- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

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

Matrix Phase-Space Representations for Gaussian Boson Sampling

Peter D. Drummond*, Alexander S. Dellios, and Margaret D. Reid

  • *Contact author: peterddrummond@protonmail.com
PDF

Phys. Rev. Lett. 136, 013601 – Published 6 January, 2026

DOI: https://doi.org/10.1103/htk1-rn8v

Abstract

We introduce coherent matrix phase-space distributions. These use conservation laws and symmetries to improve the accuracy and speed of quantum phase-space representations. As an example, this is applied to the validation of low-loss Gaussian boson sampling (GBS) quantum computational advantage experiments, where classical generation of the random photon-number counts is exponentially hard. Large improvements in sampling errors are demonstrated compared to previous methods. Matrix phase-space representations also provide a large numerical speedup due to their (at worst) quadratic scaling, compared to other methods for validating total count probabilities of large-scale, low-loss GBS networks.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (53)

  1. E. Wigner, Phys. Rev. 40, 749 (1932).
  2. K. Husimi, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940).
  3. R. J. Glauber, Phys. Rev. 131, 2766 (1963).
  4. E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
  5. E. Schrödinger, Naturwissenschaften 23, 823 (1935).
  6. P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980).
  7. P. Deuar and P. D. Drummond, Phys. Rev. A 66, 033812 (2002).
  8. I. Carusotto, Y. Castin, and J. Dalibard, Phys. Rev. A 63, 023606 (2001).
  9. S. Aaronson, Proc. R. Soc. A 467, 3393 (2011).
  10. S. Aaronson and A. Arkhipov, Theory Comput. 9, 143 (2013).
  11. C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Phys. Rev. Lett. 119, 170501 (2017).
  12. A. Deshpande, A. Mehta, T. Vincent, N. Quesada, M. Hinsche, M. Ioannou, L. Madsen, J. Lavoie, H. Qi, J. Eisert, D. Hangleiter, B. Fefferman, and I. Dhand, Sci. Adv. 8, eabi7894 (2022).
  13. A. Björklund, B. Gupt, and N. Quesada, J. Exp. Algorithmics (JEA) 24, 1 (2019).
  14. J. F. F. Bulmer, B. A. Bell, R. S. Chadwick, A. E. Jones, D. Moise, A. Rigazzi, J. Thorbecke, U.-U. Haus, T. Van Vaerenbergh, R. B. Patel, I. A. Walmsley, and A. Laing, Sci. Adv. 8, eabl9236 (2022).
  15. P. D. Drummond, B. Opanchuk, A. Dellios, and M. D. Reid, Phys. Rev. A 105, 012427 (2022).
  16. P. D. Drummond, B. Opanchuk, L. Rosales-Zárate, M. D. Reid, and P. J. Forrester, Phys. Rev. A 94, 042339 (2016).
  17. J. Martínez-Cifuentes, H. de Guise, and N. Quesada, PRX Quantum 5, 040312 (2024).
  18. A. Ehrenberg, J. T. Iosue, A. Deshpande, D. Hangleiter, and A. V. Gorshkov, Phys. Rev. Lett. 134, 140601 (2025).
  19. A. Ehrenberg, J. T. Iosue, A. Deshpande, D. Hangleiter, and A. V. Gorshkov, Phys. Rev. A 111, 042412 (2025).
  20. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert et al., NIST Special Publication 800-22rev1a (2010), http://csrc.nist.gov/publications/nistpubs/800-22-rev1a/SP800-22rev1a.pdf.
  21. J. F. Bulmer, J. Martínez-Cifuentes, B. A. Bell, and N. Quesada, arXiv:2412.17742.
  22. P. D. Drummond and B. Opanchuk, Phys. Rev. Res. 2, 033304 (2020).
  23. H.-L. Liu, H. Su, S.-Q. Gong, Y.-C. Gu, H.-Y. Tang, M.-H. Jia, Q. Wei, Y. Song, D. Wang, M. Zheng et al., arXiv:2508.09092.
  24. S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
  25. W. J. Munro and M. D. Reid, Phys. Rev. A 47, 4412 (1993).
  26. P. Deuar and P. D. Drummond, Phys. Rev. Lett. 98, 120402 (2007).
  27. S. Kiesewetter, R. Y. Teh, P. D. Drummond, and M. D. Reid, Phys. Rev. Lett. 119, 023601 (2017).
  28. S. Kiesewetter, Q. Y. He, P. D. Drummond, and M. D. Reid, Phys. Rev. A 90, 043805 (2014).
  29. L. Rosales-Zárate, B. Opanchuk, P. D. Drummond, and M. D. Reid, Phys. Rev. A 90, 022109 (2014).
  30. K. Takata, A. Marandi, and Y. Yamamoto, Phys. Rev. A 92, 043821 (2015).
  31. P. D. Drummond and S. Chaturvedi, Phys. Scr. 91, 073007 (2016).
  32. R. Y. Teh, S. Kiesewetter, P. D. Drummond, and M. D. Reid, Phys. Rev. A 98, 063814 (2018).
  33. A. Dellios, P. D. Drummond, B. Opanchuk, R. Y. Teh, and M. D. Reid, Phys. Lett. A 429, 127911 (2022).
  34. A. S. Dellios, B. Opanchuk, N. Goodman, M. D. Reid, and P. D. Drummond, Phys. Lett. A 549, 130529 (2025).
  35. A. S. Dellios, M. D. Reid, and P. D. Drummond, Quantum Sci. Technol. 10, 045030 (2025).
  36. J. F. Corney and P. D. Drummond, Phys. Rev. A 68, 063822 (2003).
  37. J. F. Corney and P. D. Drummond, Phys. Rev. Lett. 93, 260401 (2004).
  38. P. D. Drummond and M. D. Reid, Phys. Rev. A 94, 063851 (2016).
  39. P. Deuar and P. D. Drummond, J. Phys. A 39, 2723 (2006).
  40. B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13 (1986).
  41. M. Wolinsky and H. J. Carmichael, Phys. Rev. Lett. 60, 1836 (1988).
  42. L. Krippner, W. J. Munro, and M. D. Reid, Phys. Rev. A 50, 4330 (1994).
  43. L. Gravina, F. Minganti, and V. Savona, PRX Quantum 4, 020337 (2023).
  44. S. Aaronson and A. Arkhipov, Quantum Inf. Comput. 14, 1383 (2014).
  45. E. Schrödinger, Naturwissenschaften 14, 664 (1926).
  46. P. Adam, I. Földesi, and J. Janszky, Phys. Rev. A 49, 1281 (1994).
  47. J. Huang and P. Kumar, Phys. Rev. A 40, 1670 (1989).
  48. C. Zhu and C. M. Caves, Phys. Rev. A 42, 6794 (1990).
  49. B. Opanchuk, L. Rosales-Zárate, M. D. Reid, and P. D. Drummond, Phys. Rev. A 97, 042304 (2018).
  50. All phase-space simulations were performed using the publicly available software package, xqsim, which is written in the matlab programming language, P. D. Drummond, A. S. Dellios, and N. Goodman, xqsim3, https://github.com/peterddrummond/xqsim (2025).
  51. S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
  52. P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, Nature (London) 365, 307 (1993).
  53. K. L. Ng, B. Opanchuk, M. D. Reid, and P. D. Drummond, Phys. Rev. Lett. 122, 203604 (2019).

Outline

Information

Phys. Rev. Lett. 136, 013601– Published 6 January, 2026
  • Received 2 June 2025
  • Accepted 10 December 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