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  • Letter
  • Open Access

Smallest quantum codes for amplitude-damping noise

Sourav Dutta1,*, Aditya Jain2,†, and Prabha Mandayam1

  • *Contact author: sourav@physics.iitm.ac.in
  • †Contact author: aj722@cam.ac.uk
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Phys. Rev. Research 8, L032004 – Published 10 July, 2026

DOI: https://doi.org/10.1103/6g5l-m4r1

Abstract

We describe the smallest quantum error correcting (QEC) code to correct for amplitude-damping (AD) noise, namely, a three-qubit code that corrects all the single-qubit damping errors. We generalize this construction to a family of codes that correct AD noise up to any fixed order of the damping strength. We underpin the fundamental connection between the structure of our codes and the noise structure, via a relaxed form of the Knill-Laflamme conditions, different from existing formulations of approximate QEC conditions. Although the recovery procedure for this code is nondeterministic, our codes are optimal with respect to overheads and outperform existing codes to tackle AD noise in terms of entanglement fidelity. This formulation of probabilistic QEC further leads us to a family of quantum codes tailored to AD noise and also gives rise to a noise-adapted quantum Hamming bound for AD noise. Finally, we construct a set of universal logical gates for the three-qubit code, thus providing a potential pathway to fault tolerance via this class of codes.

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References (37)

  1. B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).
  2. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  3. N. Cao, J. Lin, D. Kribs, Y.T. Poon, B. Zeng and R. Laflamme, NISQ Error correction, mitigation, and noise simulation, arXiv:2111.02345 (2021).
  4. B. Koczor, Exponential error suppression for near-term quantum devices, Phys. Rev. X 11, 031057 (2021).
  5. A. Jayashankar and P. Mandayam, Quantum error correction: Noise-adapted techniques and applications, J. Indian Inst. Sci. 103, 497 (2023).
  6. D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, Approximate quantum error correction can lead to better codes, Phys. Rev. A 56, 2567 (1997).
  7. A. S. Fletcher, P. W. Shor, and M. Z. Win, Channel-adapted quantum error correction for the amplitude damping channel, IEEE Trans. Inf. Theory 54, 5705 (2008).
  8. P. W. Shor, G. Smith, J. A. Smolin, and B. Zeng, High performance single-error-correcting quantum codes for amplitude damping, IEEE Trans. Inf. Theory 57, 7180 (2011).
  9. M. Grassl, L. Kong, Z. Wei, Z.-Q. Yin, and B. Zeng, Quantum error-correcting codes for qudit amplitude damping, IEEE Trans. Inf. Theory 64, 4674 (2018).
  10. A. Jayashankar, A. M Babu, H. K. Ng, and P. Mandayam, Finding good quantum codes using the Cartan form, Phys. Rev. A 101, 042307 (2020).
  11. S. Dutta, D. Biswas, and P. Mandayam, Noise-adapted qudit codes for amplitude-damping noise, Phys. Rev. A 111, 032438 (2025).
  12. Y. Ouyang and C.-Y. Lai, Linear programming bounds for approximate quantum error correction over arbitrary quantum channels, IEEE Trans. Inf. Theory 68, 5234 (2022).
  13. E. Knill, R. Laflamme, and L. Viola, Theory of quantum error correction for general noise, Phys. Rev. Lett. 84, 2525 (2000).
  14. C. Bény and O. Oreshkov, General conditions for approximate quantum error correction and near-optimal recovery channels, Phys. Rev. Lett. 104, 120501 (2010).
  15. H. K. Ng and P. Mandayam, Simple approach to approximate quantum error correction based on the transpose channel, Phys. Rev. A 81, 062342 (2010).
  16. A. Nayak and P. Sen, Invertible quantum operations and perfect encryption of quantum states, arXiv:quant-ph/0605041.
  17. M. A. Alhejji, Some problems concerning quantum channels and entropies, Ph.D. thesis, University of Colorado at Boulder, 2023.
  18. J. Fern and J. Terilla, Probabilistic quantum error correction, arXiv:quant-ph/0209058.
  19. R. Kukulski, Ł. Pawela, and Z. Puchała, On the probabilistic quantum error correction, IEEE Trans. Inf. Theory 69, 4620 (2023).
  20. P. Barberis-Blostein, D. G. Norris, L. A. Orozco, and H. J. Carmichael, From quantum feedback to probabilistic error correction: Manipulation of quantum beats in cavity QED, New J. Phys. 12, 023002 (2010).
  21. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
  22. I. L. Chuang, D. W. Leung, and Y. Yamamoto, Bosonic quantum codes for amplitude damping, Phys. Rev. A 56, 1114 (1997).
  23. R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Perfect quantum error correcting code, Phys. Rev. Lett. 77, 198 (1996).
  24. See Supplemental Material at http://link.aps.org/supplemental/10.1103/6g5l-m4r1 for detailed calculations, which includes Refs. [22, 31, 32, 33, 34, 35, 36, 37].
  25. A. S. Fletcher, P. W. Shor, and M. Z. Win, Optimum quantum error recovery using semidefinite programming, Phys. Rev. A 75, 012338 (2007).
  26. D. Eric Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology, 1997.
  27. Y. Ouyang and R. Chao, Permutation-invariant constant-excitation quantum codes for amplitude damping, IEEE Trans. Inf. Theory 66, 2921 (2020).
  28. V. Joshi, A. Rudra, S. Dutta, S. Dhomkar, and P. Mandayam, Demonstrating noise-adapted quantum error correction with break-even performance, arXiv:2603.04564.
  29. A. Jayashankar, M. D. H. Long, H. K. Ng, and P. Mandayam, Achieving fault tolerance against amplitude-damping noise, Phys. Rev. Res. 4, 023034 (2022).
  30. L. D. H. My, A. Jayashankar, P. Mandayam, and H. K. Ng, Fault tolerance against amplitude-damping noise using Bacon-Shor codes, Phys. Rev. Res. 7, 033275 (2025).
  31. C. Cafaro and P. van Loock, Approximate quantum error correction for generalized amplitude-damping errors, Phys. Rev. A 89, 022316 (2014).
  32. D. Biswas, G. M. Vaidya, and P. Mandayam, Noise-adapted recovery circuits for quantum error correction, Phys. Rev. Res. 6, 043034 (2024).
  33. H. Terashima and M. Ueda, Nonunitary quantum circuit, Int. J. Quantum Inf. 03, 633 (2005).
  34. B. E. Sagan, Combinatorics: The Art of Counting (American Mathematical Society, Providence, Rhode Island, 2020), Vol. 210.
  35. Victor V. Albert, Bosonic coding introduction and use cases, in Quantum Fluids of Light and Matter (IOS Press, 2025), pp. 79–107.
  36. A. Joshi, K. Noh, and Y. Y. Gao, Quantum information processing with Bosonic qubits in circuit QED, Quantum Sci. Technol. 6, 033001 (2021).
  37. 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).

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Phys. Rev. Research 8, L032004– Published 10 July, 2026
  • Received 12 March 2025
  • Accepted 10 June 2026
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