- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

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

High-Temperature Partition Functions and Classical Simulatability of Long-Range Quantum Systems

Jorge Sánchez-Segovia1,2,*, Jan T. Schneider2,†, and Álvaro M. Alhambra1,‡

  • *Contact author: jorge.sanchezsegovia@estudiante.uam.es
  • †Contact author: jan.schneider@iff.csic.es
  • ‡Contact author: alvaro.alhambra@csic.es
PDF

PRX Quantum 6, 040366 – Published 17 December, 2025

DOI: https://doi.org/10.1103/cww3-j1vd

Abstract

Long-range quantum systems, in which the interactions decay as 1/rα, are of increasing interest due to the variety of experimental setups in which they naturally appear. Motivated by this, we study fundamental properties of long-range spin systems in thermal equilibrium, focusing on the weak regime of α>D. Our main result is a proof of analyticity of their partition functions at high temperatures, which allows us to construct a classical algorithm with subexponential runtime exp(O(log2(N/ϵ))) that approximates the log-partition function to small additive error ϵ. As by-products, we establish the equivalence of ensembles and the Gaussianity of the density of states, which we verify numerically in both the weak and strong long-range regimes. This also yields constraints on the appearance of various classes of phase transitions, including thermal, dynamical, and excited-state ones. Our main technical contribution is the extension to the quantum long-range regime of the convergence criterion for cluster expansions of Kotecký and Preiss.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

With the advent of quantum technologies, and platforms such as Rydberg atoms, trapped ions, and cold molecules, the physics of systems of many strongly interacting quantum particles is becoming increasingly accessible to experiments. In these experiments, the interactions between those particles are mediated by electromagnetic forces, which to a good approximation often decay slowly with their distance, making the models that describe them “long-range interacting.”

In this work, we study the thermal and equilibrium properties of general families of such models. We prove that, if their temperature is high enough, they are thermally stable in a qualitatively similar way as if those long-range interactions were not present. This potentially surprising conclusion has important consequences for the appearance of different kinds of phase transitions in those models. We also prove, through an explicit classical algorithm, how that stability implies that many of their equilibrium properties can be calculated efficiently in subexponential time.

Our results open the door to a deeper theoretical understanding of the physics of long-range systems and also of their computational complexity. The efficiency of our scheme is still behind what can be achieved with quantum simulation algorithms that are able to run similar tasks in polynomial time. This suggests two possibilities: either that there is a genuine quantum advantage in this setting or, perhaps more likely, that there is still room to improve classical approaches. A conclusion either way would greatly improve our understanding of these general families of long-range models.

Article Text

References (100)

  1. M. Foss-Feig, G. Pagano, A. C. Potter, and N. Y. Yao, Progress in trapped-ion quantum simulation, Annu. Rev. Condens. Matter Phys. 16, 145 (2025).
  2. A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys. 16, 132 (2020).
  3. J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, Quantum many-body models with cold atoms coupled to photonic crystals, Nat. Photonics 9, 326 (2015).
  4. C.-L. Hung, A. González-Tudela, J. I. Cirac, and H. J. Kimble, Quantum spin dynamics with pairwise-tunable, long-range interactions, Proc. Natl. Acad. Sci. USA 113, E4946 (2016).
  5. S. L. Cornish, M. R. Tarbutt, and K. R. A. Hazzard, Quantum computation and quantum simulation with ultracold molecules, Nat. Phys. 20, 730 (2024).
  6. R. Islam, C. Senko, W. C. Campbell, S. Korenblit, J. Smith, A. Lee, E. E. Edwards, C.-C. J. Wang, J. K. Freericks, and C. Monroe, Emergence and frustration of magnetism with variable-range interactions in a quantum simulator, Science 340, 583 (2013).
  7. P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, and C. Monroe, Non-local propagation of correlations in quantum systems with long-range interactions, Nature 511, 198 (2014).
  8. M. K. Joshi, F. Kranzl, A. Schuckert, I. Lovas, C. Maier, R. Blatt, M. Knap, and C. F. Roos, Observing emergent hydrodynamics in a long-range quantum magnet, Science 376, 720 (2022).
  9. M. K. Joshi, A. Elben, B. Vermersch, T. Brydges, C. Maier, P. Zoller, R. Blatt, and C. F. Roos, Quantum information scrambling in a trapped-ion quantum simulator with tunable range interactions, Phys. Rev. Lett. 124, 240505 (2020).
  10. J. Barré, D. Mukamel, and S. Ruffo, Inequivalence of ensembles in a system with long-range interactions, Phys. Rev. Lett. 87, 030601 (2001).
  11. M. Kastner, Nonequivalence of ensembles for long-range quantum spin systems in optical lattices, Phys. Rev. Lett. 104, 240403 (2010).
  12. A. Russomanno, M. Fava, and M. Heyl, Quantum chaos and ensemble inequivalence of quantum long-range Ising chains, Phys. Rev. B 104, 094309 (2021).
  13. N. Defenu, D. Mukamel, and S. Ruffo, Ensemble inequivalence in long-range quantum systems, Phys. Rev. Lett. 133, 050403 (2024).
  14. A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo, Physics of Long-Range Interacting Systems, in Physics of Long-Range Interacting Systems, edited by A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo (Oxford University Press, Oxford, 2014), p. 32.
  15. N. Defenu, Metastability and discrete spectrum of long-range systems, Proc. Natl. Acad. Sci. 118, e2101785118 (2021).
  16. G. Bornet, G. Emperauger, C. Chen, B. Ye, M. Block, M. Bintz, J. A. Boyd, D. Barredo, T. Comparin, F. Mezzacapo, T. Roscilde, T. Lahaye, N. Y. Yao, and A. Browaeys, Scalable spin squeezing in a dipolar Rydberg atom array, Nature 621, 728 (2023).
  17. W. J. Eckner, N. Darkwah Oppong, A. Cao, A. W. Young, W. R. Milner, J. M. Robinson, J. Ye, and A. M. Kaufman, Realizing spin squeezing with Rydberg interactions in an optical clock, Nature 621, 734 (2023).
  18. M. Kastner, Diverging equilibration times in long-range quantum spin models, Phys. Rev. Lett. 106, 130601 (2011).
  19. P. Hauke and L. Tagliacozzo, Spread of correlations in long-range interacting quantum systems, Phys. Rev. Lett. 111, 207202 (2013).
  20. J. T. Schneider, J. Despres, S. J. Thomson, L. Tagliacozzo, and L. Sanchez-Palencia, Spreading of correlations and entanglement in the long-range transverse Ising chain, Phys. Rev. Res. 3, L012022 (2021).
  21. T. Kuwahara and K. Saito, Strictly linear light cones in long-range interacting systems of arbitrary dimensions, Phys. Rev. X 10, 031010 (2020).
  22. M. C. Tran, A. Y. Guo, C. L. Baldwin, A. Ehrenberg, A. V. Gorshkov, and A. Lucas, Lieb-Robinson light cone for power-law interactions, Phys. Rev. Lett. 127, 160401 (2021).
  23. S. Pappalardi, A. Russomanno, B. Žunkovič, F. Iemini, A. Silva, and R. Fazio, Scrambling and entanglement spreading in long-range spin chains, Phys. Rev. B 98, 134303 (2018).
  24. T. Mori, Prethermalization in the transverse-field Ising chain with long-range interactions, J. Phys. A: Math. Theor. 52, 054001 (2019).
  25. 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).
  26. N. Defenu, A. Lerose, and S. Pappalardi, Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions, Phys. Rep. 1074, 1 (2024).
  27. R. Mattes, I. Lesanovsky, and F. Carollo, Long-range interacting systems are locally noninteracting, Phys. Rev. Lett. 134, 070402 (2025).
  28. Z.-X. Gong, M. Foss-Feig, F. G. Brandão, and A. V. Gorshkov, Entanglement area laws for long-range interacting systems, Phys. Rev. Lett. 119, 050501 (2017).
  29. T. Kuwahara and K. Saito, Area law of noncritical ground states in 1D long-range interacting systems, Nat. Commun. 11, 4478 (2020).
  30. J. T. Schneider, S. J. Thomson, and L. Sanchez-Palencia, Entanglement spectrum and quantum phase diagram of the long-range XXZ chain, Phys. Rev. B 106, 014306 (2022).
  31. N. Defenu, T. Donner, T. Macrì, G. Pagano, S. Ruffo, and A. Trombettoni, Long-range interacting quantum systems, Rev. Mod. Phys. 95, 035002 (2023).
  32. D. Poulin and P. Wocjan, Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer, Phys. Rev. Lett. 103, 220502 (2009).
  33. S. Bravyi, A. Chowdhury, D. Gosset, and P. Wocjan, Quantum Hamiltonian complexity in thermal equilibrium, Nat. Phys. 18, 1367 (2022).
  34. T. Helmuth, W. Perkins, and G. Regts, Algorithmic Pirogov-Sinai theory, Probab. Theory Relat. Fields 176, 851 (2020).
  35. R. L. Mann and T. Helmuth, Efficient algorithms for approximating quantum partition functions, J. Math. Phys. 62, 022201 (2021).
  36. A. W. Harrow, S. Mehraban, and M. Soleimanifar, Classical algorithms, correlation decay, and complex zeros of partition functions of Quantum many-body systems, in Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 (Association for Computing Machinery, New York, NY, USA, 2020), pp. 378–386.
  37. J. Haah, R. Kothari, and E. Tang, Learning quantum Hamiltonians from high-temperature Gibbs states and real-time evolutions, Nat. Phys. 20, 1027 (2024).
  38. T. Kuwahara and K. Saito, Gaussian concentration bound and Ensemble equivalence in generic quantum many-body systems including long-range interactions, Ann. Phys. 421, 168278 (2020).
  39. M. Hartmann, G. mahler, and O. Hess, Gaussian quantum fluctuations in interacting many particle systems, Lett. Math. Phys. 68, 103 (2004).
  40. K. S. Rai, J. I. Cirac, and Á. M. Alhambra, Matrix product state approximations to quantum states of low energy variance, Quantum 8, 1401 (2024).
  41. P. Cejnar, P. Stránský, M. Macek, and M. Kloc, Excited-state quantum phase transitions, J. Phys. A: Math. Theor. 54, 133001 (2021).
  42. R. Kotecký and D. Preiss, Cluster expansion for abstract polymer models, Commun. Math. Phys. 103, 491 (1986).
  43. T. Helmuth and R. L. Mann, Efficient algorithms for approximating quantum partition functions at low temperature, Quantum 7, 1155 (2023).
  44. D. S. Wild and A. M. Alhambra, Classical simulation of short-time quantum dynamics, PRX Quantum 4, 020340 (2023).
  45. R. L. Mann and R. M. Minko, Algorithmic cluster expansions for quantum problems, PRX Quantum 5, 010305 (2024).
  46. M. Kliesch, C. Gogolin, M. Kastoryano, A. Riera, and J. Eisert, Locality of temperature, Phys. Rev. X 4, 031019 (2014).
  47. A. Molnar, N. Schuch, F. Verstraete, and J. I. Cirac, Approximating Gibbs states of local Hamiltonians efficiently with projected entangled pair states, Phys. Rev. B 91, 045138 (2015).
  48. C. Yin and A. Lucas, Polynomial-time classical sampling of high-temperature quantum gibbs states, arXiv:2305.18514.
  49. A. Bluhm, Ángela Capel, and A. Pérez-Hernández, Strong decay of correlations for Gibbs states in any dimension, J. Stat. Phys. 192, 134 (2025).
  50. A. Bakshi, A. Liu, A. Moitra, and E. Tang, High-temperature Gibbs states are unentangled and efficiently preparable, arXiv:2403.16850.
  51. Y. Kimura and T. Kuwahara, Clustering theorem in 1D long-range interacting systems at arbitrary temperatures, Commun. Math. Phys. 406, 65 (2025).
  52. D. Kim, T. Kuwahara, and K. Saito, Thermal area law in long-range interacting systems, Phys. Rev. Lett. 134, 020402 (2025).
  53. S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction (Cambridge University Press, Cambridge, 2017).
  54. R. Dobrushin, in Topics in statistical and theoretical physics: F.A. Berezin memorial volume (American Mathematical Society, Providence R.I., 1996).
  55. D. Ueltschi, Cluster expansions and correlation functions, Mosc. Math. J. 4, 511 (2004).
  56. R. Fernández and A. Procacci, Cluster expansion for abstract polymer models. New bounds from an old approach, Commun. Math. Phys. 274, 123 (2007).
  57. B. Nachtergaele, Y. Ogata, and R. Sims, Propagation of correlations in quantum lattice systems, J. Stat. Phys. 124, 1 (2006).
  58. M. B. Hastings, Quasi-adiabatic continuation for disordered systems: Applications to correlations, Lieb-Schultz-Mattis, and Hall conductance, arXiv:1001.5280.
  59. Z. Chen, A. Galanis, L. A. Goldberg, W. Perkins, J. Stewart, and E. Vigoda, Fast algorithms at low temperatures via markov chains, arXiv:1901.06653.
  60. A. Blanca, S. Cannon, and W. Perkins, Fast and perfect sampling of subgraphs and polymer systems, arXiv:2202.05907.
  61. R. Achutha, D. Kim, Y. Kimura, and T. Kuwahara, Provably efficient simulation of 1D long-range interacting systems at any temperature, Phys. Rev. Lett. 134, 190404 (2025).
  62. The convergence proof of the cluster expansion it relies on has recently been found to be flawed (see [37, 44]), although a fix has been reported [100].
  63. R. Lima, Equivalence of ensembles in quantum lattice systems, Ann. I.H.P. Phys. Théor. 15, 61 (1971).
  64. M. P. Müller, E. Adlam, L. Masanes, and N. Wiebe, Thermalization and canonical typicality in translation-invariant quantum lattice systems, Commun. Math. Phys. 340, 499 (2015).
  65. T. Kuwahara and K. Saito, Eigenstate thermalization from the clustering property of correlation, Phys. Rev. Lett. 124, 200604 (2020).
  66. A. M. Alhambra, Quantum many-body systems in thermal equilibrium, PRX Quantum 4, 040201 (2023).
  67. F. G. S. L. Brandao and M. Cramer, Equivalence of statistical mechanical ensembles for non-critical quantum systems, arXiv:1502.03263.
  68. C. Bertoni, C. Wassner, G. Guarnieri, and J. Eisert, Typical thermalization of low-entanglement states, Commun. Phys. 8, 301 (2025).
  69. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 2011), 2nd ed.
  70. 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).
  71. B. Pirvu, G. Vidal, F. Verstraete, and L. Tagliacozzo, Matrix product states for critical spin chains: Finite-size versus finite-entanglement scaling, Phys. Rev. B 86, 075117 (2012).
  72. P. Bruno, Absence of spontaneous magnetic order at nonzero temperature in one- and two-dimensional Heisenberg and XY systems with long-range interactions, Phys. Rev. Lett. 87, 137203 (2001).
  73. J. Zhao, M. Song, Y. Qi, J. Rong, and Z. Y. Meng, Finite-temperature critical behaviors in 2D long-range quantum Heisenberg model, Npj Quantum Mater. 8, 59 (2023).
  74. G. Giachetti, N. Defenu, S. Ruffo, and A. Trombettoni, Berezinskii-Kosterlitz-Thouless phase transitions with long-range couplings, Phys. Rev. Lett. 127, 156801 (2021).
  75. G. Giachetti, A. Trombettoni, S. Ruffo, and N. Defenu, Berezinskii-Kosterlitz-Thouless transitions in classical and quantum long-range systems, Phys. Rev. B 106, 014106 (2022).
  76. T. Xiao, D. Yao, C. Zhang, Z. Fan, and Y. Deng, Two-dimensional XY ferromagnet induced by long-range interaction, Chin. Phys. Lett. 42, 070002 (2025).
  77. J. C. Halimeh and V. Zauner-Stauber, Dynamical phase diagram of quantum spin chains with long-range interactions, Phys. Rev. B 96, 134427 (2017).
  78. J. C. Halimeh, M. Van Damme, L. Guo, J. Lang, and P. Hauke, Dynamical phase transitions in quantum spin models with antiferromagnetic long-range interactions, Phys. Rev. B 104, 115133 (2021).
  79. 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).
  80. P. Ribeiro, J. Vidal, and R. Mosseri, Exact spectrum of the Lipkin-Meshkov-Glick model in the thermodynamic limit and finite-size corrections, Phys. Rev. E 78, 021106 (2008).
  81. L. F. Santos, M. Távora, and F. Pérez-Bernal, Excited-state quantum phase transitions in many-body systems with infinite-range interaction: Localization, dynamics, and bifurcation, Phys. Rev. A 94, 012113 (2016).
  82. P. Ribeiro, J. Vidal, and R. Mosseri, Thermodynamical limit of the Lipkin-Meshkov-Glick Model, Phys. Rev. Lett. 99, 050402 (2007).
  83. T. Brandes, Excited-state quantum phase transitions in Dicke superradiance models, Phys. Rev. E 88, 032133 (2013).
  84. C. Rouzé, D. S. França, and Álvaro M. Alhambra, Optimal quantum algorithm for Gibbs state preparation, arXiv:2411.04885.
  85. E. Luijten and H. W. Blöte, Monte Carlo method for spin models with long-range interactions, Int. J. Mod. Phys. C 06, 359 (1995).
  86. E. Flores-Sola, M. Weigel, R. Kenna, and B. Berche, Cluster Monte Carlo and dynamical scaling for long-range interactions, Eur. Phys. J. Spec. Top. 226, 581 (2017).
  87. T. Kuwahara, Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures, Phys. Rev. X 15, 041010 (2025).
  88. C.-F. Chen and C. Rouzé, Quantum gibbs states are locally markovian, arXiv:2504.02208.
  89. K. Kato and T. Kuwahara, On the clustering of conditional mutual information via dissipative dynamics, arXiv:2504.02235.
  90. Ángela Capel, Álvaro M. Alhambra, P. Gondolf, A. R. de Alarcón, and S. O. Scalet, Conditional independence of 1D gibbs states with applications to efficient learning, arXiv:2402.18500.
  91. A. Bakshi, A. Liu, A. Moitra, and E. Tang, Learning quantum Hamiltonians at any temperature in polynomial time, arXiv:2310.02243.
  92. C.-F. Chen, A. Anshu, and Q. T. Nguyen, Learning quantum gibbs states locally and efficiently, arXiv:2504.02706.
  93. B. T. McDonough, C. Yin, A. Lucas, and C. Zhang, Lieb-robinson bounds with exponential-in-volume tails, arXiv:2502.02652.
  94. J. Sánchez Segovia, High-temperature partition functions and classical simulability of long-range quantum systems, Zenodo, 2025, https://doi.org/10.48550/arXiv.2504.20901.
  95. B. Nachtergaele, R. Sims, and A. Young, Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms, J. Math. Phys. 60, 061101 (2019).
  96. R. Otter, The number of trees, Ann. Math. 49, 583 (1948).
  97. B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, Matrix product operator representations, New J. Phys. 12, 025012 (2010).
  98. 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).
  99. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases, 4 (2022).
  100. T. Kuwahara (private communication).

Outline

Information

PRX Quantum 6, 040366– Published 17 December, 2025
  • Received 23 May 2025
  • Accepted 17 November 2025
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