- 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

Simulating dynamics of the two-dimensional transverse-field Ising model: A comparative study of large-scale classical numerics

Joseph Vovrosh1,*, Sergi Julià-Farré1, Wladislaw Krinitsin2,3, Michael Kaicher1, Fergus Hayes1, Emmanuel Gottlob1, Augustine Kshetrimayum1, Kemal Bidzhiev1, Simon B. Jäger4 et al.

Markus Schmitt2,3, Joseph Tindall5, Constantin Dalyac1, Tiago Mendes-Santos1,†, and Alexandre Dauphin1

  • *Contact author: joseph.vovrosh@pasqal.com
  • †Contact author: tiago.mendes-santos@pasqal.com
PDF

Phys. Rev. Research 8, 023311 – Published 17 June, 2026

DOI: https://doi.org/10.1103/x4gs-g63p

Abstract

The quantum dynamics of many-qubit systems is an outstanding problem that has recently driven significant advances in both numerical methods and programmable quantum processing units (QPUs). In this work, we employ a comprehensive toolbox of state-of-the-art numerical approaches to classically simulate the dynamics of the two-dimensional transverse-field Ising model. Our methods include three different tensor-network techniques—matrix product states, tree tensor networks, and two-dimensional tensor networks under the belief propagation approximation—as well as time-dependent variational Monte Carlo with neural quantum states. We focus on two paradigmatic dynamical protocols: (1) quantum annealing through a critical point and (2) postquench dynamics. Our extensive results show the quantitative predictions of various state-of-the-art numerical methods providing a benchmark for future numerical investigations and experimental studies with the aim to push the limitations on classical and QPUs. In particular, our work connects classical simulability to different regimes associated with quantum dynamics in Rydberg arrays—namely, quasiadiabatic dynamics, the Kibble-Zurek mechanism, and quantum quenches.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (120)

  1. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  2. S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
  3. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).
  4. R. Haghshenas, E. Chertkov, M. Mills, W. Kadow, S.-H. Lin, Y.-H. Chen, C. Cade, I. Niesen, T. Begušić, M. S. Rudolph, et al., Digital quantum magnetism at the frontier of classical simulations, Nature (London) 653, 56 (2026).
  5. A. D. King, A. Nocera, M. M. Rams, J. Dziarmaga, R. Wiersema, W. Bernoudy, J. Raymond, N. Kaushal, N. Heinsdorf, R. Harris, K. Boothby, F. Altomare, et al., Beyond-classical computation in quantum simulation, Science 388, 199 (2025).
  6. T. I. Andersen, N. Astrakhantsev, A. H. Karamlou, J. Berndtsson, J. Motruk, A. Szasz, J. A. Gross, A. Schuckert, T. Westerhout, Y. Zhang, E. Forati, D. Rossi, et al., Thermalization and criticality on an analogue–digital quantum simulator, Nature (London) 638, 79 (2025).
  7. J. Choi, A. L. Shaw, I. S. Madjarov, X. Xie, R. Finkelstein, J. P. Covey, J. S. Cotler, D. K. Mark, H.-Y. Huang, A. Kale, H. Pichler, F. G. S. L. Brandão, S. Choi, and M. Endres, Preparing random states and benchmarking with many-body quantum chaos, Nature (London) 613, 468 (2023).
  8. A. L. Shaw, Z. Chen, J. Choi, D. K. Mark, P. Scholl, R. Finkelstein, A. Elben, S. Choi, and M. Endres, Benchmarking highly entangled states on a 60-atom analogue quantum simulator, Nature (London) 628, 71 (2024).
  9. J. Tindall, M. Fishman, E. M. Stoudenmire, and D. Sels, Efficient tensor network simulation of IBM’s Eagle kicked Ising experiment, PRX Quantum 5, 010308 (2024).
  10. L. Mauron and G. Carleo, Challenging the quantum advantage frontier with large-scale classical simulations of annealing dynamics, arXiv:2503.08247.
  11. J. Tindall, A. Mello, M. Fishman, M. Stoudenmire, and D. Sels, Dynamics of disordered quantum systems with two- and three-dimensional tensor networks, Science 392, 868 (2026).
  12. E. Stoudenmire and S. R. White, Studying two-dimensional systems with the density matrix renormalization group, Annu. Rev. Condens. Matter Phys. 3, 111 (2012).
  13. L. Tagliacozzo, G. Evenbly, and G. Vidal, Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law, Phys. Rev. B 80, 235127 (2009).
  14. V. Murg, F. Verstraete, O. Legeza, and R. M. Noack, Simulating strongly correlated quantum systems with tree tensor networks, Phys. Rev. B 82, 205105 (2010).
  15. J. Schuhmacher, M. Ballarin, A. Baiardi, G. Magnifico, F. Tacchino, S. Montangero, and I. Tavernelli, Hybrid tree tensor networks for quantum simulation, PRX Quantum 6, 010320 (2025).
  16. F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv:cond-mat/0407066.
  17. J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac, Classical simulation of infinite-size quantum lattice systems in two spatial dimensions, Phys. Rev. Lett. 101, 250602 (2008).
  18. J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Phys. Rev. Lett. 107, 070601 (2011).
  19. G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
  20. F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  21. S. R. White and A. E. Feiguin, Real-time evolution using the density matrix renormalization group, Phys. Rev. Lett. 93, 076401 (2004).
  22. H. C. Jiang, Z. Y. Weng, and T. Xiang, Accurate determination of tensor network state of quantum lattice models in two dimensions, Phys. Rev. Lett. 101, 090603 (2008).
  23. R. Alkabetz and I. Arad, Tensor networks contraction and the belief propagation algorithm, Phys. Rev. Res. 3, 023073 (2021).
  24. J. Tindall and M. Fishman, Gauging tensor networks with belief propagation, SciPost Phys. 15, 222 (2023).
  25. M. C. Bañuls, M. B. Hastings, F. Verstraete, and J. I. Cirac, Matrix product states for dynamical simulation of infinite chains, Phys. Rev. Lett. 102, 240603 (2009).
  26. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  27. A. Lerose, M. Sonner, and D. A. Abanin, Influence matrix approach to many-body Floquet dynamics, Phys. Rev. X 11, 021040 (2021).
  28. G. Park, J. Gray, and G. K.-L. Chan, Simulating quantum dynamics in two-dimensional lattices with tensor network influence functional belief propagation, Phys. Rev. B 112, 174310 (2025).
  29. M. Lubasch, J. I. Cirac, and M.-C. Bañuls, Algorithms for finite projected entangled pair states, Phys. Rev. B 90, 064425 (2014).
  30. C. Guo, D. Poletti, and I. Arad, Block belief propagation algorithm for two-dimensional tensor networks, Phys. Rev. B 108, 125111 (2023).
  31. T. Nishino and K. Okunishi, Corner transfer matrix renormalization group method, J. Phys. Soc. Jpn. 65, 891 (1996).
  32. R. Orus and G. Vidal, Simulation of two dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction, Phys. Rev. B 80, 094403 (2009).
  33. P. Rall, D. Liang, J. Cook, and W. Kretschmer, Simulation of qubit quantum circuits via Pauli propagation, Phys. Rev. A 99, 062337 (2019).
  34. M. S. Rudolph, E. Fontana, Z. Holmes, and L. Cincio, Classical surrogate simulation of quantum systems with LOWESA, arXiv:2308.09109.
  35. L. Broers and L. Mathey, Exclusive-or encoded algebraic structure for efficient quantum dynamics, New J. Phys. 28, 044508 (2026).
  36. L. Broers, R.-Y. Sun, and S. Yunoki, Scalable simulation of quantum many-body dynamics with or-represented quantum algebra, arXiv:2506.13241.
  37. T. Begušić and G. K.-L. Chan, Real-time operator evolution in two and three dimensions via sparse Pauli dynamics, PRX Quantum 6, 020302 (2025).
  38. M. S. Rudolph, T. Jones, Y. Teng, A. Angrisani, and Z. Holmes, Pauli propagation: A computational framework for simulating quantum systems, arXiv:2505.21606.
  39. A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020).
  40. M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010).
  41. L. Henriet, L. Beguin, A. Signoles, T. Lahaye, A. Browaeys, G.-O. Reymond, and C. Jurczak, Quantum computing with neutral atoms, Quantum 4, 327 (2020).
  42. P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, A. M. Läuchli, and A. Browaeys, Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms, Nature (London) 595, 233 (2021).
  43. S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, and W. W. Ho, Quantum phases of matter on a 256-atom programmable quantum simulator, Nature (London) 595, 227 (2021).
  44. 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).
  45. A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pichler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, P. Zoller, M. Endres, M. Greiner, V. Vuletic, and M. D. Lukin, Quantum Kibble-Zurek mechanism and critical dynamics on a programmable Rydberg simulator, Nature (London) 568, 207 (2019).
  46. T. Manovitz, S. H. Li, S. Ebadi, R. Samajdar, A. A. Geim, S. J. Evered, D. Bluvstein, H. Zhou, N. U. Koyluoglu, J. Feldmeier, P. E. Dolgirev, N. Maskara, M. Kalinowski, S. Sachdev, D. A. Huse, M. Greiner, V. Vuletić, and M. D. Lukin, Quantum coarsening and collective dynamics on a programmable simulator, Nature (London) 638, 86 (2025).
  47. A. L. Shaw, D. K. Mark, J. Choi, R. Finkelstein, P. Scholl, S. Choi, and M. Endres, Experimental signatures of Hilbert-space ergodicity: Universal bitstring distributions and applications in noise learning, Phys. Rev. X 15, 031001 (2025).
  48. S. Darbha, A. Khudorozhkov, P. L. S. Lopes, F. Liu, E. Rrapaj, J. Balewski, M. Hamdan, P. E. Dolgirev, A. Schuckert, K. Klymko, S.-T. Wang, M. D. Lukin, D. Camps, and M. Kornjača, Probing emergent prethermal dynamics and resonant melting on a programmable quantum simulator, arXiv:2510.11706.
  49. J. Vovrosh, J. d. Hond, S. Julià-Farré, J. Knolle, and A. Dauphin, Meson spectroscopy of exotic symmetries of Ising criticality in Rydberg atom arrays, arXiv:2506.21299.
  50. D. K. Mark, F. M. Surace, T. Schuster, A. L. Shaw, W. Gong, S. Choi, and M. Endres, Observation of ballistic plasma and memory in high-energy gauge theory dynamics, arXiv:2510.11679.
  51. D. González-Cuadra, M. Hamdan, T. V. Zache, B. Braverman, M. Kornjača, A. Lukin, S. H. Cantú, F. Liu, S.-T. Wang, A. Keesling, M. D. Lukin, P. Zoller, and A. Bylinskii, Observation of string breaking on a (2 + 1)D Rydberg quantum simulator, Nature (London) 642, 321 (2025).
  52. M. Schmitt, M. M. Rams, J. Dziarmaga, M. Heyl, and W. H. Zurek, Quantum phase transition dynamics in the two-dimensional transverse-field Ising model, Sci. Adv. 8, eabl6850 (2022).
  53. T. Mendes-Santos, M. Schmitt, A. Angelone, A. Rodriguez, P. Scholl, H. Williams, D. Barredo, T. Lahaye, A. Browaeys, M. Heyl, and M. Dalmonte, Wave-function network description and Kolmogorov complexity of quantum many-body systems, Phys. Rev. X 14, 021029 (2024).
  54. S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cambridge, 2011).
  55. C. Laumann, R. Moessner, A. Scardicchio, and S. Sondhi, Quantum annealing: The fastest route to quantum computation? Eur. Phys. J. Spec. Top. 224, 75 (2015).
  56. T. W. B. Kibble, Topology of cosmic domains and strings, J. Phys. A: Math. Gen. 9, 1387 (1976).
  57. W. H. Zurek, Cosmological experiments in superfluid helium? Nature (London) 317, 505 (1985).
  58. W. H. Zurek, U. Dorner, and P. Zoller, Dynamics of a quantum phase transition, Phys. Rev. Lett. 95, 105701 (2005).
  59. See Supplemental Material at http://link.aps.org/supplemental/10.1103/x4gs-g63p for details on numerical methods and simulations, including implementation, convergence and benchmarking, and presents additional results for quenches and ramps with supporting analyses of observables, scaling, and method comparisons, which also includes Refs. [116, 117, 118, 119, 120].
  60. I. Frérot, P. Naldesi, and T. Roscilde, Multi-speed prethermalization in spin models with power-law decaying interactions, Phys. Rev. Lett. 120, 050401 (2018).
  61. R. Khasseh, A. Russomanno, M. Schmitt, M. Heyl, and R. Fazio, Discrete truncated Wigner approach to dynamical phase transitions in Ising models after a quantum quench, Phys. Rev. B 102, 014303 (2020).
  62. F. Balducci, A. Gambassi, A. Lerose, A. Scardicchio, and C. Vanoni, Localization and melting of interfaces in the two-dimensional quantum Ising model, Phys. Rev. Lett. 129, 120601 (2022).
  63. J. Tindall and D. Sels, Confinement in the transverse field Ising model on the Heavy Hex lattice, Phys. Rev. Lett. 133, 180402 (2024).
  64. L. Pavešić, D. Jaschke, and S. Montangero, Constrained dynamics and confinement in the two-dimensional quantum Ising model, Phys. Rev. B 111, L140305 (2025).
  65. W. Krinitsin, N. Tausendpfund, M. Rizzi, M. Heyl, and M. Schmitt, Roughening dynamics of interfaces in the two-dimensional quantum Ising model, Phys. Rev. Lett. 134, 240402 (2025).
  66. W. Krinitsin, N. Tausendpfund, M. Heyl, M. Rizzi, and M. Schmitt, Time evolution of the quantum Ising model in two dimensions using tree tensor networks, Phys. Rev. B 112, 134310 (2025).
  67. A. Yoshinaga, H. Hakoshima, T. Imoto, Y. Matsuzaki, and R. Hamazaki, Emergence of Hilbert space fragmentation in Ising models with a weak transverse field, Phys. Rev. Lett. 129, 090602 (2022).
  68. O. Hart and R. Nandkishore, Hilbert space shattering and dynamical freezing in the quantum Ising model, Phys. Rev. B 106, 214426 (2022).
  69. R. Mondaini, K. R. Fratus, M. Srednicki, and M. Rigol, Eigenstate thermalization in the two-dimensional transverse field Ising model, Phys. Rev. E 93, 032104 (2016).
  70. R. Mondaini and M. Rigol, Eigenstate thermalization in the two-dimensional transverse field Ising model. II. Off-diagonal matrix elements of observables, Phys. Rev. E 96, 012157 (2017).
  71. B. Žunkovič, M. Heyl, M. Knap, and A. Silva, Dynamical quantum phase transitions in spin chains with long-range interactions: Merging different concepts of nonequilibrium criticality, Phys. Rev. Lett. 120, 130601 (2018).
  72. B. Blaß and H. Rieger, Test of quantum thermalization in the two-dimensional transverse-field Ising model, Sci. Rep. 6, 38185 (2016).
  73. Quantum Monte Carlo Methods in Equilibrium and Nonequilibrium Systems, Springer Series in Solid-State Sciences, edited by M. Suzuki, M. Cardona, P. Fulde, K. Von Klitzing, and H.-J. Queisser (Springer, Berlin, Heidelberg, 1987), Vol. 74.
  74. S. Suzuki, J.-I. Inoue, and B. K. Chakrabarti, Quantum Ising Phases and Transitions in Transverse Ising Models, Lecture Notes in Physics, Vol. 862 (Springer, Berlin, Heidelberg, 2013).
  75. M. B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. (2007) P08024.
  76. J. Eisert, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010).
  77. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  78. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  79. P. Secular, N. Gourianov, M. Lubasch, S. Dolgov, S. R. Clark, and D. Jaksch, Parallel time-dependent variational principle algorithm for matrix product states, Phys. Rev. B 101, 235123 (2020).
  80. S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, and C. Hubig, Time-evolution methods for matrix-product states, Ann. Phys. 411, 167998 (2019).
  81. T. Begušić, J. Gray, and G. K.-L. Chan, Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance, Sci. Adv. 10, eadk4321 (2024).
  82. H.-J. Liao, K. Wang, Z.-S. Zhou, P. Zhang, and T. Xiang, Simulation of IBM’s kicked Ising experiment with projected entangled pair operator, arXiv:2308.03082.
  83. L. Pavešić, M. D. Liberto, and S. Montangero, Scattering and induced false vacuum decay in the two-dimensional quantum Ising model, arXiv:2509.02702.
  84. M. S. Rudolph and J. Tindall, Simulating and sampling from quantum circuits with 2D tensor networks, arXiv:2507.11424.
  85. K. Bidzhiev, S. Grava, P. L. Henaff, M. Mendizabal, E. Merhej, and A. Quelle, Efficient emulation of neutral atom quantum hardware, arXiv:2510.09813.
  86. N. Tausendpfund, W. Krinitsin, M. Schmitt, and M. Rizzi, TTN.jl—A tree tensor network library for calculating groundstates and solving time evolution (0.1), Zenodo, 2024, https://zenodo.org/records/14421855.
  87. J. Tindall, TensorNetworkQuantumSimulator.jl (2025), https://github.com/JoeyT1994/TensorNetworkQuantumSimulator.jl.
  88. M. Fishman, S. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
  89. G. Carleo, F. Becca, M. Schiró, and M. Fabrizio, Localization and glassy dynamics of many-body quantum systems, Sci. Rep. 2, 243 (2012).
  90. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  91. Y. Levine, O. Sharir, N. Cohen, and A. Shashua, Quantum entanglement in deep learning architectures, Phys. Rev. Lett. 122, 065301 (2019).
  92. O. Sharir, A. Shashua, and G. Carleo, Neural tensor contractions and the expressive power of deep neural quantum states, Phys. Rev. B 106, 205136 (2022).
  93. G. Carleo, B. Bauer, and M. Troyer, Simulating adiabatic quantum computation with a variational approach, arXiv:2403.05147.
  94. L. Mauron, Z. Denis, J. Nys, and G. Carleo, Predicting topological entanglement entropy in a Rydberg analogue simulator, Nat. Phys. 21, 1332 (2025).
  95. T. Mendes-Santos, M. Schmitt, and M. Heyl, Highly resolved spectral functions of two-dimensional systems with neural quantum states, Phys. Rev. Lett. 131, 046501 (2023).
  96. M. Schmitt and M. Heyl, Simulating dynamics of correlated matter with neural quantum states, arXiv:2506.03124.
  97. M. Schmitt and M. Heyl, Quantum many-body dynamics in two dimensions with artificial neural networks, Phys. Rev. Lett. 125, 100503 (2020).
  98. A. Sinibaldi, C. Giuliani, G. Carleo, and F. Vicentini, Unbiasing time-dependent variational Monte Carlo by projected quantum evolution, Quantum 7, 1131 (2023).
  99. M. Schmitt and M. Reh, jVMC: Versatile and performant variational Monte Carlo leveraging automated differentiation and GPU acceleration, SciPost Phys. Codebases 2 (2022).
  100. Q. Yang and P. Corboz, Efficient iPEPS simulation on the honeycomb lattice via QR-based corner transfer matrix renormalization group, Phys. Rev. B 113, 085109 (2026).
  101. We omit the NQS results in Figs. 7–7 for the correlations, since the Monte Carlo error bars are larger than the y scale used in the graph.
  102. P. Weiss, L’hypothèse du champ moléculaire et la propriété ferromagnétique, J. Phys. Theor. Appl. 6, 661 (1907).
  103. M. Kaicher, J. Vovrosh, A. Dauphin, and S. B. Jäger, Quantum simulation of general spin-1/2 Hamiltonians with parity-violating fermionic Gaussian states, arXiv:2601.13811.
  104. J. Schachenmayer, A. Pikovski, and A. Rey, Many-body quantum spin dynamics with Monte Carlo trajectories on a discrete phase space, Phys. Rev. X 5, 011022 (2015).
  105. L. F. Hackl, Aspects of Gaussian states: Entanglement, squeezing and complexity, Ph.D. thesis, Pennsylvania State University, 2018.
  106. M. A. Perlin, C. Qu, and A. M. Rey, Spin squeezing with short-range spin-exchange interactions, Phys. Rev. Lett. 125, 223401 (2020).
  107. H. Hosseinabadi, O. Chelpanova, and J. Marino, User-friendly truncated Wigner approximation for dissipative spin dynamics, PRX Quantum 6, 030344 (2025).
  108. B. Kloss, Y. B. Lev, and D. Reichman, Time-dependent variational principle in matrix-product state manifolds: Pitfalls and potential, Phys. Rev. B 97, 024307 (2018).
  109. Y. Zhou, E. M. Stoudenmire, and X. Waintal, What limits the simulation of quantum computers? Phys. Rev. X 10, 041038 (2020).
  110. P. Scholl, Quantum simulation of spin models with large arrays of Rydberg atoms, Ph.D. thesis, Université Paris-Saclay, 2021.
  111. S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J. Liu, R. Samajdar, et al., Quantum optimization of maximum independent set using Rydberg atom arrays, Science 376, 1209 (2022).
  112. R. Trivedi, A. Franco Rubio, and J. I. Cirac, Quantum advantage and stability to errors in analogue quantum simulators, Nat. Commun. 15, 6507 (2024).
  113. R. Trivedi and J. I. Cirac, Noise robustness of problem-to-simulator mappings for quantum many-body physics, arXiv:2509.17579.
  114. T. Ayral, T. Louvet, Y. Zhou, C. Lambert, E. M. Stoudenmire, and X. Waintal, Density-matrix renormalization group algorithm for simulating quantum circuits with a finite fidelity, PRX Quantum 4, 020304 (2023).
  115. V. Joseph, J.-F. Sergi, K. Wladislaw, K. Michael, H. Fergus, G. Emmanuel, K. Augustine, B. Kemal, J. imon B. Schmitt, M. T. Joseph, D. Constantin, M.-S. Tiago, D. Alexandre, Simulating dynamics of the 2D transverse-field Ising model: Data and plotting scripts [Data set], Zenodo, 2025, https://zenodo.org/records/20137061.
  116. T. Zhang, H. Wang, W. Zhang, Y. Wang, A. Du, Z. Li, Y. Wu, C. Li, J. Hu, H. Zhai, and W. Chen, Observation of near-critical Kibble-Zurek scaling in Rydberg atom arrays, Phys. Rev. Lett. 135, 093403 (2025).
  117. I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A: Math. Theor. 42, 504003 (2009).
  118. M. P. Kaicher, D. Vodola, and S. B. Jäger, Mean-field treatment of the long-range transverse field Ising model with fermionic Gaussian states, Phys. Rev. B 107, 165144 (2023).
  119. M. A. Seifi Mirjafarlou, A. Jafarizadeh, and M. A. Rajabpour, Generalization of Balian–Brezin decomposition for exponentials with linear fermionic part, J. Phys. A: Math. Theor. 57, 195206 (2024).
  120. T. Shi, E. Demler, and J. I. Cirac, Variational study of fermionic and bosonic systems with non-Gaussian states: Theory and applications, Ann. Phys. 390, 245 (2018).

Outline

Information

Phys. Rev. Research 8, 023311– Published 17 June, 2026
  • Received 8 December 2025
  • Accepted 6 May 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