- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

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

Quantum-enhanced neural exchange-correlation functionals

Igor O. Sokolov1,*, Gert-Jan Both1, Art D. Bochevarov2, Pavel A. Dub3, Daniel S. Levine2, Christopher T. Brown4, Shaheen Acheche1, Panagiotis Kl. Barkoutsos1, and Vincent E. Elfving1

  • 1Pasqal, 24 av. Emile Baudot, 91120 Palaiseau, France
  • 2Schrödinger Inc., 1540 Broadway, 24th Floor, New York, New York 10036, USA
  • 3Schrödinger Inc., 9868 Scranton Rd., Suite 3200, San Diego, California 92121, USA
  • 4Schrödinger Inc., 1 Main St., 11th Floor, Cambridge, Massachusetts 02142, USA

  • *Contact author: sokolov.igor.ch@gmail.com
PDF

Phys. Rev. A 113, 012427 – Published 20 January, 2026

DOI: https://doi.org/10.1103/m51l-fys2

Abstract

Kohn-Sham density functional theory (KS-DFT) provides the exact ground-state energy and electron density of a molecule, contingent on the as-yet-unknown universal exchange-correlation (XC) functional. Recent research has demonstrated that neural networks can efficiently learn to represent approximations to that functional, offering accurate generalizations to molecules not present during the training process. With the latest advancements in quantum-enhanced machine learning (ML), evidence is growing that quantum neural network (QNN) models may offer advantages in ML applications. In this work, we explore the use of QNNs for representing XC functionals, enhancing and comparing them to classical ML techniques. We present QNNs based on differentiable quantum circuits as quantum (hybrid) models for XC in KS-DFT, implemented across various architectures. We assess their performance on one-dimensional (1D) and 3D systems. To that end, we expand existing differentiable KS-DFT frameworks and propose strategies for efficient training of such functionals, highlighting the importance of fractional orbital occupation for accurate results. We prove that, under specific assumptions, noise-induced errors do not accumulate across SCF iterations, and we confirm this result through numerical experiments. Our best QNN-based XC functional yields energy profiles of the H2 and planar H4 molecules that deviate by no more than 1 mHa from the reference DMRG and FCI/6-31G results, respectively. Moreover, they reach chemical precision on a system, H2H2, not present in the training dataset, using only a few variational parameters. This work lays the foundation for the integration of quantum models in KS-DFT, thereby opening new avenues for expressing XC functionals in a differentiable way and facilitating computations of various properties.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (112)

  1. R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, International Series of Monographs on Chemistry (Oxford University Press, Oxford, 1994).
  2. W. Koch and M. C. Holthausen, A Chemist's Guide to Density Functional Theory (Wiley, New York, 2015).
  3. E. Engel and R. M. Dreizler, Density Functional Theory: An Advanced Course, Theoretical and Mathematical Physics (Springer, Berlin, 2011).
  4. A. D. Becke, J. Chem. Phys. 140, 18A301 (2014).
  5. R. O. Jones, Rev. Mod. Phys. 87, 897 (2015).
  6. A. J. Cohen, P. Mori-Sánchez, and W. Yang, Chem. Rev. 112, 289 (2012).
  7. M. G. Medvedev, I. S. Bushmarinov, J. Sun, J. P. Perdew, and K. A. Lyssenko, Science 355, 49 (2017).
  8. K. P. Kepp, Science 356, 496 (2017).
  9. A. J. Cohen, P. Mori-Sánchez, and W. Yang, Science 321, 792 (2008).
  10. A. J. Cohen, P. Mori-Sánchez, and W. Yang, J. Chem. Phys. 129, 121104 (2008).
  11. N. Mardirossian and M. Head-Gordon, Mol. Phys. 115, 2315 (2017).
  12. L. Goerigk, A. Hansen, C. Bauer, S. Ehrlich, A. Najibi, and S. Grimme, Phys. Chem. Chem. Phys. 19, 32184 (2017).
  13. E. Caldeweyher, S. Ehlert, A. Hansen, H. Neugebauer, S. Spicher, C. Bannwarth, and S. Grimme, J. Chem. Phys. 150, 154122 (2019).
  14. P. Verma and D. G. Truhlar, Trends Chem. 2, 302 (2020).
  15. M. A. Rohrdanz, K. M. Martins, and J. M. Herbert, J. Chem. Phys. 130, 054112 (2009).
  16. S. Mohr, L. E. Ratcliff, L. Genovese, D. Caliste, P. Boulanger, S. Goedecker, and T. Deutsch, Phys. Chem. Chem. Phys. 17, 31360 (2015).
  17. M. Bursch, J.-M. Mewes, A. Hansen, and S. Grimme, Angew. Chem. Int. Ed. 61, e202205735 (2022).
  18. D. J. Tozer, V. E. Ingamells, and N. C. Handy, J. Chem. Phys. 105, 9200 (1996).
  19. D. J. Tozer and N. C. Handy, Recent Advances In Density Functional Methods: (Part II) (World Scientific, Singapore, 1997), pp. 1–14.
  20. F. Brockherde, L. Vogt, L. Li, M. E. Tuckerman, K. Burke, and K.-R. Müller, Nat. Commun. 8, 872 (2017).
  21. R. Nagai, R. Akashi, and O. Sugino, npj Comput. Mater. 6, 43 (2020).
  22. S. Dick and M. Fernandez-Serra, Nat. Commun. 11, 3509 (2020).
  23. B. Kalita, L. Li, R. J. McCarty, and K. Burke, Acc. Chem. Res. 54, 818 (2021).
  24. J. Kirkpatrick, B. McMorrow, D. H. P. Turban, A. L. Gaunt, J. S. Spencer, A. G. D. G. Matthews, A. Obika, L. Thiry, M. Fortunato, D. Pfau, L. R. Castellanos, S. Petersen, A. W. R. Nelson, P. Kohli, P. Mori-Sánchez, D. Hassabis, and A. J. Cohen, Science 374, 1385 (2021).
  25. L. Li, S. Hoyer, R. Pederson, R. Sun, E. D. Cubuk, P. Riley, and K. Burke, Phys. Rev. Lett. 126, 036401 (2021).
  26. E. Cuierrier, P.-O. Roy, R. Wang, and M. Ernzerhof, J. Chem. Phys. 157, 171103 (2022).
  27. S. Riemelmoser, C. Verdi, M. Kaltak, and G. Kresse, J. Chem. Theory Comput. 19, 7287 (2023).
  28. M. F. Kasim and S. M. Vinko, Phys. Rev. Lett. 127, 126403 (2021).
  29. R. Nagai, R. Akashi, and O. Sugino, Phys. Rev. Res. 4, 013106 (2022).
  30. J. Wu, S.-M. Pun, X. Zheng, and G. Chen, J. Chem. Phys. 159, 090901 (2023).
  31. X. Zhang and G. K.-L. Chan, J. Chem. Phys. 157, 204801 (2022).
  32. P. A. M Casares, J. S. Baker, M. Medvidović, R. d. Reis, and J. M. Arrazola, J. Chem. Phys. 160, 062501 (2024).
  33. B. Kalita, R. Pederson, J. Chen, L. Li, and K. Burke, J. Phys. Chem. Lett. 13, 2540 (2022).
  34. J. Schmidt, C. L. Benavides-Riveros, and M. A. Marques, J. Phys. Chem. Lett. 10, 6425 (2019).
  35. H. Ma, A. Narayanaswamy, P. Riley, and L. Li, Sci. Adv. 8, eabq0279 (2022).
  36. O. Kyriienko, A. E. Paine, and V. E. Elfving, Phys. Rev. A 103, 052416 (2021).
  37. A. Abbas, D. Sutter, C. Zoufal, A. Lucchi, A. Figalli, and S. Woerner, Nat. Comput. Sci. 1, 403 (2021).
  38. K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Phys. Rev. A 98, 032309 (2018).
  39. M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Phys. Rev. A 99, 032331 (2019).
  40. O. Kyriienko and V. E. Elfving, Phys. Rev. A 104, 052417 (2021).
  41. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, et al., Nat. Rev. Phys. 3, 625 (2021).
  42. I. Cong, S. Choi, and M. D. Lukin, Nat. Phys. 15, 1273 (2019).
  43. H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, et al., Science 376, 1182 (2022).
  44. M. Schuld, R. Sweke, and J. J. Meyer, Phys. Rev. A 103, 032430 (2021).
  45. M. Larocca, P. Czarnik, K. Sharma, G. Muraleedharan, P. J. Coles, and M. Cerezo, Quantum 6, 824 (2022).
  46. A. Abbas, R. King, H.-Y. Huang, W. J. Huggins, R. Movassagh, D. Gilboa, and J. R. McClean, in Proceedings of the 37th International Conference on Neural Information Processing Systems (Curran Associates Inc., Red Hook, NY, USA, 2023).
  47. K. Beer, D. Bondarenko, T. Farrelly, T. J. Osborne, R. Salzmann, D. Scheiermann, and R. Wolf, Nat. Commun. 11, 808 (2020).
  48. S. H. Sack, R. A. Medina, A. A. Michailidis, R. Kueng, and M. Serbyn, PRX Quantum 3, 020365 (2022).
  49. H.-Y. Huang, R. Kueng, and J. Preskill, Nat. Phys. 16, 1050 (2020).
  50. D. Miller, L. E. Fischer, K. Levi, E. J. Kuehnke, I. O. Sokolov, P. K. Barkoutsos, J. Eisert, and I. Tavernelli, npj Quantum Inf. 10, 122 (2024).
  51. E. Gil-Fuster, C. Gyurik, A. Pérez-Salinas, and V. Dunjko, in The Thirteenth International Conference on Learning Representations (ICLR, 2025).
  52. B. Senjean, S. Yalouz, and M. Saubanère, SciPost Phys. 14, 055 (2023).
  53. M. Rossmannek, P. K. Barkoutsos, P. J. Ollitrault, and I. Tavernelli, J. Chem. Phys. 154, 114105 (2021).
  54. R. Schade, C. Bauer, K. Tamoev, L. Mazur, C. Plessl, and T. D. Kühne, Phys. Rev. Res. 4, 033160 (2022).
  55. F. Gaitan and F. Nori, Phys. Rev. B 79, 205117 (2009).
  56. T. E. Baker and D. Poulin, Phys. Rev. Res. 2, 043238 (2020).
  57. P. Barkoutsos, I. Olegovich Sokolov, G.-J. Both, and V. E. Elfving, Determining exchange-correlation functionals for molecular systems using a quantum computer (2025), https://patents.google.com/patent/EP4629143A1/en.
  58. M. Schuld and N. Killoran, PRX Quantum 3, 030101 (2022).
  59. I. O. Sokolov and S. Acheche, QEX: Quantum-enhanced neural exchange-correlation functionals (2025), https://github.com/pasqal-io/qex.
  60. J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: Composable transformations of python+numpy programs (2018), http://github.com/jax-ml/jax.
  61. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  62. K. Hornik, M. Stinchcombe, and H. White, Neural Netw. 2, 359 (1989).
  63. M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Nat. Commun. 12, 1791 (2021).
  64. A. Pérez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, Quantum 4, 226 (2020).
  65. A. Pérez-Salinas, D. López-Núñez, A. García-Sáez, P. Forn-Díaz, and J. I. Latorre, Phys. Rev. A 104, 012405 (2021).
  66. J. Landman, S. Thabet, C. Dalyac, H. Mhiri, and E. Kashefi, in The Eleventh International Conference on Learning Representations (ICLR, 2023).
  67. K. Nakaji, S. Uno, Y. Suzuki, R. Raymond, T. Onodera, T. Tanaka, H. Tezuka, N. Mitsuda, and N. Yamamoto, Phys. Rev. Res. 4, 023136 (2022).
  68. S. Dasgupta and J. M. Herbert, J. Comput. Chem. 38, 869 (2017).
  69. X.-M. Zhang, M.-H. Yung, and X. Yuan, Phys. Rev. Res. 3, 043200 (2021).
  70. A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, in NIPS 2017 Workshop on Autodiff (ICLR, 2017).
  71. Q. Sun, T. C. Berkelbach, N. S. Blunt, G. H. Booth, S. Guo, Z. Li, J. Liu, J. D. McClain, E. R. Sayfutyarova, S. Sharma, et al., WIREs Comput. Mol. Sci. 8, e1340 (2018).
  72. J.-D. Chai, J. Chem. Phys. 136, 154104 (2012).
  73. E. Cances, K. N. Kudin, G. E. Scuseria, and G. Turinici, J. Chem. Phys. 118, 5364 (2003).
  74. T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-structure Theory (John Wiley & Sons, Hoboken, USA, 2013).
  75. R. J. Bartlett and M. Musiał, Rev. Mod. Phys. 79, 291 (2007).
  76. I. W. Bulik, T. M. Henderson, and G. E. Scuseria, J. Chem. Theory Comput. 11, 3171 (2015).
  77. D. P. Kingma and J. Ba, arXiv:1412.6980.
  78. C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, ACM Trans. Math. Softw. 23, 550 (1997).
  79. D. Seitz, N. Heim, J. P. Moutinho, R. Guichard, V. Abramavicius, A. Wennersteen, G.-J. Both, A. Quelle, C. de Groot, G. V. Velikova, V. E. Elfving, and M. Dagrada, IEEE Software, 42, 44 (2025).
  80. D. Tuna, D. Lefrancois, Ł. Wolanski, S. Gozem, I. Schapiro, T. Andruniów, A. Dreuw, and M. Olivucci, J. Chem. Theory Comput. 11, 5758 (2015).
  81. H. Douroudgari, M. Vahedpour, and F. Khouini, Sci. Rep. 11, 13220 (2021).
  82. M.-C. Kim, E. Sim, and K. Burke, Phys. Rev. Lett. 111, 073003 (2013).
  83. D. K. Park, F. Petruccione, and J.-K. K. Rhee, Sci. Rep. 9, 3949 (2019).
  84. I. O. Sokolov, P. K. Barkoutsos, P. J. Ollitrault, D. Greenberg, J. Rice, M. Pistoia, and I. Tavernelli, J. Chem. Phys. 152, 124107 (2020).
  85. X. Liu, X. Wang, Z. Wen, and Y. Yuan, SIAM J. Matrix Anal. Appl. 35, 546 (2014).
  86. X. Liu, Z. Wen, X. Wang, M. Ulbrich, and Y. Yuan, SIAM J. Numer. Anal. 53, 1758 (2015).
  87. N. D. Woods, M. Payne, and P. Hasnip, J. Phys.: Condens. Matter 31, 453001 (2019).
  88. E. Cances and C. Le Bris, ESAIM: M2AN 34, 749 (2000).
  89. Z. Bai, R.-C. Li, and D. Lu, arXiv:2009.09022.
  90. P. Pulay, Chem. Phys. Lett. 73, 393 (1980).
  91. K. N. Kudin, G. E. Scuseria, and E. Cances, J. Chem. Phys. 116, 8255 (2002).
  92. G. P. Kerker, Phys. Rev. B 23, 3082 (1981).
  93. V. R. Saunders and I. H. Hillier, Int. J. Quantum Chem. 7, 699 (1973).
  94. G. Luise, C.-W. Huang, T. Vogels, D. P. Kooi, S. Ehlert, S. Lanius, K. J. Giesbertz, A. Karton, D. Gunceler, M. Stanley, et al., arXiv:2506.14665.
  95. T. Ko and X. Li, Math. Comp. 92, 1693 (2023).
  96. T. Ko, X. Li, and C. Wang, in Proceedings of the IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Los Alamitos, CA, 2024), Vol. 1, pp. 669–680.
  97. W. Somogyi, E. Pankovets, V. Kuzmin, and A. Melnikov, arXiv:2410.19921.
  98. S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, Nature (London) 627, 778 (2024).
  99. L. Schatzki, M. Larocca, Q. T. Nguyen, F. Sauvage, and M. Cerezo, npj Quantum Inf. 10, 12 (2024).
  100. A. Sannia, F. Tacchino, I. Tavernelli, G. L. Giorgi, and R. Zambrini, npj Quantum Inf. 10, 81 (2024).
  101. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'brien, Nat. Commun. 5, 4213 (2014).
  102. S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, npj Quantum Inf. 5, 75 (2019).
  103. S. Das and V. Gavini, Phys. Rev. B 107, 125133 (2023).
  104. A. D. Becke, J. Chem. Phys. 88, 2547 (1988).
  105. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, et al., Nature (London) 618, 500 (2023).
  106. G. Torlai, G. Mazzola, G. Carleo, and A. Mezzacapo, Phys. Rev. Res. 2, 022060(R) (2020).
  107. E. Sheridan, L. Mineh, R. A. Santos, and T. Cubitt, arXiv:2402.18534.
  108. E. Koridon, F. Frohnert, E. Prehn, E. van Nieuwenburg, J. Tura, and S. Polla, Mach. Learn.: Sci. Technol. 6, 025020 (2025).
  109. W. van Dam, H. Liu, G. H. Low, A. Paetznick, A. Paz, M. Silva, A. Sundaram, K. Svore, and M. Troyer, arXiv:2409.05835.
  110. V. Lebedev, USSR Comput. Math. Math. Phys. 15, 44 (1975).
  111. D. Wecker, M. B. Hastings, and M. Troyer, Phys. Rev. A 92, 042303 (2015).
  112. J. Romero, R. Babbush, J. R. McClean, C. Hempel, P. J. Love, and A. Aspuru-Guzik, Quantum Sci. Technol. 4, 014008 (2018).

Outline

Information

Phys. Rev. A 113, 012427– Published 20 January, 2026
  • Received 12 September 2024
  • Revised 22 August 2025
  • Accepted 8 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