- 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

Electron-molecule scattering via R-matrix variational algorithms on a quantum computer

Dario Picozzi*

Jonathan Tennyson

Vincent Graves

Jimena D. Gorfinkiel

  • School of Physical Sciences, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom

  • *Contact author: picozzi.dario@gmail.com
PDF

Phys. Rev. A 114, 012407 – Published 6 July, 2026

DOI: https://doi.org/10.1103/q8ry-hlxt

Abstract

Electron-molecule collisions play a central role in both natural processes and modern technological applications, particularly in plasma processing. Conventional computational strategies such as the R-matrix method have been widely adopted yet encounter significant scaling challenges in treating more complex systems. In this work we present a quantum computational approach that utilizes the variational quantum eigensolver (VQE) and variations thereof to overcome these limitations. We explore a number of methods, including the use of number projection operators and simultaneous optimization. We demonstrate the feasibility of our method on a model problem of electron scattering from the hydrogen molecule, with numerical results obtained using a noiseless classical simulator. We recover the full spectrum of the Hamiltonian within a chosen symmetry sector. Moreover, the optimal circuit parameters directly encode the R-matrix boundary amplitudes needed for subsequent scattering computations. We formulate the R-matrix inner-region problem for electron-molecule scattering in a variational quantum-computing framework.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (59)

  1. B. I. Schneider, T. N. Rescigno, B. H. Lengsfield, and A. E. Orel, Ab initio electron-molecule collisions: The water molecule, Science 284, 1489 (1999).
  2. J. M. Carr, P. G. Galiatsatos, J. D. Gorfinkiel, A. G. Harvey, M. A. Lysaght, D. Madden, Z. Mašín, M. Plummer, J. Tennyson, and H. N. Varambhia, UKRmol: a low-energy electron- and positron-molecule scattering suite, Eur. Phys. J. D 66, 58 (2012).
  3. M. C. Zammit, J. S. Savage, D. V. Fursa, and I. Bray, Complete solution of electronic excitation and ionization in electron-hydrogen molecule scattering, Phys. Rev. Lett. 116, 233201 (2016).
  4. K. Bartschat and M. J. Kushner, Electron collisions with atoms, ions, molecules, and surfaces: Fundamental science empowering advances in technology, Proc. Natl. Acad. Sci. USA 113, 7026 (2016).
  5. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  6. S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
  7. D. S. Abrams and S. Lloyd, Simulation of many-body Fermi systems on a universal quantum computer, Phys. Rev. Lett. 79, 2586 (1997).
  8. D. S. Abrams and S. Lloyd, Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors, Phys. Rev. Lett. 83, 5162 (1999).
  9. A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Simulated quantum computation of molecular energies, Science 309, 1704 (2005).
  10. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 4213 (2014).
  11. J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New J. Phys. 18, 023023 (2016).
  12. B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, Towards quantum chemistry on a quantum computer, Nat. Chem. 2, 106 (2010).
  13. P. J. J. O'Malley et al., Scalable quantum simulation of molecular energies, Phys. Rev. X 6, 031007 (2016).
  14. C. Hempel, C. Maier, J. Romero, J. McClean, T. Monz, H. Shen, P. Jurcevic, B. P. Lanyon, P. Love, R. Babbush, A. Aspuru-Guzik, R. Blatt, and C. F. Roos, Quantum chemistry calculations on a trapped-ion quantum simulator, Phys. Rev. X 8, 031022 (2018).
  15. Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferová, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. D. Sawaya, S. Sim, L. Veis, and A. Aspuru-Guzik, Quantum chemistry in the age of quantum computing, Chem. Rev. 119, 10856 (2019).
  16. S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Quantum computational chemistry, Rev. Mod. Phys. 92, 015003 (2020).
  17. B. Bauer, S. Bravyi, M. Motta, and G. K.-L. Chan, Quantum algorithms for quantum chemistry and quantum materials science, Chem. Rev. 120, 12685 (2020).
  18. J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, and J. Tennyson, The variational quantum eigensolver: A review of methods and best practices, Phys. Rep. 986, 1 (2022).
  19. K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W. K. Mok, S. Sim, L. C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  20. S. Guo , Experimental quantum computational chemistry with optimized unitary coupled cluster ansatz, Nat. Phys. 20, 1240 (2024).
  21. P. Chawla, Shweta, K. R. Swain, T. Patel, R. Bala, D. Shetty, K. Sugisaki, S. B. Mandal, J. Riu, J. Nogue, V. S. Prasannaa, and B. P. Das, Relativistic variational-quantum-eigensolver calculations of molecular electric dipole moments on quantum hardware, Phys. Rev. A 111, 022817 (2025).
  22. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
  23. M. Cerezo, M. Larocca, D. García-Martín, N. L. Diaz, P. Braccia, E. Fontana, M. S. Rudolph, P. Bermejo, A. Ijaz, S. Thanasilp, E. R. Anschuetz, and Z. Holmes, Does provable absence of barren plateaus imply classical simulability? Nat. Commun. 16, 7907 (2025).
  24. S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum computation of scattering in scalar quantum field theories Quantum Inf. Comput. 14, 1014 (2014).
  25. S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum algorithms for quantum field theories, Science 336, 1130 (2012).
  26. S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum algorithms for fermionic quantum field theories, arXiv:1404.7115.
  27. K. Choi, D. Lee, J. Bonitati, Z. Qian, and J. Watkins, Rodeo algorithm for quantum computing, Phys. Rev. Lett. 127, 040505 (2021).
  28. N. Klco and M. J. Savage, Digitization of scalar fields for quantum computing, Phys. Rev. A 99, 052335 (2019).
  29. F. Arute et al., Hartree-Fock on a superconducting qubit quantum computer, Science 369, 1084 (2020).
  30. M. Yusf, L. Gan, C. Moffat, and G. Rupak, Elastic scattering on a quantum computer, Phys. Rev. C 111, 034001 (2025).
  31. X. Xing, A. Gomez Cadavid, A. F. Izmaylov, and T. V. Tscherbul, A hybrid quantum-classical algorithm for multichannel quantum scattering of atoms and molecules, J. Phys. Chem. Lett. 14, 6224 (2023).
  32. H. H. S. Chan, R. Meister, T. Jones, D. P. Tew, and S. C. Benjamin, Grid-based methods for chemistry simulations on a quantum computer, Sci. Adv. 9, eabo7484 (2023).
  33. S. Sharma, T. Papenbrock, and L. Platter, Scattering phase shifts from a quantum computer, Phys. Rev. C 109, L061001 (2024).
  34. K. M. Nakanishi, K. Mitarai, and K. Fujii, Subspace-search variational quantum eigensolver for excited states, Phys. Rev. Res. 1, 033062 (2019).
  35. R. M. Parrish, E. G. Hohenstein, P. L. McMahon, and T. J. Martínez, Quantum computation of electronic transitions using a variational quantum eigensolver, Phys. Rev. Lett. 122, 230401 (2019).
  36. D. Picozzi, Pascal's pyramid and number projection operators for quantum computation, arXiv:2407.16561.
  37. E. P. Wigner and L. Eisenbud, Higher angular momenta and long range interaction in resonance reactions, Phys. Rev. 72, 29 (1947).
  38. P. G. Burke, R-Matrix Theory of Atomic Collisions: Application to Atomic, Molecular and Optical Processes (Springer, New York, 2011).
  39. J. Tennyson, Electron–molecule collision calculations using the R-matrix method, Phys. Rep. 491, 29 (2010).
  40. Z. Mašín, J. Benda, J. D. Gorfinkiel, A. G. Harvey, and J. Tennyson, UKRmol+: A suite for modelling electronic processes in molecules interacting with electrons, positrons and photons using the R-matrix method, Comput. Phys. Commun. 249, 107092 (2020).
  41. A. G. Sunderland, J. W. Heggarty, C. J. Noble, and N. S. Scott, Parallelization of R-matrix propagation methods on distributed memory computers, Comput. Phys. Commun. 114, 183 (1998).
  42. J. D. Gorfinkiel and J. Tennyson, Electron–H3+ collisions at intermediate energies, J. Phys. B 37, L343 (2004).
  43. J. D. Gorfinkiel and J. Tennyson, Electron impact ionization of small molecules at intermediate energies: the molecular R-matrix with pseudostates method, J. Phys. B 38, 1607 (2005).
  44. G. Halmová and J. Tennyson, Resonances in electron-impact electron detachment of C2−, Phys. Rev. Lett. 100, 213202 (2008).
  45. R. Zhang, P. G. Galiatsatos, and J. Tennyson, Positron collisions with acetylene calculated using the R-matrix with pseudo-states method, J. Phys. B 44, 195203 (2011).
  46. C. Bloch, Une formulation unifiée de la théorie des réactions nucléaires, Nucl. Phys. 4, 503 (1957).
  47. J. Tennyson, Partitioned R-matrix theory for molecules, J. Phys. B 37, 1061 (2004).
  48. J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Simulation of electronic structure Hamiltonians using quantum computers, Mol. Phys. 109, 735 (2011).
  49. I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M.-H. Yung, and A. Aspuru-Guzik, Simulating chemistry using quantum computers, Annu. Rev. Phys. Chem. 62, 185 (2011).
  50. D. Picozzi and J. Tennyson, Symmetry-adapted encodings for qubit number reduction by point-group and other Boolean symmetries, Quantum Sci. Technol. 8, 035026 (2023).
  51. D. W. Berry, C. Gidney, M. Motta, J. R. McClean, and R. Babbush, Improved techniques for preparing eigenstates of fermionic Hamiltonians, npj Quantum Inf. 4, 22 (2018).
  52. D. Bacon, I. L. Chuang, and A. W. Harrow, Efficient quantum circuits for Schur and Clebsch-Gordan transforms, Phys. Rev. Lett. 97, 170502 (2006).
  53. L. Wang and A. Zunger, Solving Schrödinger's equation around a desired energy: Application to silicon quantum dots, J. Chem. Phys. 100, 2394 (1994).
  54. L. C. Tazi and A. J. W. Thom, Folded spectrum VQE: A quantum computing method for the calculation of molecular excited states, J. Chem. Theory Comput. 20, 2491 (2024).
  55. C. H. Bennett, Logical reversibility of computation, IBM J. Res. Dev. 17, 525 (1973).
  56. C. H. Bennett, Time/space trade-offs for reversible computation, SIAM J. Comput. 18, 766 (1989).
  57. M. J. D. Powell, in Advances in Optimization and Numerical Analysis, edited by S. Gomez and J.-P. Hennart, Mathematics and Its Applications Vol. 275 (Kluwer Academic, Dordrecht, 1994), pp. 51–67.
  58. M. J. D. Powell, Direct search algorithms for optimization calculations, Acta Numer. 7, 287 (1998).
  59. D. Picozzi, electron-molecule-rmatrix-vqe: Code and data for Electron-molecule scattering via R-matrix variational algorithms on a quantum computer, Zenodo, version v1.0.0 (2026), doi:10.5281/zenodo.20140843.

Outline

Information

Phys. Rev. A 114, 012407– Published 6 July, 2026
  • Received 7 July 2025
  • Revised 28 January 2026
  • Accepted 28 April 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