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  • Open Access

Photon Catalysis for General Multimode Multi-Photon Quantum State Preparation

Andrei Aralov1,*, Émilie Gillet1,*, Viet Nguyen2, Andrea Cosentino2, Mattia Walschaers1,†, and Massimo Frigerio1,‡

  • *These authors contributed equally to this work.
  • †Contact author: mattia.walschaers@lkb.upmc.fr
  • ‡Contact author: massimo.frigerio@lkb.upmc.fr
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PRX Quantum 7, 020323 – Published 6 May, 2026

DOI: https://doi.org/10.1103/ktc9-9rjb

Abstract

Multimode multiphoton states are at the center of many photonic quantum technologies, from photonic quantum computing to quantum sensing. In this work, we derive a procedure to generate exactly, and with a predictable number of steps, any such state by using only multiport interferometers, photon number resolving detectors, photon additions, and displacements. We achieve this goal by establishing a connection between photonic quantum state engineering and the algebraic problem of symmetric tensor decomposition. This connection allows us to solve the problem by using corresponding results from algebraic geometry and unveils a mechanism of photon catalysis, where photons are injected and subsequently retrieved in measurements, to generate entanglement that cannot be obtained through Gaussian operations. We also introduce a tensor decomposition, that generalizes our method and allows to construct circuits yielding perfect fidelity, using the minimum number of catalysis photons. As a benchmark, we numerically evaluate our method and compare its performance with state-of-the art results, confirming 100% fidelity on different classes of states.

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Popular Summary

Many promising applications of quantum photonics, such as quantum computing, quantum sensing, and quantum simulation, require the production of specific, highly entangled states of light. These states involve superpositions of different photon distribution patterns across different modes—beams, time bins, or wavelengths. There is currently no general blueprint for producing an arbitrary state, and applications rely on ad hoc procedures tailored for a narrow family of states.

In this work, we provide a step-by-step recipe to produce any multimode, multiphoton state, using only the ingredients available in a quantum optics lab: interferometers, photon addition, and photon-counting detectors. Our method translates the description of a quantum state into a mathematical object, a polynomial or tensor, and uses results from algebra to decompose it into a finite number of simpler elements. These building blocks can be translated back into individual optical operations. Crucially, a theorem from algebra guarantees we can construct any desired state in a provably bounded number of steps. Our method unveils a catalysis mechanism: some states can only be made by injecting more photons than needed, and later removing them by measurement, leaving behind a type of entanglement that is otherwise inaccessible.

Our work shows that every photonic state is, in principle, reachable with realistic experimental tools, but some require more catalytic steps than others. This provides a new way to quantify the “complexity” of a quantum state that could guide future experiments and serve as a theoretical yardstick for measuring the resources needed by photonic quantum technologies.

Article Text

References (75)

  1. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, Multiphoton entanglement and interferometry, Rev. Mod. Phys. 84, 777 (2012).
  2. E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001).
  3. 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).
  4. F. Flamini, N. Spagnolo, and F. Sciarrino, Photonic quantum information processing: A review, Rep. Prog. Phys. 82, 016001 (2019).
  5. U. Chabaud and M. Walschaers, Resources for bosonic quantum computational advantage, Phys. Rev. Lett. 130, 090602 (2023).
  6. 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).
  7. N. Maring et al., A versatile single-photon-based quantum computing platform, Nat. Photonics 18, 603 (2024).
  8. L. Monbroussou, E. Z. Mamon, H. Thomas, V. Yacoub, U. Chabaud, and E. Kashefi, Toward quantum advantage with photonic state injection, Phys. Rev. Res. 7, 033051 (2025).
  9. K. Alexander et al., A manufacturable platform for photonic quantum computing, Nature 641, 876 (2025).
  10. A. Salavrakos, N. Maring, P.-E. Emeriau, and S. Mansfield, Photon-native quantum algorithms, Mater. Quantum Technol. 5, 023001 (2025).
  11. J. P. Dowling, Quantum optical metrology—The lowdown on high-N00N states, Contemp. Phys. 49, 125 (2008).
  12. U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Optimal quantum phase estimation, Phys. Rev. Lett. 102, 040403 (2009).
  13. G. Y. Xiang, H. F. Hofmann, and G. J. Pryde, Optimal multi-photon phase sensing with a single interference fringe, Sci. Rep. 3, 2684 (2013).
  14. P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, Quantum enhanced multiple phase estimation, Phys. Rev. Lett. 111, 070403 (2013).
  15. L. Zhang and K. W. C. Chan, Quantum multiparameter estimation with generalized balanced multimode noon-like states, Phys. Rev. A 95, 032321 (2017).
  16. S. Hong, J. ur Rehman, Y.-S. Kim, Y.-W. Cho, S.-W. Lee, H. Jung, S. Moon, S.-W. Han, and H.-T. Lim, Quantum enhanced multiple-phase estimation with multi-mode n00n states, Nat. Commun. 12, 5211 (2021).
  17. M. Namkung, D.-H. Kim, S. Hong, Y.-S. Kim, C. Lee, and H.-T. Lim, Optimal multiple-phase estimation with multi-mode noon states against photon loss, New J. Phys. 26, 073028 (2024).
  18. I. Dhand, M. Engelkemeier, L. Sansoni, S. Barkhofen, C. Silberhorn, and M. B. Plenio, Proposal for quantum simulation via all-optically-generated tensor network states, Phys. Rev. Lett. 120, 130501 (2018).
  19. B. Bartlett, O. Y. Long, A. Dutt, and S. Fan, Programmable photonic system for quantum simulation in arbitrary topologies, APL Quantum 1, 016102 (2024).
  20. F. H. B. Somhorst, R. van der Meer, M. Correa Anguita, R. Schadow, H. J. Snijders, M. de Goede, B. Kassenberg, P. Venderbosch, C. Taballione, J. P. Epping, H. H. van den Vlekkert, J. Timmerhuis, J. F. F. Bulmer, J. Lugani, I. A. Walmsley, P. W. H. Pinkse, J. Eisert, N. Walk, and J. J. Renema, Quantum simulation of thermodynamics in an integrated quantum photonic processor, Nat. Commun. 14, 3895 (2023).
  21. S. Aaronson and A. Arkhipov, in Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC ’11 (Association for Computing Machinery, New York, NY, USA, 2011), pp. 333–342.
  22. L. Chakhmakhchyan and N. J. Cerf, Boson sampling with Gaussian measurements, Phys. Rev. A 96, 032326 (2017).
  23. N. Spagnolo, D. J. Brod, E. F. Galvão, and F. Sciarrino, Non-linear boson sampling, npj Quantum Inf. 9, 3 (2023).
  24. U. Chabaud and S. Mehraban, Holomorphic representation of quantum computations, Quantum 6, 831 (2022).
  25. U. Chabaud, D. Markham, and F. Grosshans, Stellar representation of non-Gaussian quantum states, Phys. Rev. Lett. 124, 063605 (2020).
  26. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Theoretical and Mathematical Physics (Springer, Berlin, Germany, 1997), 2nd ed.
  27. A. F. Verbeure, Many-Body Boson Systems, Theoretical and Mathematical Physics (Springer, London, England, 2010).
  28. M. Walschaers, Non-Gaussian quantum states and where to find them, PRX Quantum 2, 030204 (2021).
  29. U. Chabaud, G. Ferrini, F. Grosshans, and D. Markham, Classical simulation of Gaussian quantum circuits with non-Gaussian input states, Phys. Rev. Res. 3, 033018 (2021).
  30. J. Fiurášek, S. Massar, and N. J. Cerf, Conditional generation of arbitrary multimode entangled states of light with linear optics, Phys. Rev. A 68, 042325 (2003).
  31. D. T. Pegg, L. S. Phillips, and S. M. Barnett, Optical state truncation by projection synthesis, Phys. Rev. Lett. 81, 1604 (1998).
  32. M. Dušek, Discrimination of the Bell states of qudits by means of linear optics, Opt. Commun. 199, 161 (2001).
  33. D. A. Kopylov, C. Offen, L. Ares, B. Wembe, S. Ober-Blöbaum, T. Meier, P. R. Sharapova, and J. Sperling, Multiphoton, multimode state classification for nonlinear optical circuits, Phys. Rev. Res. 7, 033062 (2025).
  34. C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Gaussian Boson sampling, Phys. Rev. Lett. 119, 170501 (2017).
  35. J. Sperling, A. Perez-Leija, K. Busch, and C. Silberhorn, Mode-independent quantum entanglement for light, Phys. Rev. A 100, 062129 (2019).
  36. C. E. Lopetegui, M. Isoard, N. Treps, and M. Walschaers, Detection of mode-intrinsic quantum entanglement, Opt. Quantum 3, 312 (2025).
  37. A. I. Lvovsky and J. Mlynek, Quantum-optical catalysis: Generating nonclassical states of light by means of linear optics, Phys. Rev. Lett. 88, 250401 (2002).
  38. T. J. Bartley, G. Donati, J. B. Spring, X.-M. Jin, M. Barbieri, A. Datta, B. J. Smith, and I. A. Walmsley, Multiphoton state engineering by heralded interference between single photons and coherent states, Phys. Rev. A 86, 043820 (2012).
  39. L.-Y. Hu, J.-N. Wu, Z. Liao, and M. S. Zubairy, Multiphoton catalysis with coherent state input: Nonclassicality and decoherence, J. Phys. B: At., Mol. Opt. Phys. 49, 175504 (2016).
  40. R. J. Birrittella, M. E. Baz, and C. C. Gerry, Photon catalysis and quantum state engineering, J. Opt. Soc. Am. B 35, 1514 (2018).
  41. M. Eaton, R. Nehra, and O. Pfister, Non-Gaussian and Gottesman–Kitaev–Preskill state preparation by photon catalysis, New J. Phys. 21, 113034 (2019).
  42. T. J. Bartley and I. A. Walmsley, Directly comparing entanglement-enhancing non-Gaussian operations, New J. Phys. 17, 023038 (2015).
  43. G. De Gliniasty, P. Bagourd, S. Draux, and B. Bourdoncle, Simple rules for two-photon state preparation with linear optics, in 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE Computer Society, Los Alamitos, CA, USA, 2024), Vol. 01, pp. 706–711.
  44. D. Su, C. R. Myers, and K. K. Sabapathy, Conversion of Gaussian states to non-Gaussian states using photon-number-resolving detectors, Phys. Rev. A 100, 052301 (2019).
  45. J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su, B. Q. Baragiola, S. Guha, G. Dauphinais, K. K. Sabapathy, N. C. Menicucci, and I. Dhand, Blueprint for a scalable photonic fault-tolerant quantum computer, Quantum 5, 392 (2021).
  46. K. Fukui, S. Takeda, M. Endo, W. Asavanant, J.-i. Yoshikawa, P. van Loock, and A. Furusawa, Efficient backcasting search for optical quantum state synthesis, Phys. Rev. Lett. 128, 240503 (2022).
  47. M. V. Larsen et al., Integrated photonic source of Gottesman–Kitaev–Preskill qubits, Nature 642, 587 (2025).
  48. N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, Universal quantum computation with continuous-variable cluster states, Phys. Rev. Lett. 97, 110501 (2006).
  49. F. Arzani, R. I. Booth, and U. Chabaud, Can effective descriptions of Bosonic systems be considered complete? Nat. Commun. 16, 9744 (2025).
  50. A. Shpilka, Affine projections of symmetric polynomials, J. Comput. Syst. Sci. 65, 639 (2002).
  51. J. M. Landsberg, in Ref. [75], Chap. 2, pp. 42–43.
  52. D. A. Cox, J. Little, and D. O’Shea, in Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Springer International Publishing, Cham, 2015), pp. 346–347.
  53. D. A. Cox, J. Little, and D. O’Shea, in Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Springer International Publishing, Cham, 2015), pp. 400–401.
  54. G. Roeland, S. Kaali, V. Roman Rodriguez, N. Treps, and V. Parigi, Mode-selective single-photon addition to a multimode quantum field, New J. Phys. 24, 043031 (2022).
  55. S. L. Braunstein, Squeezing as an irreducible resource, Phys. Rev. A 71, 055801 (2005).
  56. J. M. Landsberg, in [75], Chap. 5, pp. 125–126.
  57. G. Blekherman and Z. Teitler, On maximum, typical and generic ranks, Math. Ann. 362, 1021 (2015).
  58. A. Bernardi, E. Carlini, M. V. Catalisano, A. Gimigliano, and A. Oneto, The Hitchhiker guide to: Secant varieties and tensor decomposition, Mathematics 6, 314 (2018).
  59. C. Fabre and N. Treps, Modes and states in quantum optics, Rev. Mod. Phys. 92, 035005 (2020).
  60. B. Beauzamy, E. Bombieri, P. Enflo, and H. L. Montgomery, Products of polynomials in many variables, J. Number Theory 36, 219 (1990).
  61. K. Ranestad and F.-O. Schreyer, On the rank of a symmetric form, J. Algebra 346, 340 (2011).
  62. R. A. Horn and C. R. Johnson, in Matrix Analysis (Cambridge University, 2012), pp. 263–264, 2nd ed.
  63. C. J. Hillar and L.-H. Lim, Most tensor problems are NP-hard, J. ACM 60, 1 (2013).
  64. F. Galuppi and M. Mella, Identifiability of homogeneous polynomials and cremona transformations, J. Reine Angew. Math. 2019, 279 (2019).
  65. L. Oeding and G. Ottaviani, Eigenvectors of tensors and algorithms for Waring decomposition, J. Symb. Comput. 54, 9 (2013).
  66. J. Kileel and J. M. Pereira, Subspace power method for symmetric tensor decomposition, Numer. Algorithms, 1 (2025).
  67. J. Nie, Low rank symmetric tensor approximations, SIAM J. Matrix Anal. Appl. 38, 1517 (2017).
  68. J. Berthomieu, C. Eder, and M. Safey El Din, in 2021 International Symposium on Symbolic and Algebraic Computation, 46th International Symposium on Symbolic and Algebraic Computation (ACM, Saint Petersburg, Russia, 2021), pp. 51–58.
  69. The DeepMind JAX Ecosystem, 2020, http://github.com/google-deepmind.
  70. https://github.com/EQ15T/photon-catalysis.
  71. N. Heurtel, A. Fyrillas, G. d. Gliniasty, R. Le Bihan, S. Malherbe, M. Pailhas, E. Bertasi, B. Bourdoncle, P.-E. Emeriau, R. Mezher, L. Music, N. Belabas, B. Valiron, P. Senellart, S. Mansfield, and J. Senellart, Perceval: A software platform for discrete variable photonic quantum computing, Quantum 7, 931 (2023).
  72. J. M. Landsberg, in Ref. [75], Chap. 4, pp. 114–115.
  73. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experimental realization of any discrete unitary operator, Phys. Rev. Lett. 73, 58 (1994).
  74. 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).
  75. J. M. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics Vol. 128 (American Mathematical Society, Providence, R.I., 2012).

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PRX Quantum 7, 020323– Published 6 May, 2026
  • Received 8 August 2025
  • Revised 5 December 2025
  • Accepted 4 February 2026
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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.

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