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Skeleton of Isometric Tensor Network States for Abelian String-Net Models

Julian Boesl1,2, Yu-Jie Liu3, Frank Pollmann1,2, and Michael Knap1,2

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Phys. Rev. Lett. 137, 020405 – Published 7 July, 2026

DOI: https://doi.org/10.1103/d3fz-5755

Abstract

We construct parametrized isometric tensor network states—referred to as “skeletons”—that allow us to explore phases of Abelian topological order and can be efficiently implemented on quantum processors. We obtain stable finite correlation length deformations of string-net fixed points, which are constructed both by conserving virtual symmetries of the tensor and by imposing local isometry constraints. They connect distinct topological phases via a shared critical point, thereby providing analytically tractable examples of phase transitions beyond anyon condensation. By mapping such classes of 2D tensor networks to 1D stochastic automata with local update rules, we show that expectation values of generalized Pauli strings of arbitrary weight can be efficiently computed using classical methods. Therefore these skeletons not only serve as an organizing principle for Abelian topological order but also provide a nontrivial test bed for quantum processors.

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References (70)

  1. 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).
  2. F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Criticality, the area law, and the computational power of projected entangled pair states, Phys. Rev. Lett. 96, 220601 (2006).
  3. N. Schuch, I. Cirac, and D. Pérez-García, Peps as ground states: Degeneracy and topology, Ann. Phys. (Amsterdam) 325, 2153 (2010).
  4. M. B. Şahinoğlu, D. Williamson, N. Bultinck, M. Mariën, J. Haegeman, N. Schuch, and F. Verstraete, Characterizing topological order with matrix product operators, in Annales Henri Poincaré (Springer, New York, 2021), Vol. 22, pp. 563–592.
  5. D. J. Williamson, N. Bultinck, and F. Verstraete, Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation, arXiv:1711.07982.
  6. M. M. Wolf, G. Ortiz, F. Verstraete, and J. I. Cirac, Quantum phase transitions in matrix product systems, Phys. Rev. Lett. 97, 110403 (2006).
  7. N. G. Jones, J. Bibo, B. Jobst, F. Pollmann, A. Smith, and R. Verresen, Skeleton of matrix-product-state-solvable models connecting topological phases of matter, Phys. Rev. Res. 3, 033265 (2021).
  8. I. Camp and N. G. Jones, Matrix-product state skeletons in Onsager-integrable quantum chains, arXiv:2511.07212.
  9. A. Smith, B. Jobst, A. G. Green, and F. Pollmann, Crossing a topological phase transition with a quantum computer, Phys. Rev. Res. 4, L022020 (2022).
  10. X. Chen, B. Zeng, Z.-C. Gu, I. L. Chuang, and X.-G. Wen, Tensor product representation of a topological ordered phase: Necessary symmetry conditions, Phys. Rev. B 82, 165119 (2010).
  11. W.-T. Xu and G.-M. Zhang, Tensor network state approach to quantum topological phase transitions and their criticalities of Z2 topologically ordered states, Phys. Rev. B 98, 165115 (2018).
  12. L. Haller, W.-T. Xu, Y.-J. Liu, and F. Pollmann, Quantum phase transition between symmetry enriched topological phases in tensor-network states, Phys. Rev. Res. 5, 043078 (2023).
  13. J. Haegeman, V. Zauner, N. Schuch, and F. Verstraete, Shadows of anyons and the entanglement structure of topological phases, Nat. Commun. 6, 8284 (2015).
  14. M. Iqbal, K. Duivenvoorden, and N. Schuch, Study of anyon condensation and topological phase transitions from a Z4 topological phase using the projected entangled pair states approach, Phys. Rev. B 97, 195124 (2018).
  15. W.-T. Xu and N. Schuch, Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters, Phys. Rev. B 104, 155119 (2021).
  16. N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Computational complexity of projected entangled pair states, Phys. Rev. Lett. 98, 140506 (2007).
  17. M. P. Zaletel and F. Pollmann, Isometric tensor network states in two dimensions, Phys. Rev. Lett. 124, 037201 (2020).
  18. T. Soejima, K. Siva, N. Bultinck, S. Chatterjee, F. Pollmann, and M. P. Zaletel, Isometric tensor network representation of string-net liquids, Phys. Rev. B 101, 085117 (2020).
  19. L. Slattery and B. K. Clark, Quantum circuits for two-dimensional isometric tensor networks, arXiv:2108.02792.
  20. Z.-Y. Wei, D. Malz, and J. I. Cirac, Sequential generation of projected entangled-pair states, Phys. Rev. Lett. 128, 010607 (2022).
  21. Y.-J. Liu, K. Shtengel, A. Smith, and F. Pollmann, Methods for simulating string-net states and anyons on a digital quantum computer, PRX Quantum 3, 040315 (2022).
  22. Y.-J. Liu, K. Shtengel, and F. Pollmann, Simulating two-dimensional topological quantum phase transitions on a digital quantum computer, Phys. Rev. Res. 6, 043256 (2024).
  23. X.-H. Yu, J. I. Cirac, P. Kos, and G. Styliaris, Dual-isometric projected entangled pair states, Phys. Rev. Lett. 133, 190401 (2024).
  24. J. Boesl, Y.-J. Liu, W.-T. Xu, F. Pollmann, and M. Knap, Quantum phase transitions between symmetry-enriched fracton phases, Phys. Rev. B 112, 125152 (2025).
  25. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  26. Z.-C. Gu, M. Levin, B. Swingle, and X.-G. Wen, Tensor-product representations for string-net condensed states, Phys. Rev. B 79, 085118 (2009).
  27. O. Buerschaper, M. Aguado, and G. Vidal, Explicit tensor network representation for the ground states of string-net models, Phys. Rev. B 79, 085119 (2009).
  28. L.-Y. Hung and Y. Wan, String-net models with ZN fusion algebra, Phys. Rev. B 86, 235132 (2012).
  29. T. Lan and X.-G. Wen, Topological quasiparticles and the holographic bulk-edge relation in (2+1)-dimensional string-net models, Phys. Rev. B 90, 115119 (2014).
  30. C.-H. Lin and M. Levin, Generalizations and limitations of string-net models, Phys. Rev. B 89, 195130 (2014).
  31. C.-H. Lin, M. Levin, and F. J. Burnell, Generalized string-net models: A thorough exposition, Phys. Rev. B 103, 195155 (2021).
  32. Z.-C. Gu, M. Levin, and X.-G. Wen, Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions, Phys. Rev. B 78, 205116 (2008).
  33. J. Vidal, S. Dusuel, and K. P. Schmidt, Low-energy effective theory of the toric code model in a parallel magnetic field, Phys. Rev. B 79, 033109 (2009).
  34. S. Dusuel, M. Kamfor, R. Orús, K. P. Schmidt, and J. Vidal, Robustness of a perturbed topological phase, Phys. Rev. Lett. 106, 107203 (2011).
  35. F. Wu, Y. Deng, and N. Prokof’ev, Phase diagram of the toric code model in a parallel magnetic field, Phys. Rev. B 85, 195104 (2012).
  36. F. J. Burnell, Anyon condensation and its applications, Annu. Rev. Condens. Matter Phys. 9, 307 (2018).
  37. C.-H. Lin and F. J. Burnell, Anyon condensation in string-net models, Phys. Rev. B 110, 115127 (2024).
  38. T. A. Cochran, B. Jobst, E. Rosenberg, Y. D. Lensky, G. Gyawali, N. Eassa, M. Will, A. Szasz, D. Abanin, R. Acharya et al., Visualizing dynamics of charges and strings in (2+1)D lattice gauge theories, Nature (London) 642, 315 (2025).
  39. D. Perez-Garcia, F. Verstraete, J. I. Cirac, and M. M. Wolf, Peps as unique ground states of local Hamiltonians, Quantum Inf. Comput. 8, 0650 (2008).
  40. See Supplemental Material at http://link.aps.org/supplemental/10.1103/d3fz-5755 which includes Refs. [41–47].
  41. P. Bonderson, K. Shtengel, and J. Slingerland, Interferometry of non-Abelian anyons, Ann. Phys. (Amsterdam) 323, 2709 (2008).
  42. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (Amsterdam) 303, 2 (2003).
  43. I. Klich, On the stability of topological phases on a lattice, Ann. Phys. (Amsterdam) 325, 2120 (2010).
  44. T. Nishino and K. Okunishi, Corner transfer matrix renormalization group method, J. Phys. Soc. Jpn. 65, 891 (1996).
  45. A. Gromov, A. Lucas, and R. M. Nandkishore, Fracton hydrodynamics, Phys. Rev. Res. 2, 033124 (2020).
  46. J. Feldmeier, P. Sala, G. De Tomasi, F. Pollmann, and M. Knap, Anomalous diffusion in dipole- and higher-moment-conserving systems, Phys. Rev. Lett. 125, 245303 (2020).
  47. P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity breaking arising from hilbert space fragmentation in dipole-conserving Hamiltonians, Phys. Rev. X 10, 011047 (2020).
  48. S. K. Shukla, M. B. Şahinoğlu, F. Pollmann, and X. Chen, Boson condensation and instability in the tensor network representation of string-net states, Phys. Rev. B 98, 125112 (2018).
  49. M. Levin and Z.-C. Gu, Braiding statistics approach to symmetry-protected topological phases, Phys. Rev. B 86, 115109 (2012).
  50. Q. Zhang, W.-T. Xu, Z.-Q. Wang, and G.-M. Zhang, Non-Hermitian effects of the intrinsic signs in topologically ordered wavefunctions, Commun. Phys. 3, 209 (2020).
  51. M. P. Zaletel, Detecting two-dimensional symmetry-protected topological order in a ground-state wave function, Phys. Rev. B 90, 235113 (2014).
  52. K. J. Satzinger, Y.-J. Liu, A. Smith, C. Knapp, M. Newman, C. Jones, Z. Chen, C. Quintana, X. Mi, A. Dunsworth et al., Realizing topologically ordered states on a quantum processor, Science 374, 1237 (2021).
  53. A. Gammon-Smith, M. Knap, and F. Pollmann, Simulating topological order on quantum processors, Nat. Rev. Phys. 8, 160 (2026).
  54. W. Zhang, Sequential generation of two-dimensional super-area-law states with local parent Hamiltonian, PRX Quantum 7, 010311 (2026).
  55. S. Papanikolaou, K. S. Raman, and E. Fradkin, Topological phases and topological entropy of two-dimensional systems with finite correlation length, Phys. Rev. B 76, 224421 (2007).
  56. C. Castelnovo and C. Chamon, Quantum topological phase transition at the microscopic level, Phys. Rev. B 77, 054433 (2008).
  57. A. M. Essin and M. Hermele, Classifying fractionalization: Symmetry classification of gapped Z2 spin liquids in two dimensions, Phys. Rev. B 87, 104406 (2013).
  58. X. Chen, Symmetry fractionalization in two dimensional topological phases, Rev. Phys. 2, 3 (2017).
  59. F. Pollmann and A. M. Turner, Detection of symmetry-protected topological phases in one dimension, Phys. Rev. B 86, 125441 (2012).
  60. C.-Y. Huang, X. Chen, and F. Pollmann, Detection of symmetry-enriched topological phases, Phys. Rev. B 90, 045142 (2014).
  61. P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, England, 1995).
  62. H. Watanabe, M. Cheng, and Y. Fuji, Ground state degeneracy on torus in a family of ZN toric code, J. Math. Phys. (N.Y.) 64, 051901 (2023).
  63. D. Haag, F. Baccari, and G. Styliaris, Typical correlation length of sequentially generated tensor network states, PRX Quantum 4, 030330 (2023).
  64. D. Malz and R. Trivedi, Computational complexity of isometric tensor-network states, PRX Quantum 6, 020310 (2025).
  65. J. Boesl, F. Pollmann, and M. Knap, Entanglement pattern transition of quantum states from directed percolation, arXiv:2605.26219.
  66. H. Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states, Adv. Phys. 49, 815 (2000).
  67. G. Ódor, Universality classes in nonequilibrium lattice systems, Rev. Mod. Phys. 76, 663 (2004).
  68. N. Ziereis, S. Moudgalya, and M. Knap, Strong-to-weak symmetry breaking phases in steady states of quantum operations, arXiv:2509.09669.
  69. T.-C. Lu, Y.-J. Liu, S. Gopalakrishnan, and Y. You, Holographic duality between bulk topological order and boundary mixed-state order, arXiv:2511.19597.
  70. Numerical codes are available at 10.5281/zenodo.17427675.

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Phys. Rev. Lett. 137, 020405– Published 7 July, 2026
  • Received 28 November 2025
  • Revised 16 April 2026
  • Accepted 5 June 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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