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  • Open Access

Loop series expansions for tensor networks

Glen Evenbly*, Nicola Pancotti, and Ashley Milsted

Johnnie Gray and Garnet Kin-Lic Chan

  • *Contact author: evenbly@amazon.com
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Phys. Rev. Research 8, 013245 – Published 5 March, 2026

DOI: https://doi.org/10.1103/vqks-cr6x

Abstract

Belief propagation (BP) can be a useful tool to approximately contract a tensor network, provided that the contributions from any closed loops in the network are sufficiently weak. In this article, we describe how a loop series expansion can be applied to systematically improve the accuracy of a BP approximation to a tensor network contraction, in principle converging arbitrarily close to the exact result. More generally, our result provides a framework for expanding a tensor network as a sum of component networks in a hierarchy of increasing complexity. We benchmark this proposal for the contraction of infinite projected entangled pair states, either representing the ground state of an Affleck-Kennedy-Lieb-Tasaki (AKLT) model or with randomly defined tensors, where it is shown to improve in accuracy over standard BP by several orders of magnitude while incurring only a minor increase in computational cost. These results indicate that the proposed series expansions could be a useful tool to accurately evaluate tensor networks in cases that otherwise exceed the limits of established contraction routines.

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References (72)

  1. J. I. Cirac and F. Verstraete, Renormalization and tensor product states in spin chains and lattices, J. Phys. A: Math. Theor. 42, 504004 (2009).
  2. T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev. 51, 455 (2009).
  3. G. Evenbly and G. Vidal, Tensor network states and geometry, J. Stat. Phys. 145, 891 (2011).
  4. R. Orus, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  5. J. C. Bridgeman and C. T. Chubb, Hand-waving and interpretive dance: An introductory course on tensor networks, J. Phys. A: Math. Theor. 50, 223001 (2017).
  6. S. Montangero, Introduction to Tensor Network Methods: Numerical Simulations of Low-Dimensional Many-Body Quantum Systems (Springer, Cham, 2018).
  7. R. Orus, Tensor networks for complex quantum systems, Nat. Rev. Phys. 1, 538 (2019).
  8. P. Silvi, F. Tschirsich, M. Gerster, J. Jünemann, D. Jaschke, M. Rizzi, and S. Montangero, The Tensor Networks Anthology Simulation techniques for many-body quantum lattice systems, SciPost Phys. Lect. Notes, 8 (2019).
  9. S.-J. Ran, E. Tirrito, C. Peng, X. Chen, L. Tagliacozzo, G. Su, and M. Lewenstein, Tensor Network Contractions: Methods and Applications to Quantum Many-Body Systems (Springer, Cham, 2020).
  10. 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).
  11. E. S. Fried, N. P. D. Sawaya, Y. Cao, I. D. Kivlichan, J. Romero, and A. Aspuru-Guzik, qTorch: The quantum tensor contraction handler, PLoS One 13, e0208510 (2018).
  12. B. Villalonga, S. Boixo, B. Nelson, C. Henze, E. Rieffel, R. Biswas, and S. Mandrà, A flexible high-performance simulator for verifying and benchmarking quantum circuits implemented on real hardware, npj Quantum Inf. 5, 86 (2019).
  13. R. Schutski, D. Lykov, and I. Oseledets, Adaptive algorithm for quantum circuit simulation, Phys. Rev. A 101, 042335 (2020).
  14. Y. Zhou, E. M. Stoudenmire, and X. Waintal, What limits the simulation of quantum computers? Phys. Rev. X 10, 041038 (2020).
  15. M. Levental, Tensor networks for simulating quantum circuits on FPGAs, arXiv:2108.06831.
  16. T. Vincent, L. J. O’Riordan, M. Andrenkov, J. Brown, N. Killoran, H. Qi, and I. Dhand, Jet: Fast quantum circuit simulations with parallel task-based tensor-network contraction, Quantum 6, 709 (2022).
  17. F. Pan and P. Zhang, Simulation of quantum circuits using the big-batch tensor network method, Phys. Rev. Lett. 128, 030501 (2022).
  18. F. Pan, K. Chen, and P. Zhang, Solving the sampling problem of the Sycamore quantum circuits, Phys. Rev. Lett. 129, 090502 (2022).
  19. 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).
  20. 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).
  21. 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).
  22. M. Fannes, B. Nachtergaele, and R. F. Werner, Finitely correlated states on quantum spin chains, Commun. Math. Phys. 144, 443 (1992).
  23. S. Ostlund and S. Rommer, Thermodynamic limit of density matrix renormalization, Phys. Rev. Lett. 75, 3537 (1995).
  24. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  25. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  26. U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
  27. F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many-body systems in two and higher dimensions, arXiv:cond-mat/0407066.
  28. F. Verstraete, J. I. Cirac, and V. Murg, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).
  29. L. Cincio, J. Dziarmaga, and M. M. Rams, Multiscale entanglement renormalization ansatz in two dimensions: Quantum Ising model, Phys. Rev. Lett. 100, 240603 (2008).
  30. G. Evenbly and G. Vidal, Entanglement renormalization in two spatial dimensions, Phys. Rev. Lett. 102, 180406 (2009).
  31. N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Computational complexity of projected entangled pair states, Phys. Rev. Lett. 98, 140506 (2007).
  32. T. Nishino and K. Okunishi, Corner transfer matrix renormalization group method, J. Phys. Soc. Jpn. 65, 891 (1996).
  33. 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).
  34. R. Orús 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).
  35. P. Corboz, T. M. Rice, and M. Troyer, Competing states in the t−J model: Uniform d-wave state versus stripe state, Phys. Rev. Lett. 113, 046402 (2014).
  36. M. Lubasch, J. I. Cirac, and M.-C. Bañuls, Unifying projected entangled pair state contractions, New J. Phys. 16, 033014 (2014).
  37. M. Lubasch, J. I. Cirac, and M.-C. Bañuls, Algorithms for finite projected entangled pair states, Phys. Rev. B 90, 064425 (2014).
  38. M. T. Fishman, L. Vanderstraeten, V. Zauner-Stauber, J. Haegeman, and F. Verstraete, Faster methods for contracting infinite two-dimensional tensor networks, Phys. Rev. B 98, 235148 (2018).
  39. H. A. Bethe, Statistical theory of superlattices, Proc. R. Soc. London A 150, 552 (1935).
  40. J. Pearl, Reverend bayes on inference engines a distributed hierarchical approach, in Proceedings of the Second AAAI Conference on Artificial Intelligence, AAAI’82 (AAAI Press, Pittsburgh, Pennsylvania, 1982), pp. 133–136.
  41. J. Pearl, Fusion, propagation, and structuring in belief networks, Artif. Intell. 29, 241 (1986).
  42. M. Mézard, G. Parisi, and R. Zecchina, Analytic and algorithmic solution of random satisfiability problems, Science 297, 812 (2002).
  43. J. S. Yedidia, W. Freeman, and Y. Weiss, Generalized belief propagation, in Advances in Neural Information Processing Systems, edited by T. Leen, T. Dietterich, and V. Tresp (MIT Press, Cambridge, MA, USA, 2000), Vol. 13, pp. 668–674.
  44. J. S. Yedidia, W. T. Freeman, and Y. Weiss, Understanding belief propagation and its generalizations, in Exploring Artificial Intelligence in the New Millennium, edited by G. Lakemeyer and B. Nebel (Morgan Kaufmann, San Francisco, CA, 2003), pp. 239–269.
  45. J. S. Yedidia, W. T. Freeman, and Y. Weiss, Constructing free-energy approximations and generalized belief propagation algorithms, IEEE Trans. Inf. Theory 51, 2282 (2005).
  46. M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and its Applications (World Scientific, Singapore, 1987), Vol. 9.
  47. S. Yoon, A. V. Goltsev, S. N. Dorogovtsev, and J. F. F. Mendes, Belief-propagation algorithm and the Ising model on networks with arbitrary distributions of motifs, Phys. Rev. E 84, 041144 (2011).
  48. B. Karrer, M. E. J. Newman, and L. Zdeborová, Percolation on sparse networks, Phys. Rev. Lett. 113, 208702 (2014).
  49. M. S. Leifer and D. Poulin, Quantum graphical models and belief propagation, Ann. Phys. 323, 1899 (2008).
  50. D. Poulin and E. Bilgin, Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs, Phys. Rev. A 77, 052318 (2008).
  51. A. Wrigley, W. S. Lee, and N. Ye, Tensor belief propagation, in Proceedings of the 34th International Conference on Machine Learning, edited by D. Precup and Y. W. Teh (PMLR, 2017), Vol. 70, pp. 3771–3779.
  52. E. Robeva and A. Seigal, Duality of graphical models and tensor networks, Inf. Inference 8, 273 (2018).
  53. R. Alkabetz and I. Arad, Tensor networks contraction and the belief propagation algorithm, Phys. Rev. Res. 3, 023073 (2021).
  54. S. Sahu and B. Swingle, Efficient tensor network simulation of quantum many-body physics on sparse graphs, arXiv:2206.04701.
  55. C. Guo, D. Poletti, and I. Arad, Block belief propagation algorithm for two-dimensional tensor networks, Phys. Rev. B 108, 125111 (2023).
  56. N. Pancotti and J. Gray, One-step replica symmetry breaking in the language of tensor networks, arXiv:2306.15004.
  57. Y. Wang, Y. E. Zhang, F. Pan, and P. Zhang, Tensor network message passing, Phys. Rev. Lett. 132, 117401 (2024).
  58. J. Tindall and M. Fishman, Gauging tensor networks with belief propagation, SciPost Phys. 15, 222 (2023).
  59. 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).
  60. M. P. Zaletel and F. Pollmann, Isometric tensor network states in two dimensions, Phys. Rev. Lett. 124, 037201 (2020).
  61. M. Chertkov and V. Y. Chernyak, Loop calculus in statistical physics and information science, Phys. Rev. E 73, 065102(R), (2006).
  62. M. Chertkov and V. Chernyak, Loop series for discrete statistical models on graphs, J. Stat. Mech. (2006) P06009.
  63. V. Gómez, J. Mooij, and H. Kappen, Truncating the loop series expansion for belief propagation, J. Mach. Learn. Res. 8, 1987 (2007).
  64. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799 (1987).
  65. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets, Commun. Math. Phys. 115, 477 (1988).
  66. Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, and T. Xiang, Tensor renormalization of quantum many-body systems using projected entangled simplex states, Phys. Rev. X 4, 011025 (2014).
  67. M. Levin and C. P. Nave, Tensor renormalization group approach to two-dimensional classical lattice models, Phys. Rev. Lett. 99, 120601 (2007).
  68. G. Evenbly and G. Vidal, Tensor network renormalization, Phys. Rev. Lett. 115, 180405 (2015).
  69. S. Yang, Z.-C. Gu, and X.-G. Wen, Loop optimization for tensor network renormalization, Phys. Rev. Lett. 118, 110504 (2017).
  70. G. Evenbly, Gauge fixing, canonical forms, and optimal truncations in tensor networks with closed loops, Phys. Rev. B 98, 085155 (2018).
  71. M. Hauru, C. Delcamp, and S. Mizera, Renormalization of tensor networks using graph-independent local truncations, Phys. Rev. B 97, 045111 (2018).
  72. I. Lukin and A. Sotnikov (private communication).

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Phys. Rev. Research 8, 013245– Published 5 March, 2026
  • Received 26 March 2025
  • Accepted 13 January 2026
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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.

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