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Variational preparation of quantum many-body scars with shallow circuits

Eoin Carolan1,2, Nathan Keenan3, Gabriele Cenedese3, and Giuliano Benenti1,2

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Phys. Rev. A 114, 012613 – Published 20 July, 2026

DOI: https://doi.org/10.1103/5l8p-42fv

Abstract

We present and benchmark a type of variational quantum eigensolver (VQE) which we denote σ-VQE. It is designed to target midspectrum eigenstates and prepare quantum many-body scar states. The approach leverages the fact that noisy intermediate-scale quantum devices are limited in their ability to generate generic highly entangled states. This modified VQE pairs a low-depth circuit with an energy-selective objective that explicitly penalizes energy variance around a chosen target energy. The cost function exploits the limited expressibility of the shallow circuit as atypical low-entanglement eigenstates such as scar states are preferentially selected. We validate this mechanism across two complementary families of models that contain many-body scar states: the Shiraishi-Mori embedding approach and a matrix product state parent Hamiltonian construction. We define an unbiased estimation scheme for the nonlinear cost function that is compatible with qubit-wise commuting grouping and bitstring reuse. A proof-of-principle demonstration using a small-system instance was performed on IBM Fez in December 2025 (Heron r2 QPU). These results motivate its use as a practical algorithm for detecting quantum many-body scars and variationally generating states with appreciable scar state overlap.

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References (66)

  1. W. Ritz, Über eine neue methode zur lösung gewisser variationsprobleme der mathematischen physik, J. Reine Angew. Math. 1909, 1 (1909).
  2. J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, et al., The variational quantum eigensolver: A review of methods and best practices, Phys. Rep. 986, 1 (2022).
  3. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  4. J. Eisert and J. Preskill, Mind the gaps: The fraught road to quantum advantage, arXiv:2510.19928.
  5. M. da Silva Fonseca, C. Moraes Porto, N. A. Cabrera Carpio, G. de Souza Tavares de Morais, N. H. Morgon, R. A. Nome, and C. J. Villas-Boas, An introduction to variational quantum eigensolver applied to chemistry, Braz. J. Phys. 56, 8 (2026).
  6. J. Hu, J. Li, Y. Lin, H. Long, X.-S. Xu, Z. Su, W. Zhang, Y. Zhu, and M.-H. Yung, Benchmarking variational quantum eigensolvers for quantum chemistry, arXiv:2211.12775.
  7. N. Yoshioka, T. Sato, Y. O. Nakagawa, Y.-y. Ohnishi, and W. Mizukami, Variational quantum simulation for periodic materials, Phys. Rev. Res. 4, 013052 (2022).
  8. S. Stanisic, J. L. Bosse, F. M. Gambetta, R. A. Santos, W. Mruczkiewicz, T. E. O'Brien, E. Ostby, and A. Montanaro, Observing ground-state properties of the Fermi-Hubbard model using a scalable algorithm on a quantum computer, Nat. Commun. 13, 5743 (2022).
  9. O. Huijgen, L. Coopmans, P. Najafi, M. Benedetti, and H. J. Kappen, Training quantum Boltzmann machines with the β-variational quantum eigensolver, Mach. Learn.: Sci. Technol. 5, 025017 (2024).
  10. 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).
  11. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, et al., Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  12. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  13. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  14. L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  15. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, et al., Probing many-body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017).
  16. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018).
  17. N. Shiraishi and T. Mori, Systematic construction of counterexamples to the eigenstate thermalization hypothesis, Phys. Rev. Lett. 119, 030601 (2017).
  18. A. Chandran, T. Iadecola, V. Khemani, and R. Moessner, Quantum many-body scars: A quasiparticle perspective, Annu. Rev. Condens. Matter Phys. 14, 443 (2023).
  19. S. Moudgalya, B. A. Bernevig, and N. Regnault, Quantum many-body scars and Hilbert space fragmentation: A review of exact results, Rep. Prog. Phys. 85, 086501 (2022).
  20. G. Cenedese, M. Bondani, A. Andreanov, M. Carrega, G. Benenti, and D. Rosa, Shallow quantum circuits are robust hunters for quantum many-body scars, Eur. Phys. J. Plus 140, 517 (2025).
  21. L. Cadi 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).
  22. F. Zhang, N. Gomes, Y. Yao, P. P. Orth, and T. Iadecola, Adaptive variational quantum eigensolvers for highly excited states, Phys. Rev. B 104, 075159 (2021).
  23. O. Higgott, D. Wang, and S. Brierley, Variational quantum computation of excited states, Quantum 3, 156 (2019).
  24. K. M. Nakanishi, K. Mitarai, and K. Fujii, Subspace-search variational quantum eigensolver for excited states, Phys. Rev. Res. 1, 033062 (2019).
  25. J. R. McClean, M. E. Kimchi-Schwartz, J. Carter, and W. A. de Jong, Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states, Phys. Rev. A 95, 042308 (2017).
  26. S. Gocho, H. Nakamura, S. Kanno, Q. Gao, T. Kobayashi, T. Inagaki, and M. Hatanaka, Excited state calculations using variational quantum eigensolver with spin-restricted ansätze and automatically-adjusted constraints, npj Comput. Mater. 9, 13 (2023).
  27. S. Sim, P. D. Johnson, and A. Aspuru-Guzik, Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms, Adv. Quantum Technol. 2, 1900070 (2019).
  28. V. Verteletskyi, T.-C. Yen, and A. F. Izmaylov, Measurement optimization in the variational quantum eigensolver using a minimum clique cover, J. Chem. Phys. 152, 124114 (2020).
  29. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  30. H.-Y. Huang, R. Kueng, and J. Preskill, Efficient estimation of Pauli observables by derandomization, Phys. Rev. Lett. 127, 030503 (2021).
  31. K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).
  32. M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).
  33. P. G. Larsen and A. E. B. Nielsen, Phase transitions in quantum many-body scars, Phys. Rev. Res. 6, L042007 (2024).
  34. J. Kim, J. Kim, and D. Rosa, Universal effectiveness of high-depth circuits in variational eigenproblems, Phys. Rev. Res. 3, 023203 (2021).
  35. A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature (London) 549, 242 (2017).
  36. S. Liu, S.-X. Zhang, C.-Y. Hsieh, S. Zhang, and H. Yao, Probing many-body localization by excited-state variational quantum eigensolver, Phys. Rev. B 107, 024204 (2023).
  37. Q. Lu, Comparative studies of optimizer performance in a variational quantum eigensolver, J. Chem. Phys. 163, 234106 (2025).
  38. W. Lavrijsen, A. Tudor, J. Müller, C. Iancu, and W. de Jong, Classical optimizers for noisy intermediate-scale quantum devices, in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, New York, 2020), pp. 267–277.
  39. J. C. Spall, Implementation of the simultaneous perturbation algorithm for stochastic optimization, IEEE Trans. Aerosp. Electron. Syst. 34, 817 (1998).
  40. A. Pellow-Jarman, I. Sinayskiy, A. Pillay, and F. Petruccione, A comparison of various classical optimizers for a variational quantum linear solver, Quantum Inf. Process 20, 202 (2021).
  41. T.-C. Yen, V. Verteletskyi, and A. F. Izmaylov, Measuring all compatible operators in one series of single-qubit measurements using unitary transformations, J. Chem. Theory Comput. 16, 2400 (2020).
  42. B. DalFavero, R. Sarkar, J. Rowland, D. Camps, N. P. D. Sawaya, and R. LaRose, Measurement reduction for expectation values via fine-grained commutativity, Phys. Rev. A 112, 052407 (2025).
  43. S. Ross, A First Course in Probability, 10th ed. (Pearson, Hoboken, 2023).
  44. P. C. Burke and S. Dooley, Taking the temperature of quantum many-body scars, Phys. Rev. Res. 7, 043054 (2025).
  45. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations, Phys. Rev. B 98, 155134 (2018).
  46. S. Dooley, S. Pappalardi, and J. Goold, Entanglement enhanced metrology with quantum many-body scars, Phys. Rev. B 107, 035123 (2023).
  47. A. Hudomal, I. Vasić, N. Regnault, and Z. Papić, Quantum scars of bosons with correlated hopping, Commun. Phys. 3, 99 (2020).
  48. S. Hillmich, C. Hadfield, R. Raymond, A. Mezzacapo, and R. Wille, Decision diagrams for quantum measurements with shallow circuits, in 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, New York, 2021), pp. 24–34.
  49. A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The randomized measurement toolbox, Nat. Rev. Phys. 5, 9 (2023).
  50. A. F. Izmaylov, T.-C. Yen, R. A. Lang, and V. Verteletskyi, Unitary partitioning approach to the measurement problem in the variational quantum eigensolver method, J. Chem. Theory Comput. 16, 190 (2020).
  51. J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum natural gradient, Quantum 4, 269 (2020).
  52. 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).
  53. E. Grant, L. Wossnig, M. Ostaszewski, and M. Benedetti, An initialization strategy for addressing barren plateaus in parametrized quantum circuits, Quantum 3, 214 (2019).
  54. M. Ostaszewski, E. Grant, and M. Benedetti, Structure optimization for parameterized quantum circuits, Quantum 5, 391 (2021).
  55. J. M. Kübler, A. Arrasmith, L. Cincio, and P. J. Coles, An adaptive optimizer for measurement-frugal variational algorithms, Quantum 4, 263 (2020).
  56. A. Gu, A. Lowe, P. A. Dub, P. J. Coles, and A. Arrasmith, Adaptive shot allocation for fast convergence in variational quantum algorithms, arXiv:2108.10434.
  57. A. Arrasmith, L. Cincio, R. D. Somma, and P. J. Coles, Operator sampling for shot-frugal optimization in variational algorithms, arXiv:2004.06252.
  58. L. Zhu, S. Liang, C. Yang, and X. Li, Optimizing shot assignment in variational quantum eigensolver measurement, J. Chem. Theory Comput. 20, 2390 (2024).
  59. S. Liang, L. Zhu, X. Liu, C. Yang, and X. Li, Artificial-intelligence-driven shot reduction in quantum measurement, Chem. Phys. Rev. 5, 041403 (2024).
  60. 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).
  61. 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).
  62. Q. Hummel, K. Richter, and P. Schlagheck, Genuine many-body quantum scars along unstable modes in Bose-Hubbard systems, Phys. Rev. Lett. 130, 250402 (2023).
  63. D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Semeghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, et al., Controlling quantum many-body dynamics in driven Rydberg atom arrays, Science 371, 1355 (2021).
  64. S. Dooley, Robust quantum sensing in strongly interacting systems with many-body scars, PRX Quantum 2, 020330 (2021).
  65. 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).
  66. S. Wang, P. Czarnik, A. Arrasmith, M. Cerezo, L. Cincio, and P. J. Coles, Can error mitigation improve trainability of noisy variational quantum algorithms? Quantum 8, 1287 (2024).

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Phys. Rev. A 114, 012613– Published 20 July, 2026
  • Received 8 March 2026
  • Accepted 17 June 2026
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