- 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

Observation of Lie algebraic invariants in quantum linear optics

Giovanni Rodari1, Tommaso Francalanci1, Eugenio Caruccio1, Francesco Hoch1, Gonzalo Carvacho1, Taira Giordani1, Nicolò Spagnolo1, Riccardo Albiero2, Niki Di Giano2,3 et al.

Francesco Ceccarelli2, Giacomo Corrielli2, Andrea Crespi2,3, Roberto Osellame2, Ulysse Chabaud4, and Fabio Sciarrino1,*

  • *Contact author: fabio.sciarrino@uniroma1.it
PDF

Phys. Rev. Research 7, 043325 – Published 22 December, 2025

DOI: https://doi.org/10.1103/7961-hg2q

Abstract

Over the past few years, various methods have been developed to engineer and to exploit the dynamics of photonic quantum states as they evolve through linear optical networks. Recent theoretical works have shown that the underlying Lie algebraic structure plays a crucial role in the description of linear optical Hamiltonians, as such formalism identifies intrinsic symmetries within photonic systems subject to linear optical dynamics. Here, we experimentally investigate the role of Lie algebras applied to the context of Boson sampling, a pivotal model to the current understanding of computational complexity regimes in photonic quantum information. Performing experiments of increasing complexity, realized within a fully reconfigurable photonic circuit, we show that sampling experiments do indeed fulfill the constraints implied by a Lie algebraic structure. In addition, we provide a comprehensive picture about how the concept of Lie algebraic invariant can be interpreted from the point of view of nth-order correlation functions in quantum optics. Our work shows how Lie algebraic invariants can be used as a benchmark tool for the correctness of an underlying linear optical dynamics within typical interferometric architectures for photonic quantum information. This opens avenues for the use of algebraic-inspired methods as analysis tools for photon-based quantum computing protocols.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (72)

  1. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys. 79, 135 (2007).
  2. K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, Linear optical quantum metrology with single photons: Exploiting spontaneously generated entanglement to beat the shot-noise limit, Phys. Rev. Lett. 114, 170802 (2015).
  3. F. Ewert, M. Bergmann, and P. van Loock, Ultrafast long-distance quantum communication with static linear optics, Phys. Rev. Lett. 117, 210501 (2016).
  4. S. Slussarenko and G. J. Pryde, Photonic quantum information processing: A concise review, Appl. Phys. Rev. 6, 041303 (2019).
  5. F. Flamini, N. Spagnolo, and F. Sciarrino, Photonic quantum information processing: A review, Rep. Prog. Phys. 82, 016001 (2018).
  6. S. Aaronson and A. Arkhipov, The computational complexity of linear optics, in Proceedings of the 43rdAnnual ACM Symposium on Theory of Computing, STOC’11 (ACM, New York, 2011), p. 333.
  7. D. J. Brod, E. F. Galvão, A. Crespi, R. Osellame, N. Spagnolo, and F. Sciarrino, Photonic implementation of boson sampling: A review, Adv. Photon. 1, 034001 (2019).
  8. A. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. O’Brien, and T. Ralph, Boson sampling from a Gaussian state, Phys. Rev. Lett. 113, 100502 (2014).
  9. C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Gaussian boson sampling, Phys. Rev. Lett. 119, 170501 (2017).
  10. R. Kruse, C. S. Hamilton, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Detailed study of Gaussian boson sampling, Phys. Rev. A 100, 032326 (2019).
  11. U. Chabaud, D. Markham, and A. Sohbi, Quantum machine learning with adaptive linear optics, Quantum 5, 496 (2021).
  12. N. Spagnolo, D. J. Brod, E. F. Galvão, and F. Sciarrino, Non-linear boson sampling, npj Quantum Inf. 9, 3 (2023).
  13. S. Aaronson and L. Chen, Complexity-theoretic foundations of quantum supremacy experiments, in Proceedings of the 32nd Computational Complexity Conference (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, DEU, 2017).
  14. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  15. T. Heindel, J.-H. Kim, N. Gregersen, A. Rastelli, and S. Reitzenstein, Quantum dots for photonic quantum information technology, Adv. Opt. Photon. 15, 613 (2023).
  16. T. Giordani, F. Hoch, G. Carvacho, N. Spagnolo, and F. Sciarrino, Integrated photonics in quantum technologies, La Rivista del Nuovo Cimento 46, 71 (2023).
  17. J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, Integrated photonic quantum technologies, Nat. Photon. 14, 273 (2019).
  18. E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature (London) 409, 46 (2001).
  19. S. Bartolucci, P. Birchall, H. Bombín, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, F. Pastawski, T. Rudolph, and C. Sparrow, Fusion-based quantum computation, Nat. Commun. 14, 912 (2023).
  20. M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, Photonic boson sampling in a tunable circuit, Science 339, 794 (2013).
  21. J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, Boson sampling on a photonic chip, Science 339, 798 (2013).
  22. M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, Experimental boson sampling, Nat. Photon. 7, 540 (2013).
  23. A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, Integrated multimode interferometers with arbitrary designs for photonic boson sampling, Nat. Photon. 7, 545 (2013).
  24. J. Loredo, M. Broome, P. Hilaire, O. Gazzano, I. Sagnes, A. Lemaitre, M. Almeida, P. Senellart, and A. White, Boson sampling with single-photon Fock states from a bright solid-state source, Phys. Rev. Lett. 118, 130503 (2017).
  25. Y. He, X. Ding, Z.-E. Su, H.-L. Huang, J. Qin, C. Wang, S. Unsleber, C. Chen, H. Wang, Y.-M. He, X.-L. Wang, W.-J. Zhang, S.-J. Chen, C. Schneider, M. Kamp, L.-X. You, Z. Wang, S. Höfling, C.-Y. Lu, and J.-W. Pan, Time-bin-encoded boson sampling with a single-photon device, Phys. Rev. Lett. 118, 190501 (2017).
  26. J. Wang, S. Paesani, R. Santagati, S. Knauer, A. A. Gentile, N. Wiebe, M. Petruzzella, J. L. O’Brien, J. G. Rarity, A. Laing, and M. G. Thompson, Experimental quantum Hamiltonian learning, Nat. Phys. 13, 551 (2017).
  27. H.-S. Zhong, Y. Li, W. Li, L.-C. Peng, Z.-E. Su, Y. Hu, Y.-M. He, X. Ding, W. Zhang, H. Li, L. Zhang, Z. Wang, L. You, X.-L. Wang, X. Jiang, L. Li, Y.-A. Chen, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, 12-photon entanglement and scalable scattershot boson sampling with optimal entangled-photon pairs from parametric down-conversion, Phys. Rev. Lett. 121, 250505 (2018).
  28. H. Wang, J. Qin, X. Ding, M.-C. Chen, S. Chen, X. You, Y.-M. He, X. Jiang, L. You, Z. Wang, C. Schneider, J. J. Renema, S. Höfling, C.-Y. Lu, and J.-W. Pan, Boson sampling with 20 input photons and a 60-mode interferometer in a 1014-dimensional Hilbert space, Phys. Rev. Lett. 123, 250503 (2019).
  29. H.-S. Zhong, et al., Quantum computational advantage using photons, Science 370, 1460 (2020).
  30. L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, A. E. Lita, T. Gerrits, S. W. Nam, V. D. Vaidya, M. Menotti, I. Dhand, Z. Vernon, N. Quesada, and J. Lavoie, Quantum computational advantage with a programmable photonic processor, Nature (London) 606, 75 (2022).
  31. Y.-H. Deng, et al., Gaussian boson sampling with pseudo-photon-number-resolving detectors and quantum computational advantage, Phys. Rev. Lett. 131, 150601 (2023).
  32. F. Hoch, et al., Quantum machine learning with adaptive boson sampling via post-selection, Nat. Commun. 16, 902 (2025).
  33. G. Rodari, L. Novo, R. Albiero, A. Suprano, C. T. Tavares, E. Caruccio, F. Hoch, T. Giordani, G. Carvacho, M. Gardina, N. D. Giano, S. D. Giorgio, G. Corrielli, F. Ceccarelli, R. Osellame, N. Spagnolo, E. F. Galvão, and F. Sciarrino, Semi-device independent characterization of multiphoton indistinguishability, PRX Quantum 6, 020340 (2025).
  34. R. van der Meer, P. Hooijschuur, F. H. Somhorst, P. Venderbosch, M. de Goede, B. Kassenberg, H. Snijders, C. T. Taballione, J. Epping, H. van den Vlekkert, N. Walk, P. W. H. Pinkse, and J. J. Renema, Experimental demonstration of an efficient, semi-device-independent photonic indistinguishability witness, arXiv:2112.00067.
  35. M. Walschaers, J. Kuipers, J.-D. Urbina, K. Mayer, M. C. Tichy, K. Richter, and A. Buchleitner, Statistical benchmark for bosonsampling, New J. Phys. 18, 032001 (2016).
  36. T. Giordani, F. Flamini, M. Pompili, N. Viggianiello, N. Spagnolo, A. Crespi, R. Osellame, N. Wiebe, M. Walschaers, A. Buchleitner, and F. Sciarrino, Experimental statistical signature of many-body quantum interference, Nat. Photon. 12, 173 (2018).
  37. S. Aaronson and A. Arkhipov, Bosonsampling is far from uniform, Quantum Info. Comput. 14, 1383 (2014).
  38. M. C. Tichy, K. Mayer, A. Buchleitner, and K. Mølmer, Stringent and efficient assessment of boson-sampling devices, Phys. Rev. Lett. 113, 020502 (2014).
  39. A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, Suppression law of quantum states in a 3D photonic fast fourier transform chip, Nat. Commun. 7, 10469 (2016).
  40. N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, Experimental validation of photonic boson sampling, Nat. Photon. 8, 615 (2014).
  41. M. Pont, R. Albiero, S. E. Thomas, N. Spagnolo, F. Ceccarelli, G. Corrielli, A. Brieussel, N. Somaschi, H. Huet, A. Harouri, A. Lemaître, I. Sagnes, N. Belabas, F. Sciarrino, R. Osellame, P. Senellart, and A. Crespi, Quantifying n-photon indistinguishability with a cyclic integrated interferometer, Phys. Rev. X 12, 031033 (2022).
  42. I. Agresti, N. Viggianiello, F. Flamini, N. Spagnolo, A. Crespi, R. Osellame, N. Wiebe, and F. Sciarrino, Pattern recognition techniques for boson sampling validation, Phys. Rev. X 9, 011013 (2019).
  43. M. L. Goh, M. Larocca, L. Cincio, M. Cerezo, and F. Sauvage, Lie-algebraic classical simulations for quantum computing, Phys. Rev. Res. 7, 033266 (2025).
  44. P. V. Parellada, V. G. i Garcia, J. J. Moyano-Fernández, and J. C. Garcia-Escartin, Lie algebraic invariants in quantum linear optics, arXiv:2409.12223.
  45. P. V. Parellada, V. G. i Garcia, J. J. Moyano-Fernández, and J. C. Garcia-Escartin, No-go theorems for photon state transformations in quantum linear optics, Results Phys. 54, 107108 (2023).
  46. J. C. Garcia-Escartin, V. Gimeno, and J. J. Moyano-Fernández, Multiple photon effective Hamiltonians in linear quantum optical networks, Opt. Commun. 430, 434 (2019).
  47. N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, Near-optimal single-photon sources in the solid state, Nat. Photon. 10, 340 (2016).
  48. P. Senellart, G. Solomon, and A. White, High-performance semiconductor quantum-dot single-photon sources, Nat. Nanotechnol. 12, 1026 (2017).
  49. G. Corrielli, A. Crespi, and R. Osellame, Femtosecond laser micromachining for integrated quantum photonics, Nanophotonics 10, 3789 (2021).
  50. C. Pentangelo, F. Ceccarelli, S. Piacentini, R. Albiero, E. Urbinati, N. Di Giano, S. Atzeni, A. Crespi, and R. Osellame, Universal photonic processors fabricated by femtosecond laser writing, in Integrated Optics: Devices, Materials, and Technologies XXVI (SPIE, San Francisco, California, 2022), p. 19.
  51. C. Pentangelo, N. Di Giano, S. Piacentini, R. Arpe, F. Ceccarelli, A. Crespi, and R. Osellame, High-fidelity and polarization-insensitive universal photonic processors fabricated by femtosecond laser writing, Nanophotonics 13, 2259 (2024).
  52. W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, Optimal design for universal multiport interferometers, Optica 3, 1460 (2016).
  53. R. A. Campos, B. E. A. Saleh, and M. C. Teich, Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics, Phys. Rev. A 40, 1371 (1989).
  54. See Supplemental Material at http://link.aps.org/supplemental/10.1103/7961-hg2q for further theoretical derivations, details on the experiment, and data analysis, which also includes Refs. [66, 67, 68, 69, 70, 71, 72].
  55. C. Fabre and N. Treps, Modes and states in quantum optics, Rev. Mod. Phys. 92, 035005 (2020).
  56. R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, 2529 (1963).
  57. G. Rodari, C. Fernandes, E. Caruccio, A. Suprano, F. Hoch, T. Giordani, G. Carvacho, R. Albiero, N. D. Giano, G. Corrielli, F. Ceccarelli, R. Osellame, D. J. Brod, L. Novo, N. Spagnolo, E.F. Galvão, and F. Sciarrino, Experimental observation of counter-intuitive features of photonic bunching, arXiv:2410. 15883.
  58. G. Rodari, F. Hoch, A. Suprano, T. Giordani, E. Negro, G. Carvacho, N. Spagnolo, E. F. Galvão, and F. Sciarrino, Polarization-encoded photonic quantum-to-quantum bernoulli factory based on a quantum dot source, Sci. Adv. 10, eado6244 (2024).
  59. H. Ollivier, I. Maillette de Buy Wenniger, S. Thomas, S. C. Wein, A. Harouri, G. Coppola, P. Hilaire, C. Millet, A. Lemaître, I. Sagnes, O. Krebs, L. Lanco, J. C. Loredo, C. Antón, N. Somaschi, and P. Senellart, Reproducibility of high-performance quantum dot single-photon sources, ACS Photon. 7, 1050 (2020).
  60. M. Pont, G. Corrielli, A. Fyrillas, I. Agresti, G. Carvacho, N. Maring, P.-E. Emeriau, F. Ceccarelli, R. Albiero, P. H. Dias Ferreira, N. Somaschi, J. Senellart, I. Sagnes, M. Morassi, A. Lemaître, P. Senellart, F. Sciarrino, M. Liscidini, N. Belabas, and R. Osellame, High-fidelity four-photon GHz states on chip, npj Quantum Inf. 10, 50 (2024).
  61. R. Albiero, C. Pentangelo, M. Gardina, S. Atzeni, F. Ceccarelli, and R. Osellame, Toward higher integration density in femtosecond-laser-written programmable photonic circuits, Micromachines 13, 1145 (2022).
  62. K. Mayer, M. C. Tichy, F. Mintert, T. Konrad, and A. Buchleitner, Counting statistics of many-particle quantum walks, Phys. Rev. A 83, 062307 (2011).
  63. A. Leverrier and R. Garcia-Patron, Analysis of circuit imperfections in boson sampling, Quantum Inf. Comput. 15, 0489 (2015).
  64. A. Arkhipov, Bosonsampling is robust against small errors in the network matrix, Phys. Rev. A 92, 062326 (2015).
  65. https://www.polifab.polimi.it/.
  66. D. C. Kay, Schaum’s Outline of Theory and Problems of Tensor Calculus (McGraw-Hill, 1988).
  67. A. K. Nowak, S. L. Portalupi, V. Giesz, O. Gazzano, C. Dal Savio, P.-F. Braun, K. Karrai, C. Arnold, L. Lanco, I. Sagnes, A. Lemaître, and P. Senellart, Deterministic and electrically tunable bright single-photon source, Nat. Commun. 5, 3240 (2014).
  68. O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, Bright solid-state sources of indistinguishable single photons, Nat. Commun. 4, 1425 (2013).
  69. U. M. Titulaer and R. J. Glauber, Correlation functions for coherent fields, Phys. Rev. 140, B676 (1965).
  70. L. Qi, Eigenvalues and invariants of tensors, J. Math. Anal. Appl. 325, 1363 (2007).
  71. L.-B. Cui, C. Chen, W. Li, and M. K. Ng, An eigenvalue problem for even order tensors with its applications, Linear Multilinear Algebra 64, 602 (2015).
  72. K. E. Cahill and R. J. Glauber, Ordered expansions in boson amplitude operators, Phys. Rev. 177, 1857 (1969).

Outline

Information

Phys. Rev. Research 7, 043325– Published 22 December, 2025
  • Received 19 May 2025
  • Accepted 1 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