Skip to main content
SpringerLink
Log in

Efficient Trainability of Linear Optical Modules in Quantum Optical Neural Networks

  • Published:
Journal of Russian Laser Research Aims and scope

Abstract

The existence of “barren plateau landscapes” for generic discrete-variable quantum neural networks, which obstructs efficient gradient-based optimization of cost functions defined by global measurements, would be surprising in the case of generic linear optical modules in quantum optical neural networks due to the tunability of the intensity of continuous variable states and the relevant unitary group having exponentially smaller dimension. We demonstrate that coherent light in m modes can be generically compiled efficiently if the total intensity scales sublinearly with m, and extend this result to cost functions based on homodyne, heterodyne, or photon detection measurement statistics, and to noisy cost functions in the presence of attenuation. We further demonstrate efficient trainability of m mode linear optical quantum circuits for variational mean field energy estimation of positive quadratic Hamiltonians for input states that do not have energy exponentially vanishing with m.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Steinbrecher, J. Olson, D. Englund, and J. Carolan, NPJ Quantum Inf., 5, 60 (2019); DOI:https://doi.org/10.1038/s41534-019-0174-7

    Article  ADS  Google Scholar 

  2. N. Harris, J. Carolan, D. Bunandar, et al., Optica, 5, 1623 (2018); DOI:https://doi.org/10.1364/OPTICA.5.001623

    Article  ADS  Google Scholar 

  3. A. Peruzzo, J. McClean, P. Shadbolt, et al., Nat. Commun., 5, 4213 (2014); DOI:https://doi.org/10.1038/ncomms5213

    Article  ADS  Google Scholar 

  4. Y. Yamamoto, K. Aihara, T. Leleu, et al., NPJ Quantum Inf., 3, 49 (2017); DOI:https://doi.org/10.1038/s41534-017-0048-9

    Article  ADS  Google Scholar 

  5. N. Killoran, T. R. Bromley, J. M. Arrazola, et al., Phys. Rev. Res., 1, 033063 (2019); DOI:https://doi.org/10.1103/PhysRevResearch.1.033063

    Article  Google Scholar 

  6. J. Carolan, M. Mohseni, J. Olson, et al., Nat. Phys., 16, 322 (2020); DOI:https://doi.org/10.1038/s41567-019-0747-6

    Article  Google Scholar 

  7. Y. Shen, N. C. Harris, S. Skirlo, et al., Nat. Photon., 11, 441 (2017); DOI:https://doi.org/10.1038/NPHOTON.2017.93

    Article  ADS  Google Scholar 

  8. J. R. McClean, S. Boixo, V. N. Smelyanskiy, et al., Nat. Commun., 9, 4812 (2018); DOI:https://doi.org/10.1038/s41467-018-07090-4

    Article  ADS  Google Scholar 

  9. J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, New J. Phys., 18, 023023 (2016); DOI:https://doi.org/10.1088/1367-2630/18/2/023023/meta

    Article  ADS  Google Scholar 

  10. T. Volkoff and P. Coles, Quantum Sci. Technol., 6, 025008 (2021); DOI:https://doi.org/10.1088/2058-9565/abd891

    Article  ADS  Google Scholar 

  11. F. Brandao, A. Harrow, and M. Horodecki, Commun. Math. Phys., 346, 397 (2016); DOI:https://doi.org/10.1007/s00220-016-2706-8

    Article  ADS  Google Scholar 

  12. M. Cerezo, A. Sone, T. Volkoff, et al., Nat. Commun., 12, 1791 (2021); DOI:https://doi.org/10.1038/s41467-021-21728-w

    Article  ADS  Google Scholar 

  13. T. Volkoff, J. Math. Phys., 58, 122202 (2017); DOI:https://doi.org/10.1063/1.4998512

    Article  ADS  MathSciNet  Google Scholar 

  14. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam (1982).

    MATH  Google Scholar 

  15. F. Centrone, N. Kumar, E. Diamanti, and I. Kerenidis, arXiv:2007.15876 (2020).

  16. S. Aaronson, S. Beigi, A. Drucker, and P. Shor, Theory Comput., 5, 1 (2009); DOI:https://doi.org/10.4086/toc.2009.v005a001

    Article  MathSciNet  Google Scholar 

  17. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York (1965).

    MATH  Google Scholar 

  18. M. Guta and J. Kahn, Phys. Rev. A, 73, 052108 (2006); DOI:https://doi.org/10.1103/PhysRevA.73.052108

    Article  ADS  MathSciNet  Google Scholar 

  19. C. Dankert, R. Cleve, J. Emerson, and E. Livine, Phys. Rev. A, 80, 012304 (2009); DOI:https://doi.org/10.1103/PhysRevA.80.012304

    Article  ADS  Google Scholar 

  20. S. Khatri, R. LaRose, A. Poremba, et al., Quantum, 3, 140 (2019); DOI: https://doi.org/10.22331/q-2019-05-13-140.

    Article  Google Scholar 

  21. A. S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, de Gruyter, Berlin/Boston (2012); DOI:https://doi.org/10.1515/9783110642490

  22. S. Wang, E. Fontana, M. Cerezo, et al., arXiv:2007.14384 (2020).

  23. J. Derezinski, J. Math. Phys., 58, 121101 (2017); DOI:https://doi.org/10.1063/1.5017931

    Article  ADS  MathSciNet  Google Scholar 

  24. C. Weedbrook, S. Pirandola, R. García-Patrón, et al., Rev. Mod. Phys., 84, 621 (2012); DOI:10.1103/RevModPhys.84.621

  25. S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, Phys. Rev. Lett., 88, 097904 (2002); DOI:https://doi.org/10.1103/PhysRevLett.88.097904

    Article  ADS  Google Scholar 

  26. S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” in: Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (2011), p. 333; DOI:10.1145/1993636.1993682

  27. C. S. Hamilton, R. Kruse, L. Sansoni, et al., Phys. Rev. Lett., 119, 170501 (2017); DOI:https://doi.org/10.1103/PhysRevLett.119.170501

    Article  ADS  Google Scholar 

  28. R. Kruse, C. S. Hamilton, L. Sansoni, et al., Phys. Rev. A, 100, 032326 (2019); DOI:https://doi.org/10.1103/PhysRevA.100.032326

    Article  ADS  Google Scholar 

  29. K. Beer, D. Bondarenko, T. Farrelly, et al., Nat. Commun., 11, 808 (2020); DOI:https://doi.org/10.1038/s41467-020-14454-2

    Article  ADS  Google Scholar 

  30. K. Sharma, M. Cerezo, L. Cincio, and P. J. Coles, arXiv:2005.12458 (2020).

  31. D. Braun, J. Phys. A: Math. Gen., 39, 14581 (2006); DOI:https://doi.org/10.1088/0305-4470/39/47/005

    Article  ADS  Google Scholar 

  32. G. Ortiz, R. Somma, J. Dukelsky, and S. Rombouts, Nucl. Phys. B, 707, 421 (2005); DOI:https://doi.org/10.1016/j.nuclphysb.2004.11.008

    Article  ADS  Google Scholar 

  33. L. Ymai, A. Tonel, A. Foerster, and J. Links, J. Phys. A: Math. Theor., 50, 264001 (2017); DOI:https://doi.org/10.1088/1751-8121/aa7227

    Article  ADS  Google Scholar 

  34. R. Richardson, J. Math. Phys., 9, 1327 (1968); DOI:https://doi.org/10.1063/1.1664719

    Article  ADS  Google Scholar 

  35. T. J. Volkoff and C. M. Herdman, Phys. Rev. A, 100, 022331 (2019); DOI:https://doi.org/10.1103/PhysRevA.100.022331

    Article  ADS  MathSciNet  Google Scholar 

  36. T. J. Volkoff, Phys. Rev. A, 94, 042327 (2016); DOI:https://doi.org/10.1103/PhysRevA.94.042327

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    Tyler J. Volkoff

Authors
  1. Tyler J. Volkoff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tyler J. Volkoff.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Volkoff, T.J. Efficient Trainability of Linear Optical Modules in Quantum Optical Neural Networks. J Russ Laser Res 42, 250–260 (2021). https://doi.org/10.1007/s10946-021-09958-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10946-021-09958-1

Keywords

Search

Navigation