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This paper establishes rigorous bounds for Tree Tensor Network States (TTNS), demonstrating their ability to faithfully approximate low-energy quantum states in gapped systems on tree graphs. By linking approximation error to Schmidt spectra and Rényi entanglement entropies, the work clarifies the computational resources required for efficient simulations of specific quantum many-body problems.
Tree tensor network states represent low-energy states faithfully
arXiv:2512.20215v2 Announce Type: replace Abstract: Extending corresponding results for matrix product states [Verstraete and Cirac, PRB 73, 094423 (2006); Schuch et al. PRL 100, 030504 (2008)], it is shown how the approximation error of tree tensor network states (TTNS) can be bounded using Schmidt spectra or R\'{e}nyi entanglement entropies of the target quantum state. Conversely, one obtains bounds on TTNS bond dimensions needed to achieve a specific approximation accuracy. The result implies that efficient TTNS approximations exist if $\alpha<1$ R\'{e}nyi entanglement entropies for single-branch cuts obey an area law, as in ground and low-energy states of certain gapped systems on tree graphs.
Abstract:Extending corresponding results for matrix product states [Verstraete and Cirac, PRB 73, 094423 (2006); Schuch et al. PRL 100, 030504 (2008)], it is shown how the approximation error of tree tensor network states (TTNS) can be bounded using Schmidt spectra or Rényi entanglement entropies of the target quantum state. Conversely, one obtains bounds on TTNS bond dimensions needed to achieve a specific approximation accuracy. The result implies that efficient TTNS approximations exist if $\alpha<1$ Rényi entanglement entropies for single-branch cuts obey an area law, as in ground and low-energy states of certain gapped systems on tree graphs.
10 pages, 2 figures; corrected typo in Eq. (1), minor improvements; published version
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); Computational Physics (physics.comp-ph)
arXiv:2512.20215 [quant-ph]
(or arXiv:2512.20215v2 [quant-ph] for this version)
J. Stat. Mech. 093101 (2026)
https://doi.org/10.1088/1742-5468/ae8cea DOI(s) linking to related resources
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