- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Applied

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

Physical implementation of analytic derivatives on a photonic quantum computer

Sebastiano Corli, Giorgio Panichi, Samuele Altilia, Edoardo Suerra, Simone Cialdi, and Enrico Prati*

  • *Contact author: enrico.prati@unimi.it
PDF

Phys. Rev. Applied 26, 014107 – Published 31 July, 2026

DOI: https://doi.org/10.1103/yrst-tr19

Abstract

A limiting factor to the exploitation of continuous variables in quantum computing, in particular in the field of quantum machine learning, has been the hindrances of computing the derivatives of a model. The constraint occurs when applying the parameter-shift rule to a photonic circuit, and the number of non-Gaussian gates exceeds a logarithmic trend with respect to the length of the circuit. This condition occurs when dealing with most nonlinear functions. Indeed, it hampers subroutines, such as backpropagation, which are paramount for machine learning and, most significantly, the computation of the derivatives of the model itself. To overcome this hurdle, we propose a method to compute derivatives of functions, implemented by continuous variable circuits, directly through photonic quantum circuits. Unlike the parameter-shift rule, this method does not fail when the number of non-Gaussian gates scales more than logarithmically against the depth of the circuit. We exploit the fact that computing the derivative of a wave function is equivalent to applying the momentum operator, which in the position basis acts as its derivative. The action of this operator is physically embodied by photon-added coherent states. We prove that, up to the second order of derivation, the number of operations does not scale with the depth of the circuit, but only with the order of derivation. We showcase numerical examples of derivatives of elementary functions parametrized on simulated photonic quantum circuits. Thanks to this method, we unleash an embodiment of quantum neural networks on photonic quantum computing platforms.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (51)

  1. S. Lloyd and S. L. Braunstein, Quantum computation over continuous variables, Phys. Rev. Lett. 82, 1784 (1999).
  2. N. Killoran, T. R. Bromley, J. M. Arrazola, M. Schuld, N. Quesada, and S. Lloyd, Continuous-variable quantum neural networks, Phys. Rev. Lett. 1, 033063 (2019).
  3. S. Corli, L. Moro, D. Dragoni, M. Dispenza, and E. Prati, Quantum machine learning algorithms for anomaly detection: A review, Future Gener. Comput. Syst, 166, 107632 (2025).
  4. T. Hillmann, F. Quijandría, G. Johansson, A. Ferraro, S. Gasparinetti, and G. Ferrini, Universal gate set for continuous-variable quantum computation with microwave circuits, Phys. Rev. Lett. 125, 160501 (2020).
  5. S. Bangar, L. Sunny, K. Yeter-Aydeniz, and G. Siopsis, Experimentally realizable continuous-variable quantum neural networks, Phys. Rev. A 108, 042414 (2023).
  6. S. Takeda and A. Furusawa, Universal quantum computing with measurement-induced continuous-variable gate sequence in a loop-based architecture, Phys. Rev. Lett. 119, 120504 (2017).
  7. K. Miyata, H. Ogawa, P. Marek, R. Filip, H. Yonezawa, J.-I. Yoshikawa, and A. Furusawa, Implementation of a quantum cubic gate by an adaptive non-Gaussian measurement, Phys. Rev. A 93, 022301 (2016).
  8. E. Suerra, S. Altilia, S. Olivares, A. Ferraro, F. Canella, D. Giannotti, G. Galzerano, S. Corli, E. Prati, and S. Cialdi, Generation and detection of squeezed states via a synchronously pumped optical parametric oscillator, Appl. Phys. B 132, 26 (2026).
  9. S. Cialdi, E. Suerra, S. Olivares, S. Capra, and M. G. A. Paris, Squeezing phase diffusion, Phys. Rev. Lett. 124, 163601 (2020).
  10. S. Cialdi, C. Porto, D. Cipriani, S. Olivares, and M. G. A. Paris, Full quantum state reconstruction of symmetric two-mode squeezed thermal states via spectral homodyne detection and a state-balancing detector, Phys. Rev. A 93, 043805 (2016).
  11. S. Corli and E. Prati, Measurement-based quantum compiling via gauge invariance, Phys. Rev. Appl. 25, 044068 (2026).
  12. D. Wierichs, J. Izaac, C. Wang, and C. Y.-Y. Lin, General parameter-shift rules for quantum gradients, Quantum 6, 677 (2022).
  13. M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).
  14. J. Bowles, D. Wierichs, and C.-Y. Park, Backpropagation scaling in parameterised quantum circuits, Quantum 9, 1873 (2025).
  15. A. F. Izmaylov, R. A. Lang, and T.-C. Yen, Analytic gradients in variational quantum algorithms: Algebraic extensions of the parameter-shift rule to general unitary transformations, Phys. Rev. A 104, 062443 (2021).
  16. M. Maronese, C. Destri, and E. Prati, Quantum activation functions for quantum neural networks, Quantum Inf. Process. 21, 128 (2022).
  17. L. Banchi, N. Quesada, and J. M. Arrazola, Training Gaussian Boson sampling distributions, Phys. Rev. A 102, 012417 (2020).
  18. G. de Felice and C. Corlett, Differentiation of linear optical circuits, arXiv:2401.07997.
  19. F. Hanamura, W. Asavanant, H. Nagayoshi, A. Sakaguchi, R. Ide, K. Fukui, P. van Loock, and A. Furusawa, Implementing arbitrary multimode continuous-variable quantum gates with fixed non-Gaussian states and adaptive linear optics, Phys. Rev. A 110, 022614 (2024).
  20. S. Markidis, On physics-informed neural networks for quantum computers, Front. Appl. Math. Stat. 8, 1036711 (2022).
  21. G. Panichi, S. Corli, and E. Prati, Quantum physics-informed neural networks for multivariable partial differential equations, Phys. Rev. Appl. 25, 014001 (2026).
  22. M. Knudsen and C. B. Mendl, Solving differential equations via continuous-variable quantum computers, arXiv:2012.12220.
  23. M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686 (2019).
  24. L. Brevi, A. Mandarino, and E. Prati, Addressing the non-perturbative regime of the quantum anharmonic oscillator by physics-informed neural networks, New J. Phys. 26, 103015 (2024).
  25. L. Brevi, A. Mandarino, and E. Prati, A tutorial on the use of physics-informed neural networks to compute the spectrum of quantum systems, Technologies 12, 174 (2024).
  26. A. E. Paine, V. E. Elfving, and O. Kyriienko, Physics-informed quantum machine learning: Solving nonlinear differential equations in latent spaces without costly grid evaluations, arXiv:2308.01827.
  27. A. Sedykh, M. Podapaka, A. Sagingalieva, K. Pinto, M. Pflitsch, and A. Melnikov, Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes, Mach. Learn.: Sci. Technol. 5, 025045 (2024).
  28. C. Trahan, M. Loveland, and S. Dent, Quantum physics-informed neural networks, Entropy 26, 649 (2024).
  29. P. Siegl, S. Wassing, D. M. Mieth, S. Langer, and P. Bekemeyer, Solving transport equations on quantum computers—potential and limitations of physics-informed quantum circuits, CEAS Aeronaut. J. 16, 63 (2025).
  30. M. J. Powell, A direct search optimization method that models the objective and constraint functions by linear interpolation, in Advances in Optimization and Numerical Analysis (Springer, New York, 1994), pp. 51–67.
  31. D. M. Olsson and L. S. Nelson, The Nelder-mead simplex procedure for function minimization, Technometrics 17, 45 (1975).
  32. A. Oosterwijk, H. A. Dijkstra, and T. van Leeuwen, An adjoint-free method to determine conditional nonlinear optimal perturbations, Comput. Geosci. 106, 190 (2017).
  33. S. Singer and J. Nelder, Nelder-mead algorithm, Scholarpedia 4, 2928 (2009).
  34. M. H. Wright, Nelder, mead, and the other simplex method, in Optimization Stories (European Mathematical Society-EMS-Publishing House GmbH, Berlin, Germany, 2012), pp. 271–276.
  35. R. A. Khalek and V. Bertone, On the derivatives of feed-forward neural networks, arXiv:2005.07039.
  36. V. I. Avrutskiy, Enhancing function approximation abilities of neural networks by training derivatives, IEEE Trans. Neural Networks Learn. Syst. 32, 916 (2020).
  37. S. Rodini, Analytical derivatives of neural networks, Comput. Phys. Commun. 270, 108169 (2022).
  38. A. Zavatta, S. Viciani, and M. Bellini, Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission, Phys. Rev. A 72, 023820 (2005).
  39. F. A. Domínguez-Serna, F. J. Mendieta-Jimenez, and F. Rojas, Entangled photon-added coherent states, Quantum Inf. Process. 15, 3121 (2016).
  40. V. V. Dodonov, M. A. Marchiolli, Y. A. Korennoy, V. I. Man’ko, and Y. A. Moukhin, Dynamical squeezing of photon-added coherent states, Phys. Rev. A 58, 4087 (1998).
  41. G. S. Agarwal and K. Tara, Nonclassical properties of states generated by the excitations on a coherent state, Phys. Rev. A 43, 492 (1991).
  42. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, Quantum state reconstruction of the single-photon Fock state, Phys. Rev. Lett. 87, 050402 (2001).
  43. C. K. Hong and L. Mandel, Experimental realization of a localized one-photon state, Phys. Rev. Lett. 56, 58 (1986).
  44. P. Grangier, G. Roger, and A. Aspect, Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences, Europhys. Lett. 1, 173 (1986).
  45. A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, Quantum homodyne tomography of a two-photon Fock state, Phys. Rev. Lett. 96, 213601 (2006).
  46. B. J. Lawrie, P. D. Lett, A. M. Marino, and R. C. Pooser, Quantum sensing with squeezed light, ACS Photonics 6, 1307 (2019).
  47. Z. Y. Ou, J.-K. Rhee, and L. J. Wang, Observation of four-photon interference with a beam splitter by pulsed parametric down-conversion, Phys. Rev. Lett. 83, 959 (1999).
  48. D. Kalamidas, C. C. Gerry, and A. Benmoussa, Proposal for generating a two-photon added coherent state via down-conversion with a single crystal, Phys. Lett. A 372, 1937 (2008).
  49. J. Fadrný, M. Neset, M. Bielak, M. Ježek, J. Bílek, and J. Fiurášek, Experimental preparation of multiphoton-added coherent states of light, npj Quantum Inf. 10, 89 (2024).
  50. N. Killoran, J. Izaac, N. Quesada, V. Bergholm, M. Amy, and C. Weedbrook, Strawberry fields: A software platform for photonic quantum computing, Quantum 3, 129 (2019).
  51. M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin et al., Tensorflow: Large-scale machine learning on heterogeneous systems (2015).

Outline

Information

Phys. Rev. Applied 26, 014107– Published 31 July, 2026
  • Received 24 February 2026
  • Revised 21 April 2026
  • Accepted 9 July 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