- 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
  • Letter
  • Open Access

Deep reinforcement learning for near-deterministic preparation of cubic-phase gates and direct preparation of quartic-phase gates in photonic quantum computing

Amanuel Anteneh1, Léandre Brunel1,*, Carlos González-Arciniegas1,†, and Olivier Pfister1,2,‡

  • 1Department of Physics, University of Virginia, 382 McCormick Road, Charlottesville, Virginia 22903, USA
  • 2Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, 351 McCormick Road, Charlottesville, Virginia 22903, USA

  • *Present address: Welinq, 40 Rue des Boulangers, 75005 Paris, France.
  • †Present address: Xanadu, 777 Bay Street, Toronto, Canada.
  • ‡Contact author: olivier.pfister@gmail.com
PDF

Phys. Rev. Research 8, L012048 – Published 2 March, 2026

DOI: https://doi.org/10.1103/wkfp-tf74

Abstract

Cubic-phase states are a sufficient gate resource for universal quantum computing over continuous variables. We present results from numerical experiments in which deep neural networks are trained via reinforcement learning to control a quantum optical circuit for generating cubic-phase states by leveraging quantum interference in phase space, with an unprecedented average success rate of 96%. The only non-Gaussian resource required is photon-number-resolving measurements. We also show that the exact same resources enable the direct generation of a quartic-phase gate, with no need for a cubic gate decomposition.

    View figure in article

      Physics Subject Headings (PhySH)

      Article Text

      Supplemental Material

      References (52)

      1. D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001).
      2. M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, Parallel generation of quadripartite cluster entanglement in the optical frequency comb, Phys. Rev. Lett. 107, 030505 (2011).
      3. M. Chen, N. C. Menicucci, and O. Pfister, Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb, Phys. Rev. Lett. 112, 120505 (2014).
      4. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, Ultra-large-scale continuous-variable cluster states multiplexed in the time domain, Nat. Photon. 7, 982 (2013).
      5. J.-i. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, Invited article: Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing, APL Photonics 1, 060801 (2016).
      6. W. Asavanant, Y. Shiozawa, S. Yokoyama, B. Charoensombutamon, H. Emura, R. N. Alexander, S. Takeda, J.-i. Yoshikawa, N. C. Menicucci, H. Yonezawa, and A. Furusawa, Generation of time-domain-multiplexed two-dimensional cluster state, Science 366, 373 (2019).
      7. M. V. Larsen, X. Guo, C. R. Breum, J. S. Neergaard-Nielsen, and U. L. Andersen, Deterministic generation of a two-dimensional cluster state, Science 366, 369 (2019).
      8. C. Roh, G. Gwak, Y.-D. Yoon, and Y.-S. Ra, Generation of three-dimensional cluster entangled state, Nat. Photon. 19, 526 (2025).
      9. Z. Yang, M. Jahanbozorgi, D. Jeong, S. Sun, O. Pfister, H. Lee, and X. Yi, A squeezed quantum microcomb on a chip, Nat. Commun. 12, 4781 (2021).
      10. M. Jahanbozorgi, Z. Yang, S. Sun, H. Chen, R. Liu, B. Wang, and X. Yi, Generation of squeezed quantum microcombs with silicon nitride integrated photonic circuits, Optica 10, 1100 (2023).
      11. Z. Wang, K. Li, Y. Wang, X. Zhou, Y. Cheng, B. Jing, F. Sun, J. Li, Z. Li, B. Wu, Q. Gong, Q. He, B.-B. Li, and Q.-F. Yang, Large-scale cluster quantum microcombs, Light Sci. Appl. 14, 164 (2025).
      12. X. Jia, C. Zhai, X. Zhu, C. You, Y. Cao, X. Zhang, Y. Zheng, Z. Fu, J. Mao, T. Dai, L. Chang, X. Su, Q. Gong, and J. Wang, Continuous-variable multipartite entanglement in an integrated microcomb, Nature (London) 639, 329 (2025).
      13. S. Lloyd and S. L. Braunstein, Quantum computation over continuous variables, Phys. Rev. Lett. 82, 1784 (1999).
      14. N. C. Menicucci, Fault-tolerant measurement-based quantum computing with continuous-variable cluster states, Phys. Rev. Lett. 112, 120504 (2014).
      15. K. Marshall, R. Pooser, G. Siopsis, and C. Weedbrook, Quantum simulation of quantum field theory using continuous variables, Phys. Rev. A 92, 063825 (2015).
      16. R. A. Briceño, R. G. Edwards, M. Eaton, C. González-Arciniegas, O. Pfister, and G. Siopsis, Toward coherent quantum computation of scattering amplitudes with a measurement-based photonic quantum processor, Phys. Rev. Res. 6, 043065 (2024).
      17. R. L. Hudson, When is the Wigner quasi-probability density non-negative? Rep. Math. Phys. 6, 249 (1974).
      18. S. Sefi and P. van Loock, How to decompose arbitrary continuous-variable quantum operations, Phys. Rev. Lett. 107, 170501 (2011).
      19. T. Kalajdzievski and J. M. Arrazola, Exact gate decompositions for photonic quantum computing, Phys. Rev. A 99, 022341 (2019).
      20. N. Budinger, A. Furusawa, and P. van Loock, All-optical quantum computing using cubic phase gates, Phys. Rev. Res. 6, 023332 (2024).
      21. 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).
      22. A. Furusawa and P. van Loock, Quantum Teleportation and Entanglement: A Hybrid Approach to Optical Quantum Information Processing (John Wiley & Sons, New Jersey, 2011).
      23. O. Pfister, Continuous-variable quantum computing in the quantum optical frequency comb, J. Phys. B: At. Mol. Opt. Phys. 53, 012001 (2020).
      24. S. Bartolucci, P. Birchall, H. Bombin, 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).
      25. 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).
      26. P. Renault, P. Yard, R. C. Pooser, M. Eaton, and H. A. Zaidi, End-to-end switchless architecture for fault-tolerant photonic quantum computing, Quantum 9, 1796 (2025).
      27. Y. Zheng, O. Hahn, P. Stadler, P. Holmvall, F. Quijandría, A. Ferraro, and G. Ferrini, Gaussian conversion protocols for cubic phase state generation, PRX Quantum 2, 010327 (2021).
      28. R. Yanagimoto, T. Onodera, E. Ng, L. G. Wright, P. L. McMahon, and H. Mabuchi, Engineering a Kerr-based deterministic cubic phase gate via Gaussian operations, Phys. Rev. Lett. 124, 240503 (2020).
      29. S. Ghose and B. C. Sanders, Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps, J. Mod. Opt. 54, 855 (2007).
      30. M. Eaton, A. Hossameldin, R. J. Birrittella, P. M. Alsing, C. C. Gerry, H. Dong, C. Cuevas, and O. Pfister, Resolution of 100 photons and quantum generation of unbiased random numbers, Nat. Photon. 17, 106 (2023).
      31. K. Marshall, R. Pooser, G. Siopsis, and C. Weedbrook, Repeat-until-success cubic phase gate for universal continuous-variable quantum computation, Phys. Rev. A 91, 032321 (2015).
      32. P. Marek, R. Filip, H. Ogawa, A. Sakaguchi, S. Takeda, J.-I. Yoshikawa, and A. Furusawa, General implementation of arbitrary nonlinear quadrature phase gates, Phys. Rev. A 97, 022329 (2018).
      33. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Generation of squeezed states by parametric down conversion, Phys. Rev. Lett. 57, 2520 (1986).
      34. F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination, Nat. Photon. 7, 147 (2013).
      35. A. E. Lita, A. J. Miller, and S. W. Nam, Counting near-infrared single-photons with 95% efficiency, Opt. Express 16, 3032 (2008).
      36. C. Cahall, K. L. Nicolich, N. T. Islam, G. P. Lafyatis, A. J. Miller, D. J. Gauthier, and J. Kim, Multi-photon detection using a conventional superconducting nanowire single-photon detector, Optica 4, 1534 (2017).
      37. M. Endo, T. Sonoyama, M. Matsuyama, F. Okamoto, S. Miki, M. Yabuno, F. China, H. Terai, and A. Furusawa, Quantum detector tomography of a superconducting nanostrip photon-number-resolving detector, Opt. Express 29, 11728 (2021).
      38. R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction (The MIT Press, Cambridge, 2018).
      39. V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, et al., Human-level control through deep reinforcement learning, Nature (London) 518, 529 (2015).
      40. S. Borah, B. Sarma, M. Kewming, G. J. Milburn, and J. Twamley, Measurement-based feedback quantum control with deep reinforcement learning for a double-well nonlinear potential, Phys. Rev. Lett. 127, 190403 (2021).
      41. V. V. Sivak, A. Eickbusch, H. Liu, B. Royer, I. Tsioutsios, and M. H. Devoret, Model-free quantum control with reinforcement learning, Phys. Rev. X 12, 011059 (2022).
      42. I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio, Deep Learning (MIT Press, Cambridge, 2016), Vol. 1.
      43. J. M. Arrazola, T. R. Bromley, J. Izaac, C. R. Myers, K. Brádler, and N. Killoran, Machine learning method for state preparation and gate synthesis on photonic quantum computers, Quantum Sci. Technol. 4, 024004 (2019).
      44. M. Kudra, M. Kervinen, I. Strandberg, S. Ahmed, M. Scigliuzzo, A. Osman, D. P. Lozano, M. O. Tholén, R. Borgani, D. B. Haviland, et al., Robust preparation of Wigner-negative states with optimized SNAP-displacement sequences, PRX Quantum 3, 030301 (2022).
      45. I. Tzitrin, J. E. Bourassa, N. C. Menicucci, and K. K. Sabapathy, Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes, Phys. Rev. A 101, 032315 (2020).
      46. Y. Yao, F. Miatto, and N. Quesada, Riemannian optimization of photonic quantum circuits in phase and fock space, SciPost Phys. 17, 082 (2024).
      47. A. Anteneh, L. Brunel, and O. Pfister, Machine learning for efficient generation of universal photonic quantum computing resources, Optica Quantum 2, 296 (2024).
      48. R. Porotti, A. Essig, B. Huard, and F. Marquardt, Deep reinforcement learning for quantum state preparation with weak nonlinear measurements, Quantum 6, 747 (2022).
      49. See Supplemental Material at http://link.aps.org/supplemental/10.1103/wkfp-tf74 for more details on the machine learning model tuning process and experimental imperfections in the state preparation circuit.
      50. J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov, Proximal policy optimization algorithms, arXiv:1707.06347.
      51. W. P. Schleich, Quantum Optics in Phase Space (Wiley‐VCH Verlag Berlin GmbH, Berlin, 2001).
      52. https://github.com/amanuelanteneh/rl-loop.

      Outline

      Information

      Phys. Rev. Research 8, L012048– Published 2 March, 2026
      • Received 11 June 2025
      • Accepted 10 February 2026
      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