- 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
  • Open Access

Probing the ground state of the antiferromagnetic Heisenberg model on the kagome lattice using geometrically informed variational quantum eigensolver

Abdellah Tounsi1,2, Nacer Eddine Belaloui1,2, Abdelmouheymen Rabah Khamadja1,2, Takei Eddine Fadi Lalaoui1,2, Mohamed Messaoud Louamri1,3, David E. Bernal Neira4, and Mohamed Taha Rouabah1,2,5,*

  • *Contact author: m.taha.rouabah@umc.edu.dz
PDF

Phys. Rev. Research 8, 043080 – Published 21 October, 2026

DOI: https://doi.org/10.1103/82g5-bsvp

Abstract

This work investigates the nature of the ground state of the antiferromagnetic Heisenberg model on fundamental kagome cells, a triangle and a star, using the variational quantum eigensolver algorithm on real quantum hardware. We demonstrate that the ground-state preparation is achievable using a shallow hardware-efficient quantum circuit with a naturally Euclidean parameter space. Our custom ansatz is capable of accurately recovering meaningful properties of the ground state such as the spin-spin correlation terms and static structure factor without explicit error mitigation. These features are found to be resilient to noise. We exploited the Fubini-Study metric in constructing the ansatz, ensuring a singularity-free parameter space. With this ansatz design, the adaptive gradient descent optimizer achieves a faster convergence in terms of the number of iterations compared to simultaneous perturbation stochastic approximation. We further apply error mitigation techniques, including zero-noise extrapolation (ZNE) and qubitwise readout-error mitigation (REM). While ZNE does not obey the Rayleigh-Ritz variational principle, the conditions under which REM preserves it are discussed.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (76)

  1. J. Kempe, A. Kitaev, and O. Regev, The complexity of the local Hamiltonian problem, SIAM J. Comput. 35, 1070 (2006).
  2. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  3. S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Quantum computational chemistry, Rev. Mod. Phys. 92, 015003 (2020).
  4. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th ed. (Cambridge University Press, Cambridge, 2010).
  5. A. Somoroff, Q. Ficheux, R. A. Mencia, H. Xiong, R. Kuzmin, and V. E. Manucharyan, Millisecond coherence in a superconducting qubit, Phys. Rev. Lett. 130, 267001 (2023).
  6. S. Ganjam, Y. Wang, Y. Lu, A. Banerjee, C. U. Lei, L. Krayzman, K. Kisslinger, C. Zhou, R. Li, Y. Jia, M. Liu, L. Frunzio, and R. J. Schoelkopf, Surpassing millisecond coherence in on chip superconducting quantum memories by optimizing materials and circuit design, Nat. Commun. 15, 3687 (2024).
  7. J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  8. 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).
  9. 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).
  10. N. E. Belaloui, A. Tounsi, A. R. Khamadja, M. M. Louamri, A. Benslama, D. E. Bernal Neira, and M. T. Rouabah, Ground-state energy estimation on current quantum hardware through the variational quantum eigensolver: A practical study, J. Chem. Theory Comput. 21, 6777 (2025).
  11. 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).
  12. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  13. A. Y. Kitaev, Quantum measurements and the Abelian stabilizer problem, arXiv:quant-ph/9511026.
  14. Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O’Brien, Quantum error mitigation, Rev. Mod. Phys. 95, 045005 (2023).
  15. C. Yi, C. Zhou, and J. Takahashi, Quantum phase estimation by compressed sensing, Quantum 8, 1579 (2024).
  16. J. Robledo-Moreno, M. Motta, H. Haas, A. Javadi-Abhari, P. Jurcevic, W. Kirby, S. Martiel, K. Sharma, S. Sharma, T. Shirakawa, I. Sitdikov, R.-Y. Sun, K. J. Sung, M. Takita, M. C. Tran, S. Yunoki, and A. Mezzacapo, Chemistry beyond the scale of exact diagonalization on a quantum-centric supercomputer, Sci. Adv. 11, eadu9991 (2025).
  17. M. Mikkelsen and Y. O. Nakagawa, Quantum-selected configuration interaction with time-evolved state Phys. Rev. Res. 7, 043043 (2025).
  18. L. Balents, Spin liquids in frustrated magnets, Nature (London) 464, 199 (2010).
  19. G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletić, and M. D. Lukin, Probing topological spin liquids on a programmable quantum simulator, Science 374, 1242 (2021).
  20. J. L. Bosse and A. Montanaro, Probing ground-state properties of the kagome antiferromagnetic Heisenberg model using the variational quantum eigensolver, Phys. Rev. B 105, 094409 (2022).
  21. J. Kattemölle and J. van Wezel, Variational quantum eigensolver for the Heisenberg antiferromagnet on the kagome lattice, Phys. Rev. B 106, 214429 (2022).
  22. M. Sajjan, R. Gupta, S. S. Kale, V. Singh, K. Kumaran, and S. Kais, Physics-inspired quantum simulation of resonating valence bond states: A prototypical template for a spin-liquid ground state, J. Phys. Chem. A 127, 8751 (2023).
  23. Y. Guo, F. Guo, B. Zhong, X. Chen, X. Yuan, X. Jin, and H. Tang, Quantum-enhanced computing for the antiferromagnetic j1−j2 Heisenberg model, Adv. Quantum Technol. 8, 2300240 (2025).
  24. T. Weaving, A. Ralli, V. Wimalaweera, P. J. Love, and P. V. Coveney, Simulating the antiferromagnetic Heisenberg model on a spin-frustrated kagome lattice with the contextual subspace variational quantum eigensolver, arXiv:2506.12391.
  25. Q. Wang, H. Lei, Y. Qi, and C. Felser, Intriguing kagome topological materials, npj Quantum Mater. 10, 72 (2025).
  26. B. Bauer, L. Cincio, B. Keller, M. Dolfi, G. Vidal, S. Trebst, and A. Ludwig, Chiral spin liquid and emergent anyons in a kagome lattice Mott insulator, Nat. Commun. 5, 5137 (2014).
  27. J.-W. Mei, J.-Y. Chen, H. He, and X.-G. Wen, Gapped spin liquid with Z2 topological order for the kagome Heisenberg model, Phys. Rev. B 95, 235107 (2017).
  28. A. Tounsi, N. E. Belaloui, M. M. Louamri, A. Benslama, and M. T. Rouabah, Optimized topological quantum compilation of three-qubit controlled gates in the Fibonacci anyon model: A controlled-injection approach, Phys. Rev. A 110, 012603 (2024).
  29. P. Mendels and F. Bert, Quantum kagome antiferromagnet: ZnCu3(OH)6Cl2, J. Phys.: Conf. Ser. 320, 012004 (2011).
  30. A. S. Teixeira Pires, Theoretical Tools for Spin Models in Magnetic Systems, IOP ebooks (Institute of Physics Publishing, London, England, 2021).
  31. B. Sriram Shastry and B. Sutherland, Exact ground state of a quantum mechanical antiferromagnet, Physica B+C 108, 1069 (1981).
  32. S. Sachdev, Kagome´- and triangular-lattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons, Phys. Rev. B 45, 12377 (1992).
  33. C. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuillier, P. Sindzingre, P. Lecheminant, and L. Pierre, First excitations of the spin 12 Heisenberg antiferromagnet on the kagomé lattice, Eur. Phys. J. B 2, 501 (1998).
  34. Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, Projected-wave-function study of the spin-12 Heisenberg model on the kagomé lattice, Phys. Rev. Lett. 98, 117205 (2007).
  35. S. Yan, D. A. Huse, and S. R. White, Spin-liquid ground state of the S=12 kagome Heisenberg antiferromagnet, Science 332, 1173 (2011).
  36. Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Signatures of Dirac cones in a DMRG study of the kagome Heisenberg model, Phys. Rev. X 7, 031020 (2017).
  37. R.-Y. Sun, H.-K. Jin, H.-H. Tu, and Y. Zhou, Possible chiral spin liquid state in the S=12 kagome Heisenberg model, npj Quantum Mater. 9, 16 (2024).
  38. H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z. Huang, B. Normand, and T. Xiang, Gapless spin-liquid ground state in the S=12 kagome antiferromagnet, Phys. Rev. Lett. 118, 137202 (2017).
  39. A. M. Läuchli, J. Sudan, and R. Moessner, S=12 kagome Heisenberg antiferromagnet revisited, Phys. Rev. B 100, 155142 (2019).
  40. T. Đurić, J. H. Chung, B. Yang, and P. Sengupta, Spin-1/2 kagome Heisenberg antiferromagnet: Machine learning discovery of the spinon pair-density-wave ground state, Phys. Rev. X 15, 011047 (2025).
  41. J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum natural gradient, Quantum 4, 269 (2020).
  42. F. Dell’Anna, R. Gómez-Lurbe, A. Pérez, and E. Ercolessi, Quantum natural gradient optimizer on noisy platforms: Quantum approximate optimization algorithm as a case study, Phys. Rev. A 112, 022612 (2025).
  43. T. A. Atif, U. Chukwu, J. Berwald, and R. Dridi, Accelerating NISQ variational methods using geometry, in Quantum Computing, Communication, and Simulation III, edited by P. R. Hemmer and A. L. Migdall (SPIE, San Francisco, California, 2023).
  44. M. R. Geller and M. Sun, Toward efficient correction of multiqubit measurement errors: Pair correlation method, Quantum Sci. Technol. 6, 025009 (2021).
  45. K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).
  46. M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).
  47. M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2025).
  48. M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Geometry and non-adiabatic response in quantum and classical systems, Phys. Rep. 697, 1 (2017).
  49. M. Bukov, D. Sels, and A. Polkovnikov, Geometric speed limit of accessible many-body state preparation, Phys. Rev. X 9, 011034 (2019).
  50. N. Yamamoto, On the natural gradient for variational quantum eigensolver, arXiv:1909.05074.
  51. S.-I. Amari, Natural gradient works efficiently in learning, Neural Comput. 10, 251 (1998).
  52. C.-Y. Park and N. Killoran, Hamiltonian variational ansatz without barren plateaus, Quantum 8, 1239 (2024).
  53. P. Murali, D. C. Mckay, M. Martonosi, and A. Javadi-Abhari, Software mitigation of crosstalk on noisy intermediate-scale quantum computers, in Proceedings of the Twenty-Fifth International Conference on Architectural Support for Programming Languages and Operating Systems, ASPLOS ’20 (Association for Computing Machinery, New York, NY, 2020), pp. 1001–1016.
  54. D. C. McKay, I. Hincks, E. J. Pritchett, M. Carroll, L. C. G. Govia, and S. T. Merkel, Benchmarking quantum processor performance at scale, arXiv:2311.05933.
  55. M. A. Haddou and M. Bennai, Sculpting quantum landscapes: Fubini-study metric conditioning for geometry-aware learning in parameterized quantum circuits, arXiv:2506.21940.
  56. F. Sorourifar, M. T. Rouabah, N. E. Belaloui, M. M. Louamri, D. Chamaki, E. J. Gustafson, N. M. Tubman, J. A. Paulson, and D. E. Bernal Neira, Toward efficient quantum computation of molecular ground-state energies, AIChE J. 71, e18887 (2025).
  57. K. Lively, T. Bode, J. Szangolies, J.-X. Zhu, and B. Fauseweh, Noise robust detection of quantum phase transitions, Phys. Rev. Res. 6, 043254 (2024).
  58. Y. Kim, C. J. Wood, T. J. Yoder, S. T. Merkel, J. M. Gambetta, K. Temme, and A. Kandala, Scalable error mitigation for noisy quantum circuits produces competitive expectation values, Nat. Phys. 19, 752 (2023).
  59. T. Kurita, H. Qassim, M. Ishii, H. Oshima, S. Sato, and J. Emerson, Synergetic quantum error mitigation by randomized compiling and zero-noise extrapolation for the variational quantum eigensolver, Quantum 7, 1184 (2023).
  60. A. Tounsi, N. E. Belaloui, A. R. Khamadja, T. E. F. Lalaoui, M. M. Louamri, D. E. Bernal Neira, and M. T. Rouabah, Supplementary data and code for: “Probing the ground state of the antiferromagnetic Heisenberg model on the kagome lattice using geometrically informed variational quantum eigensolver”, Zenodo, 2026, http://doi.org/10.5281/zenodo.21264601.
  61. R. Gómez-Lurbe, Efficient protocol to estimate the quantum Fisher information matrix for commuting-block circuits, Quantum Inf. Process. 25, 278 (2026).
  62. L. Armijo, Minimization of functions having Lipschitz continuous first partial derivatives, Pac. J. Math. 16, 1 (1966).
  63. C. R. Harris, K. Jarrod Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, et al., Array programming with NumPy, Nature (London) 585, 357 (2020).
  64. D. Wecker, M. B. Hastings, and M. Troyer, Progress towards practical quantum variational algorithms, Phys. Rev. A 92, 042303 (2015).
  65. 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).
  66. J. Romero, R. Babbush, J. R. McClean, C. Hempel, P. J. Love, and A. Aspuru-Guzik, Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz, Quantum Sci. Technol. 4, 014008 (2019).
  67. O. Crawford, B. v. Straaten, D. Wang, T. Parks, E. Campbell, and S. Brierley, Efficient quantum measurement of Pauli operators in the presence of finite sampling error, Quantum 5, 385 (2021).
  68. B. Nachman, M. Urbanek, W. A. de Jong, and C. W. Bauer, Unfolding quantum computer readout noise, npj Quantum Inf. 6, 84 (2020).
  69. Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Phys. Rev. X 7, 021050 (2017).
  70. K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017).
  71. A. Kandala, K. Temme, A. D. Córcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Error mitigation extends the computational reach of a noisy quantum processor, Nature (London) 567, 491 (2019).
  72. E. F. Dumitrescu, A. J. McCaskey, G. Hagen, G. R. Jansen, T. D. Morris, T. Papenbrock, R. C. Pooser, D. J. Dean, and P. Lougovski, Cloud quantum computing of an atomic nucleus, Phys. Rev. Lett. 120, 210501 (2018).
  73. T. Giurgica-Tiron, Y. Hindy, R. LaRose, A. Mari, and W. J. Zeng, Digital zero noise extrapolation for quantum error mitigation, in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Denver, CO, 2020).
  74. A. He, B. Nachman, W. A. de Jong, and C. W. Bauer, Zero-noise extrapolation for quantum-gate error mitigation with identity insertions, Phys. Rev. A 102, 012426 (2020).
  75. C. M. Bishop, Pattern Recognition and Machine Learning, 1st ed., Information Science and Statistics (Springer, New York, NY, 2006).
  76. A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis, 3rd ed. (CRC Press, Boca Raton, FL, 2025), Chap. IV.14.7, electronic edition with errors fixed as of 20 February 2025, For non-commercial purposes only.

Outline

Information

Phys. Rev. Research 8, 043080– Published 21 October, 2026
  • Received 17 December 2025
  • Accepted 8 September 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