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Efficient Eigenstate Preparation in an Integrable Model with Hilbert Space Fragmentation

Roberto Ruiz1,*,†, Alejandro Sopena2,1,†,‡, Balázs Pozsgay3,§, and Esperanza López1,‖

  • *Contact author: roberto.ruiz@ift.csic.es
  • †The first two authors contributed equally.
  • ‡Contact author: alexsopena15@gmail.com
  • §Contact author: pozsgay.balazs@ttk.elte.hu
  • ‖Contact author: esperanza.lopez@csic.es
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PRX Quantum 6, 030316 – Published 30 July, 2025

DOI: https://doi.org/10.1103/g9f9-p8ks

Abstract

We demonstrate that eigenstates of selected interacting spin chains can be efficiently prepared using quantum circuits with polynomial depth in system size and particle number. While this is well established for free-fermionic spin chains, we extend this result to the folded XXZ model, an integrable rigid-rod deformation of the XX model with simple interactions that exhibits Hilbert space fragmentation. We construct explicit quantum circuits that efficiently prepare arbitrary eigenstates of this model on an open chain. Error-mitigated noisy simulations with up to 13 qubits and various qubit connectivities achieve a relative error below 5%. As a byproduct, we extend a recent reformulation of the Bethe ansatz as a quantum circuit to open boundary conditions.

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Popular Summary

Understanding the behavior of quantum systems with many interacting particles is a cornerstone of modern physics and a key application of quantum computing. However, efficiently preparing the quantum states that describe these systems, especially their “eigenstates,” the fundamental building blocks of quantum mechanics, has remained a challenging task. Our work tackles this problem for a specific class of quantum systems: the folded XXZ spin chain, a model that captures interactions between quantum spins while exhibiting unique behaviors like Hilbert space fragmentation.

We developed a quantum algorithm capable of preparing all eigenstates of the folded XXZ spin chain with open boundaries. Remarkably, this includes complex configurations featuring magnons (spin excitations) and domain walls (boundaries separating spin regions). To our knowledge, this is the first efficient quantum algorithm for all eigenstates of an interacting quantum spin chain. Along the way, we extended a recently developed framework for simulating spin chains based on the Bethe ansatz to systems with open boundaries.

Our algorithm is not just of theoretical interest, it is also practical. Using numerical simulations on noisy quantum devices, we demonstrate that the algorithm can prepare simple eigenstates with high fidelity on current hardware platforms with realistic error rates. This makes it an excellent candidate for benchmarking quantum computers, providing a rigorous test of their ability to handle many-body physics. Our results demonstrate that even with today’s noisy quantum hardware, significant progress can be made in simulating complex quantum systems, paving the way for new discoveries in quantum many-body physics.

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References (93)

  1. F. Verstraete, J. I. Cirac, and J. I. Latorre, Quantum circuits for strongly correlated quantum systems, J. Phys. A 79, 032316 (2009).
  2. Z. Jiang, K. J. Sung, K. Kechedzhi, V. N. Smelyanskiy, and S. Boixo, Quantum algorithms to simulate many-body physics of correlated fermions, Phys. Rev. Appl. 9, 044036 (2018).
  3. I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, G. K.-L. Chan, and R. Babbush, Quantum simulation of electronic structure with linear depth and connectivity, Phys. Rev. Lett. 120, 110501 (2018).
  4. A. Cervera-Lierta, Exact Ising model simulation on a quantum computer, Quantum 2, 114 (2018).
  5. M. Farreras and A. Cervera-Lierta, Simulation of the 1D XY model on a quantum computer, ArXiv:2410.21143.
  6. L. G. Valiant, Quantum circuits that can be simulated classically in polynomial time, SIAM J. Comput. 31, 1229 (2002).
  7. B. M. Terhal and D. P. DiVincenzo, Classical simulation of noninteracting-fermion quantum circuits, Phys. Rev. A 65, 032325 (2002).
  8. R. Jozsa and A. Miyake, Matchgates and classical simulation of quantum circuits, Proc. R. Soc. A 464, 3089 (2008).
  9. B. Sutherland, Beautiful Models (World Scientific Publishing Company, 2004).
  10. V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1993).
  11. L. D. Faddeev, How algebraic Bethe ansatz works for integrable model, ArXiv:hep-th/9605187.
  12. B. Pozsgay, Excited state correlations of the finite Heisenberg chain, J. Phys. A 50, 074006 (2017).
  13. J. Sato, B. Aufgebauer, H. Boos, F. Göhmann, A. Klümper, M. Takahashi, and C. Trippe, Computation of static Heisenberg-chain correlators: Control over length and temperature dependence, Phys. Rev. Lett. 106, 257201 (2011).
  14. K. K. Kozlowski, Asymptotic analysis and quantum integrable models, ArXiv:1508.06085.
  15. E. Bianchi, L. Hackl, M. Kieburg, M. Rigol, and L. Vidmar, Volume-law entanglement entropy of typical pure quantum states, PRX Quantum 3, 030201 (2022).
  16. J. S. Van Dyke, G. S. Barron, N. J. Mayhall, E. Barnes, and S. E. Economou, Preparing Bethe ansatz eigenstates on a quantum computer, PRX Quantum 2, 040329 (2021).
  17. A. Sopena, M. H. Gordon, D. García-Martín, G. Sierra, and E. López, Algebraic Bethe circuits, Quantum 6, 796 (2022).
  18. R. Ruiz, A. Sopena, M. H. Gordon, G. Sierra, and E. López, The Bethe ansatz as a quantum circuit, Quantum 8, 1356 (2024).
  19. D. Raveh and R. I. Nepomechie, Deterministic Bethe state preparation, Quantum 8, 1510 (2024).
  20. J. S. Van Dyke, E. Barnes, S. E. Economou, and R. I. Nepomechie, Preparing exact eigenstates of the open XXZ chain on a quantum computer, J. Phys. A 55, 055301 (2022).
  21. W. Li, M. Okyay, and R. I. Nepomechie, Bethe states on a quantum computer: Success probability and correlation functions, J. Phys. A 55, 355305 (2022).
  22. C. Trippe, F. Göhmann, and A. Klümper, Thermodynamics and short-range correlations of the XXZ chain close to its triple point, J. Stat. Mech. 2010, P01021 (2010).
  23. L. Zadnik and M. Fagotti, The folded spin-1/2 XXZ model: I. Diagonalisation, jamming, and ground state properties, SciPost Phys. Core 4, 010 (2021).
  24. L. Zadnik, K. Bidzhiev, and M. Fagotti, The folded spin-1/2 XXZ model: II. Thermodynamics and hydrodynamics with a minimal set of charges, SciPost Phys. 10, 099 (2021).
  25. B. Pozsgay, T. Gombor, A. Hutsalyuk, Y. Jiang, L. Pristyák, and E. Vernier, Integrable spin chain with Hilbert space fragmentation and solvable real-time dynamics, Phys. Rev. E 104, 044106 (2021).
  26. G. I. Menon, M. Barma, and D. Dhar, Conservation laws and integrability of a one-dimensional model of diffusing dimers, J. Stat. Phys. 86, 1237 (1997).
  27. Z.-C. Yang, F. Liu, A. V. Gorshkov, and T. Iadecola, Hilbert-space fragmentation from strict confinement, Phys. Rev. Lett. 124, 207602 (2020).
  28. Z. Papić, in Entanglement in Spin Chains: From Theory to Quantum Technology Applications, edited by A. Bayat, S. Bose, H. Johannesson (Springer International Publishing, Cham, 2022), p. 341.
  29. 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).
  30. K. Bidzhiev, M. Fagotti, and L. Zadnik, Macroscopic effects of localised measurements in jammed states of quantum spin chains, Phys. Rev. Lett. 128, 130603 (2022).
  31. L. Zadnik, S. Bocini, K. Bidzhiev, and M. Fagotti, Measurement catastrophe and ballistic spread of charge density with vanishing current, J. Phys. A 55, 474001 (2022).
  32. B. Pozsgay, A Yang-Baxter integrable cellular automaton with a four site update rule, J. Phys. A 54, 384001 (2021).
  33. S. Gopalakrishnan, A. Morningstar, R. Vasseur, and V. Khemani, Distinct universality classes of diffusive transport from full counting statistics, Phys. Rev. B 109, 024417 (2024).
  34. T. Gombor and B. Pozsgay, Integrable spin chains and cellular automata with medium-range interaction, Phys. Rev. E 104, 054123 (2021).
  35. F. C. Alcaraz and R. Z. Bariev, An exactly solvable constrained XXZ chain, ArXiv:cond-mat/9904042.
  36. F. C. Alcaraz and R. Z. Bariev, Integrable models of strongly correlated particles with correlated hopping, Phys. Rev. B 59, 3373 (1999).
  37. I. N. Karnaukhov and A. A. Ovchinnikov, One-dimensional strongly interacting Luttinger liquid of lattice spinless fermions, Europhys. Lett. 57, 540 (2002).
  38. F. C. Alcaraz and M. J. Lazo, Exactly solvable interacting vertex models, J. Stat. Mech. 2007, P08008 (2007).
  39. N. Abarenkova and A. Pronko, Temperature correlation function in the absolutely anisotropic XXZ Heisenberg magnet, Theor. Math. Phys. 131, 690 (2002).
  40. N. M. Bogoliubov and C. Malyshev, The correlation functions of the XXZ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers, St. Petersburg Math. J. 22, 359 (2011).
  41. N. M. Bogoliubov and C. L. Malyshev, Ising limit of a Heisenberg XXZ magnet and some temperature correlation functions, Theor. Math. Phys. 169, 1517 (2011).
  42. N. M. Bogolyubov and C. L. Malyshev, Integrable models and combinatorics, Russ. Math. Surv. 70, 789 (2015).
  43. M. Ganahl, M. Haque, and H. G. Evertz, Quantum bowling: Particle-hole transmutation in one-dimensional strongly interacting lattice models, ArXiv:1302.2667.
  44. B. Pozsgay, T. Gombor, and A. Hutsalyuk, Integrable hard rod deformation of the Heisenberg spin chains, Phys. Rev. E 104, 064124 (2021).
  45. M. Borsi, L. Pristyák, and B. Pozsgay, Matrix product symmetries and breakdown of thermalization from hard rod deformations, Phys. Rev. Lett. 131, 037101 (2023).
  46. M. Fagotti, Quantum jamming brings quantum mechanics to macroscopic scales, Phys. Rev. X 14, 021015 (2024).
  47. S. Scherg, T. Kohlert, P. Sala, F. Pollmann, B. Hebbe Madhusudhana, I. Bloch, and M. Aidelsburger, Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains, Nat. Commun. 12, 4490 (2021).
  48. T. Kohlert, S. Scherg, P. Sala, F. Pollmann, B. H. Madhusudhana, I. Bloch, and M. Aidelsburger, Experimental realization of fragmented models in tilted Fermi-Hubbard chains, ArXiv:2106.15586.
  49. Y.-Y. Wang, et al., Exploring Hilbert-space fragmentation on a superconducting processor, PRX Quantum 6, 010325 (2025).
  50. K. Kim, F. Yang, K. Mølmer, and J. Ahn, Realization of an extremely anisotropic Heisenberg magnet in Rydberg atom arrays, Phys. Rev. X 14, 011025 (2024).
  51. F. Yang, H. Yarloo, H.-C. Zhang, K. Mølmer, and A. E. B. Nielsen, Probing Hilbert space fragmentation with strongly interacting Rydberg atoms, ArXiv:2403.13790.
  52. C. M. Langlett and S. Xu, Hilbert space fragmentation and exact scars of generalized Fredkin spin chains, Phys. Rev. B 103, L220304 (2021).
  53. S. Moudgalya and O. I. Motrunich, Hilbert space fragmentation and commutant algebras, Phys. Rev. X 12, 011050 (2022).
  54. S. K. Singh, A. Dhar, H. Spohn, and A. Kundu, Thermalization and hydrodynamics in an interacting integrable system: The case of hard rods, ArXiv:2310.18684.
  55. A. Sopena and R. Ruiz, XXZ_folded 2.0, Zenodo (2024).
  56. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  57. R. Raussendorf and J. Harrington, Fault-tolerant quantum computation with high threshold in two dimensions, Phys. Rev. Lett. 98, 190504 (2007).
  58. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  59. S. Efthymiou, S. Ramos-Calderer, C. Bravo-Prieto, A. Pérez-Salinas, D. García-Martín, A. Garcia-Saez, J. I. Latorre, and S. Carrazza, Qibo: A framework for quantum simulation with hardware acceleration, Quantum Sci. Technol. 7, 015018 (2022).
  60. C. R. Clark, H. N. Tinkey, B. C. Sawyer, A. M. Meier, K. A. Burkhardt, C. M. Seck, C. M. Shappert, N. D. Guise, C. E. Volin, S. D. Fallek, H. T. Hayden, W. G. Rellergert, and K. R. Brown, High-fidelity Bell-state preparation with 40Ca+ optical qubits, Phys. Rev. Lett. 127, 130505 (2021).
  61. L. Ding, M. Hays, Y. Sung, B. Kannan, J. An, A. Di Paolo, A. H. Karamlou, T. M. Hazard, K. Azar, D. K. Kim, B. M. Niedzielski, A. Melville, M. E. Schwartz, J. L. Yoder, T. P. Orlando, S. Gustavsson, J. A. Grover, K. Serniak, and W. D. Oliver, High-fidelity, frequency-flexible two-qubit fluxonium gates with a transmon coupler, Phys. Rev. X 13, 031035 (2023).
  62. S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletić, and M. D. Lukin, High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622, 268 (2023).
  63. H. Zhang, C. Ding, D.K. Weiss, Z. Huang, Y. Ma, C. Guinn, S. Sussman, S. P. Chitta, D. Chen, A. A. Houck, J. Koch, and D. I. Schuster, Tunable inductive coupler for high-fidelity gates between fluxonium qubits, PRX Quantum 5, 020326 (2024).
  64. J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Phys. Rev. A 94, 052325 (2016).
  65. A. Hashim, R. K. Naik, A. Morvan, J.-L. Ville, B. Mitchell, J. M. Kreikebaum, M. Davis, E. Smith, C. Iancu, K. P. O’Brien, I. Hincks, J. J. Wallman, J. Emerson, and I. Siddiqi, Randomized compiling for scalable quantum computing on a noisy superconducting quantum processor, Phys. Rev. X 11, 041039 (2021).
  66. E. Magesan, J. M. Gambetta, and J. Emerson, Scalable and robust randomized benchmarking of quantum processes, Phys. Rev. Lett. 106, 180504 (2011).
  67. H.-L. Huang, D. Wu, D. Fan, and X. Zhu, Superconducting quantum computing: A review, Sci. China Inf. Sci. 63, 180501 (2020).
  68. D. Kielpinski, C. Monroe, and D. J. Wineland, Architecture for a large-scale ion-trap quantum computer, Nature 417, 709 (2002).
  69. J. P. Home, D. Hanneke, J. D. Jost, J. M. Amini, D. Leibfried, and D. J. Wineland, Complete methods set for scalable ion trap quantum information processing, Science 325, 1227 (2009).
  70. V. Kaushal, B. Lekitsch, A. Stahl, J. Hilder, D. Pijn, C. Schmiegelow, A. Bermudez, M. Müller, F. Schmidt-Kaler, and U. Poschinger, Shuttling-based trapped-ion quantum information processing, AVS Quantum Sci. 2, 014101 (2020).
  71. F. Arute, et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  72. G. Li, Y. Ding, and Y. Xie, in Proceedings of the Twenty-Fourth International Conference on Architectural Support for Programming Languages and Operating Systems (ACM, RI, USA, 2019), p. 1001.
  73. 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).
  74. H. Pashayan, O. Reardon-Smith, K. Korzekwa, and S. D. Bartlett, Fast estimation of outcome probabilities for quantum circuits, PRX Quantum 3, 020361 (2022).
  75. P. Czarnik, A. Arrasmith, P. J. Coles, and L. Cincio, Error mitigation with Clifford quantum-circuit data, Quantum 5, 592 (2021).
  76. A. Bärtschi and S. Eidenbenz, in Fundamentals of Computation Theory (Springer International Publishing, Berlin, Heidelberg, 2019), p. 126.
  77. A. N. Ivanov and O. I. Motrunich, Volume-entangled exact scar states in the PXP and related models in any dimension, Phys. Rev. Lett. 134, 050403 (2025).
  78. A. Elben, B. Vermersch, R. van Bijnen, C. Kokail, T. Brydges, C. Maier, M. K. Joshi, R. Blatt, C. F. Roos, and P. Zoller, Cross-platform verification of intermediate scale quantum devices, Phys. Rev. Lett. 124, 010504 (2020).
  79. J. Carrasco, A. Elben, C. Kokail, B. Kraus, and P. Zoller, Theoretical and experimental perspectives of quantum verification, PRX Quantum 2, 010102 (2021).
  80. A. Anshu, Z. Landau, and Y. Liu, in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC ’22 (ACM, New York, NY, 2022), p. 44.
  81. J. Denzler, S. Varona, T. Guaita, and J. Carrasco, Highly-entangled, highly-doped states that are efficiently cross-device verifiable, ArXiv:2501.11688.
  82. https://github.com/AlejandroSopena/XXZ_folded.
  83. F. C. Alcaraz and M. J. Lazo, The Bethe ansatz as a matrix product ansatz, J. Phys. A 37, L1 (2004).
  84. F. C. Alcaraz and M. J. Lazo, Exact solutions of exactly integrable quantum chains by a matrix product ansatz, J. Phys. A 37, 4149 (2004).
  85. F. C. Alcaraz and M. J. Lazo, Generalization of the matrix product ansatz for integrable chains, J. Phys. A 39, 11335 (2006).
  86. H. Katsura and I. Maruyama, Derivation of the matrix product ansatz for the Heisenberg chain from the algebraic Bethe ansatz, J. Phys. A 43, 175003 (2010).
  87. V. Murg, V. E. Korepin, and F. Verstraete, Algebraic Bethe ansatz and tensor networks, Phys. Rev. B 86, 045125 (2012).
  88. J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  89. Formula Eq. (A5) matches Eqs. (55) and (57) of Ref. [18] if the number of qubits is 2M, xa replace ya, and sab replace Bab.
  90. R. Ruiz, A. Sopena, E. López, G. Sierra, and B. Pozsgay, Bethe ansatz, quantum circuits, and the F-basis, SciPost Phys. 18, 187 (2025).
  91. If scattering amplitudes were not removed from the outset, scattering phases would appear in the multiqubit unitaries Pk. These phases are crucial for establishing a connection between the unitaries at Δ>0 and the decomposition into layers of matchgates at Δ=0. Nonetheless, the phases become superfluous if the analysis focuses on the free-fermion point from the outset.
  92. Note that Fk,k can be redefined as Fk,k(1⊗Uk)↦Fk,k for 1≤k<2M to eliminate the need for gates. However, the resulting two-qubit gate is no longer a matchgate. This adjustment can be assumed in the gate counting of Table 1, effectively replacing the number of matchgates with the number of two-qubit gates.
  93. A. Sopena, M. H. Gordon, G. Sierra, and E. López, Simulating quench dynamics on a digital quantum computer with data-driven error mitigation, Quantum Sci. Technol. 6, 045003 (2021).

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PRX Quantum 6, 030316– Published 30 July, 2025
  • Received 6 December 2024
  • Revised 2 April 2025
  • Accepted 10 June 2025
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