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  • Letter
  • Open Access

Unified boson sampling

Luca Bianchi1,*, Carlo Marconi2, Laura Ares3, Davide Bacco1, and Jan Sperling3

  • *Contact author: luca.bianchi@unifi.it
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Phys. Rev. Research 7, L042068 – Published 26 December, 2025

DOI: https://doi.org/10.1103/8hy1-m5gg

Abstract

Boson sampling is a key candidate for demonstrating quantum advantage and has already yielded significant advances in quantum simulation, machine learning, and graph theory. In this work, a unification and extension of distinct forms of boson sampling is developed. The devised protocol merges discrete-variable scattershot boson sampling with continuous-variable Gaussian boson sampling. Therefore, it is rendered possible to harness the complexity of more interesting states, such as squeezed photons, in advanced sampling protocols. A generating function formalism is developed for the joint description of multiphoton and multimode light undergoing Gaussian transformations. The resulting analytical tools enable one to explore interfaces of different photonic quantum-information-processing platforms. A numerical simulation of unified sampling is carried out, benchmarking its performance, complexity, and scalability. Entanglement is characterized to exemplify the generation of quantum correlations from the nonlinear interactions of a unified sampler.

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References (71)

  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2001), Vol. 2.
  2. C. H. Bennett and D. P. DiVincenzo, Quantum information: An introduction, Nature (London) 404, 247 (2000).
  3. M. A. Khan, M. N. Aman, and B. Sikdar, Beyond bits: A review of quantum embedding techniques for efficient information processing, IEEE Access 12, 46118 (2024).
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  5. W. Vogel and D.-G. Welsch, Quantum Optics (John Wiley & Sons, 2006).
  6. S. Slussarenko and G. J. Pryde, Photonic quantum information processing: A concise review, Appl. Phys. Rev. 6, 041303 (2019).
  7. F. Flamini, N. Spagnolo, and F. Sciarrino, Photonic quantum information processing: A review, Rep. Prog. Phys. 82, 016001 (2018).
  8. J. W. Silverstone, et al., Silicon quantum photonics, IEEE J. Sel. Top. Quantum Electron. 22, 390 (2016).
  9. J. L. O’Brien, A. Furusawa, and J. Vučković, Photonic quantum technologies, Nat. Photonics 3, 687 (2009).
  10. 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).
  11. R. Schnabel, Squeezed states of light and their applications in laser interferometers, Phys. Rep. 684, 1 (2017).
  12. S. Lloyd and S. L. Braunstein, Quantum computation over continuous variables, Phys. Rev. Lett. 82, 1784 (1999).
  13. S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005).
  14. 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).
  15. F. Lenzini, et al., Integrated photonic platform for quantum information with continuous variables, Sci. Adv. 4, aat9331 (2018).
  16. S. Aaronson and A. Arkhipov, The computational complexity of linear optics, in Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (2011), pp. 333–342.
  17. A. M. Ben-Amram, The Church-Turing thesis and its look-alikes, ACM SIGACT News 36, 113 (2005).
  18. L. G. Valiant, The complexity of computing the permanent, Theor. Comput. Sci. 8, 189 (1979).
  19. M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, Experimental boson sampling, Nat. Photon. 7, 540 (2013).
  20. A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O’Brien, and T. C. Ralph, Boson sampling from a Gaussian state, Phys. Rev. Lett. 113, 100502 (2014).
  21. M. Bentivegna, et al., Experimental scattershot boson sampling, Sci. Adv. 1, e1400255 (2015).
  22. S. Barkhofen, T. J. Bartley, L. Sansoni, R. Kruse, C. S. Hamilton, I. Jex, and C. Silberhorn, Driven boson sampling, Phys. Rev. Lett. 118, 020502 (2017).
  23. C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Gaussian boson sampling, Phys. Rev. Lett. 119, 170501 (2017).
  24. 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).
  25. H.-S. Zhong, et al., Experimental Gaussian boson sampling, Sci. Bull. 64, 511 (2019).
  26. D. Callan, A combinatorial survey of identities for the double factorial, arXiv:0906.1317.
  27. M. Rudelson, A. Samorodnitsky, and O. Zeitouni, Hafnians, perfect matchings and Gaussian matrices, Ann. Probab. 44, 2858 (2016).
  28. J. Huh, G. G. Guerreschi, B. Peropadre, J. R. McClean, and A. Aspuru-Guzik, Boson sampling for molecular vibronic spectra, Nat. Photon. 9, 615 (2015).
  29. Y.-H. Deng, et al., Solving graph problems using Gaussian boson sampling, Phys. Rev. Lett. 130, 190601 (2023).
  30. J. M. Arrazola and T. R. Bromley, Using Gaussian boson sampling to find dense subgraphs, Phys. Rev. Lett. 121, 030503 (2018).
  31. K. K. Sabapathy, H. Qi, J. Izaac, and C. Weedbrook, Production of photonic universal quantum gates enhanced by machine learning, Phys. Rev. A 100, 012326 (2019).
  32. J. E. Bourassa, et al., Blueprint for a scalable photonic fault-tolerant quantum computer, Quantum 5, 392 (2021).
  33. F. Hoch, et al., Quantum machine learning with adaptive boson sampling via post-selection, Nat. Commun. 16, 902 (2025).
  34. 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).
  35. U. L. Andersen, J. S. Neergaard-Nielsen, P. Van Loock, and A. Furusawa, Hybrid discrete- and continuous-variable quantum information, Nat. Phys. 11, 713 (2015).
  36. M. Walschaers, Non-Gaussian quantum states and where to find them, PRX Quantum 2, 030204 (2021).
  37. A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods (CRC Press, 2023).
  38. Arvind, B. Dutta, N. Mukunda, and R. Simon, The real symplectic groups in quantum mechanics and optics, Pramana 45, 471 (1995).
  39. B. C. Hall, Lie Groups, Lie Algebras, and Representations (Springer, 2013), pp. 333–366.
  40. J. B. Brask, Gaussian states and operations—A quick reference, arXiv:2102.05748.
  41. M. Houde, W. McCutcheon, and N. Quesada, Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson, Can. J. Phys. 102, 497 (2024).
  42. R. Simon, S. Chaturvedi, and V. Srinivasan, Congruences and canonical forms for a positive matrix: Application to the Schweinler–Wigner extremum principle, J. Math. Phys. 40, 3632 (1999).
  43. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 2012).
  44. S. L. Braunstein, Squeezing as an irreducible resource, Phys. Rev. A 71, 055801 (2005).
  45. E. B. McBride, Obtaining Generating Functions (Springer Science & Business Media, 2012), Vol. 21.
  46. P. Blasiak, A. Horzela, K. A. Penson, A. I. Solomon, and G. H. Duchamp, Combinatorics and boson normal ordering: A gentle introduction, Am. J. Phys. 75, 639 (2007).
  47. J. Sperling, W. Vogel, and G. S. Agarwal, True photocounting statistics of multiple on-off detectors, Phys. Rev. A 85, 023820 (2012).
  48. J. Sperling, W. Vogel, and G. S. Agarwal, Quantum state engineering by click counting, Phys. Rev. A 89, 043829 (2014).
  49. L. Frappat, A. Sciarrino, and P. Sorba, Dictionary on Lie superalgebras, arXiv:hep-th/9607161.
  50. W.-M. Zhang, et al., Coherent states: Theory and some applications, Rev. Mod. Phys. 62, 867 (1990).
  51. A. W. Knapp and A. W. Knapp, Lie Groups Beyond an Introduction (Springer, 1996), Vol. 140.
  52. See Supplemental Material at http://link.aps.org/supplemental/10.1103/8hy1-m5gg for additional considerations pertaining to the method of Lie algebrae and generating functions, as well as simulation details, which included Refs. [42, 68, 69, 70, 71].
  53. J. Schwinger, A Quantum Legacy: Seminal Papers of Julian Schwinger, edited by A. M. Kimball (World Scientific, Singapore, 2000), Vol. 26.
  54. C. M. Caves, Reframing SU(1,1) interferometry, Adv. Quantum Technol. 3, 1900138 (2020).
  55. A. DasGupta, Disentanglement formulas: An alternative derivation and some applications to squeezed coherent states, Am. J. Phys. 64, 1422 (1996).
  56. L. Bianchi, C. Marconi, J. Sperling, and D. Bacco, Predetection squeezing as a resource for high-dimensional Bell-state measurements, Phys. Rev. Res. 7, 023038 (2025).
  57. M. Engelkemeier, L. Lorz, S. De, B. Brecht, I. Dhand, M. B. Plenio, C. Silberhorn, and J. Sperling, Quantum photonics with active feedback loops, Phys. Rev. A 102, 023712 (2020).
  58. N. Quesada, Franck-condon factors by counting perfect matchings of graphs with loops, J. Chem. Phys. 150, 164113 (2019).
  59. F. M. Miatto and N. Quesada, Fast optimization of parametrized quantum optical circuits, Quantum 4, 366 (2020).
  60. Y. Cardin and N. Quesada, Photon-number moments and cumulants of Gaussian states, Quantum 8, 1521 (2024).
  61. V. V. Kocharovsky, V. V. Kocharovsky, and S. V. Tarasov, The Hafnian master theorem, Linear Algebra Appl. 651, 144 (2022).
  62. A. Björklund, B. Gupt, and N. Quesada, A faster Hafnian formula for complex matrices and its benchmarking on a supercomputer, ACM J. Exp. Algorithmics 24, 1 (2019).
  63. E. S. Meckes, The Random Matrix Theory of the Classical Compact Groups (Cambridge University Press, 2019), Vol. 218.
  64. C. C. Margossian, A review of automatic differentiation and its efficient implementation, WIREs Data Min. Knowl. Discovery 9, e1305 (2019).
  65. A. Griewank, A mathematical view of automatic differentiation, Acta Numer. 12, 321 (2003).
  66. E. Shchukin and W. Vogel, Inseparability criteria for continuous bipartite quantum states, Phys. Rev. Lett. 95, 230502 (2005).
  67. L. F. Richardson, IX. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Philos. Trans. R. Soc. London, Ser. A 210, 307 (1911).
  68. J. E. Humphreys, Introduction to Lie Algebras and Represen- tation Theory (Springer Science & Business Media, 2012), Vol. 9.
  69. U. Leonhardt, Quantum physics of simple optical instruments, Rep. Prog. Phys. 66, 1207 (2003).
  70. N. Quesada, J. M. Arrazola, and N. Killoran, Gaussian boson sampling using threshold detectors, Phys. Rev. A 98, 062322 (2018).
  71. J. F. Bulmer, S. Paesani, R. S. Chadwick, and N. Quesada, Threshold detection statistics of bosonic states, Phys. Rev. A 106, 043712 (2022).

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Phys. Rev. Research 7, L042068– Published 26 December, 2025
  • Received 2 September 2025
  • Accepted 13 November 2025
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