise and reduce residual variance, while the outer analog Steane code corrects discrete lattice-crossing events that exceed the GKP correctable range. It is precisely this separation of error-mitigation roles that enables CV error correction. In contrast to prior work on concatenating GKP and repetition codes to establish error correction for discrete qubit/qudit encoding, we provide correction in the continuous encoding space. Analytical studies show that, under infinite squeezing, the concatenated code suppresses the variance of Gaussian displacement errors acting on all qumodes by up to 50%, while enabling unbiased correction of lattice-crossing errors with a success probability determined by the ratio between the residual Gaussian error standard deviation and the lattice-crossing magnitude. Even with finite squeezing, the proposed architecture still provides Gaussian-error suppression and lattice-crossing correction. Moreover, the presence of the outer analog Steane code relaxes the squeezing requirement of the inner GKP states, indicating near-term experimental feasibility. This work establishes a viable route toward fault-tolerant continuous-variable quantum computation and provides insight into the design of concatenated CV error-correcting architectures.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (55)

  1. P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493(R) (1995).
  2. A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett. 77, 793 (1996).
  3. D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology, 1997.
  4. W. Cai, Y. Ma, W. Wang, C.-L. Zou, and L. Sun, Bosonic quantum error correction codes in superconducting quantum circuits, Fundam. Res. 1, 50 (2021).
  5. A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).
  6. G. Q. AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature (London) 614, 676 (2023).
  7. Y. Xu, Y. Wang, E.-J. Kuo, and V. V. Albert, Qubit-oscillator concatenated codes: Decoding formalism and code comparison, PRX Quantum 4, 020342 (2023).
  8. M. P. Stafford and N. C. Menicucci, Biased Gottesman-Kitaev-Preskill repetition code, Phys. Rev. A 108, 052428 (2023).
  9. Z. Li and D. Su, Correcting biased noise using Gottesman-Kitaev-Preskill repetition code with noisy ancilla, Phys. Rev. A 109, 052420 (2024).
  10. S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005).
  11. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
  12. A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  13. J. Niset, J. Fiurášek, and N. J. Cerf, No-go theorem for Gaussian quantum error correction, Phys. Rev. Lett. 102, 120501 (2009).
  14. B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).
  15. D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001).
  16. S. Glancy and E. Knill, Error analysis for encoding a qubit in an oscillator, Phys. Rev. A 73, 012325 (2006).
  17. 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).
  18. K. Noh, S. M. Girvin, and L. Jiang, Encoding an oscillator into many oscillators, Phys. Rev. Lett. 125, 080503 (2020).
  19. S. L. Braunstein, Error correction for continuous quantum variables, Phys. Rev. Lett. 80, 4084 (1998).
  20. T. Aoki, G. Takahashi, T. Kajiya, J. Yoshikawa, S. L. Braunstein, P. van Loock, and A. Furusawa, Quantum error correction beyond qubits, Nat. Phys. 5, 541 (2009).
  21. A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, Fault-tolerant fermionic quantum computing, arXiv:2411.08955.
  22. K. Fukui, T. Matsuura, and N. C. Menicucci, Efficient concatenated bosonic code for additive Gaussian noise, Phys. Rev. Lett. 131, 170603 (2023).
  23. C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T. Campbell, C. T. Hann, J. Iverson, H. Putterman, T. C. Bohdanowicz, S. T. Flammia, et al., Building a fault-tolerant quantum computer using concatenated cat codes, PRX Quantum 3, 010329 (2022).
  24. A. Steane, Multiple-particle interference and quantum error correction, Proc. R. Soc. London A 452, 2551 (1996).
  25. T. Kalajdzievski and J. M. Arrazola, Exact gate decompositions for photonic quantum computing, Phys. Rev. A 99, 022341 (2019).
  26. D. Su, C. Weedbrook, and K. Brádler, Correcting finite squeezing errors in continuous-variable cluster states, Phys. Rev. A 98, 042304 (2018).
  27. Y. Liu, S. Singh, K. C. Smith, E. Crane, J. M. Martyn, A. Eickbusch, A. Schuckert, R. D. Li, J. Sinanan-Singh, M. B. Soley, T. Tsunoda, I. L. Chuang, N. Wiebe, and S. M. Girvin, Hybrid oscillator-qubit quantum processors: Instruction set architectures, abstract machine models, and applications, PRX Quantum 7, 010201 (2026).
  28. J.-i. Yoshikawa, Y. Miwa, A. Huck, U. L. Andersen, P. van Loock, and A. Furusawa, Demonstration of a quantum nondemolition sum gate, Phys. Rev. Lett. 101, 250501 (2008).
  29. E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55, 900 (1997).
  30. S. Lloyd and J.-J. E. Slotine, Analog quantum error correction, Phys. Rev. Lett. 80, 4088 (1998).
  31. E. T. Hockings, A. C. Doherty, and R. Harper, Scalable noise characterization of syndrome-extraction circuits with averaged circuit eigenvalue sampling, PRX Quantum 6, 010334 (2025).
  32. D.-G. Welsch, W. Vogel, and T. Opatrný, Homodyne detection and quantum state reconstruction, Prog. Opt. 39, 63 (1999).
  33. C. González-Arciniegas, P. Nussenzveig, M. Martinelli, and O. Pfister, Cluster states from Gaussian states: Essential diagnostic tools for continuous-variable one-way quantum computing, PRX Quantum 2, 030343 (2021).
  34. B. W. Walshe, B. Q. Baragiola, R. N. Alexander, and N. C. Menicucci, Continuous-variable gate teleportation and bosonic-code error correction, Phys. Rev. A 102, 062411 (2020).
  35. L. Hu, Y. Ma, W. Cai, X. Mu, Y. Xu, W. Wang, Y. Wu, H. Wang, Y. Song, C. Zou, S. M. Girvin, L.-M. Duan, and L. Sun, Demonstration of quantum error correction and universal gate set on a binomial bosonic logical qubit, Nat. Phys. 15, 503 (2019).
  36. S. Hao, X. Deng, X. Su, X. Jia, C. Xie, and K. Peng, Gates for one-way quantum computation based on Einstein-Podolsky-Rosen entanglement, Phys. Rev. A 89, 032311 (2014).
  37. T. Kalajdzievski and N. Quesada, Exact and approximate continuous-variable gate decompositions, Quantum 5, 394 (2021).
  38. M. Walschaers, Non-Gaussian quantum states and where to find them, PRX Quantum 2, 030204 (2021).
  39. J. Wu and Q. Zhuang, Continuous-variable error correction for general Gaussian noises, Phys. Rev. Appl. 15, 034073 (2021).
  40. T. C. Ralph, Quantum error correction of continuous-variable states against Gaussian noise, Phys. Rev. A 84, 022339 (2011).
  41. M. Jiang, T. Wu, J. W. Blanchard, G. Feng, X. Peng, and D. Budker, Experimental benchmarking of quantum control in zero-field nuclear magnetic resonance, Sci. Adv. 4, eaar6327 (2018).
  42. X. Li, J. Wang, Y.-Y. Jiang, G.-M. Xue, X. Cai, J. Zhou, M. Gong, Z.-F. Liu, S.-Y. Zheng, D.-K. Ma, M. Chen, W.-J. Sun, S. Yang, F. Yan, Y.-R. Jin, S.-P. Zhao, X.-F. Ding, and H.-F. Yu, Cosmic-ray-induced correlated errors in superconducting qubit array, Nat. Commun. 16, 4677 (2025).
  43. A. Kessy, A. Lewin, and K. Strimmer, Optimal whitening and decorrelation, Am. Stat.172, 309 (2018).
  44. F. Guo, Dual-displacement CVQEC simulation code (2026), available at: https://doi.org/10.5281/zenodo.19596790.
  45. R. Schnabel, Squeezed states of light and their applications in laser interferometers, Phys. Rep. 684, 1 (2017).
  46. J. Conrad, J. Eisert, and F. Arzani, Gottesman-Kitaev-Preskill codes: A lattice perspective, Quantum 6, 648 (2022).
  47. G. Huang, A. Beccari, N. J. Engelsen, and T. J. Kippenberg, Room-temperature quantum optomechanics using an ultralow noise cavity, Nature (London) 626, 512 (2024).
  48. R. J. Epstein, S. Seidelin, D. Leibfried, J. H. Wesenberg, J. J. Bollinger, J. M. Amini, R. B. Blakestad, J. Britton, J. P. Home, D. Lucas, D. Stick, P. K. Ghosh, C. Monroe, and D. J. Wineland, Simplified motional heating rate measurements of trapped ions, Phys. Rev. A 76, 033411 (2007).
  49. N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Extending the lifetime of a quantum bit with error correction in superconducting circuits, Nature (London) 536, 441 (2016).
  50. N. Wang, S. Du, W. Liu, X. Wang, Y. Li, and K. Peng, Long-distance continuous-variable quantum key distribution with entangled states, Phys. Rev. Appl. 10, 064028 (2018).
  51. Y. Lu, A. Maiti, J. W. O. Garmon, S. Ganjam, Y. Zhang, J. Claes, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, High-fidelity parametric beamsplitting with a parity-protected converter, Nat. Commun. 14, 5767 (2023).
  52. Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis, A. Petrenko, K. M. Sliwa, A. Narla, S. Shankar, and M. H. Devoret, Confining the state of light to a quantum manifold by engineered two-photon loss, Science 347, 853 (2015).
  53. C. C. Bultink, B. Tarasinski, N. Haandbæk, S. Poletto, N. Haider, D. J. Michalak, A. Bruno, and L. DiCarlo, General method for extracting the quantum efficiency of dispersive qubit readout in circuit QED, Appl. Phys. Lett. 112, 092601 (2018).
  54. L. Hänggli and R. König, Oscillator-to-oscillator codes do not have a threshold, IEEE Trans. Inf. Theory 68, 1068 (2022).
  55. P. C. Encinar, A. Agustí, and C. Sabín, Digital quantum simulation of beam splitters and squeezing with IBM quantum computers, Phys. Rev. A 104, 052609 (2021).

Outline

Information

Phys. Rev. Research 8, 023158– Published 12 May, 2026
  • Received 28 November 2025
  • Accepted 12 April 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