Tree tensor network states represent low-energy states faithfully

TL;DR

This work derives rigorous bounds on the fidelity and resource requirements of tree tensor network states (TTNS) for approximating quantum states on tree graphs. By relating truncation errors to edge Schmidt spectra and to α-Rényi entanglement entropies, it provides both lower and upper bounds on the achievable accuracy and on the necessary bond dimensions, including explicit formulas and conditions on Rényi parameters. A key insight is that if edge entanglement satisfies an area (or log-area) law, TTNS can approximate low-energy states efficiently with bond dimensions growing only polynomially in system size. The results thus establish when TTNS on trees are faithful and practically efficient, guiding both theoretical understanding and computational strategies for tree-structured quantum systems.

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. For tree lattices, the result implies that efficient TTNS approximations exist if $α<1$ Rényi entanglement entropies for single-branch cuts obey an area law, as in ground and low-energy states of certain gapped systems.

Figures (2)

• Figure 1: For an exact TTNS representation of a target state \ref{['eq:psi']} with expansion coefficients $\Psi^{\sigma_1,\dotsc,\sigma_N}$, (a) assign the physical sites to vertices of a directed tree graph, where cutting edge $i$ yields a bipartition into subsystem $\mathcal{L}_i$ and its complement $\mathcal{R}_i$. (b) A sequence of $N-1$ SVDs \ref{['eq:SVD']} or, equivalently, QR decompositions then splits off one TTNS tensor $A_i$ in each step. (c) The resulting tensors are isometries with the orthonormality property \ref{['eq:A-ON']}.
• Figure 2: (a) The SVD \ref{['eq:SVD']} for edge $i$ yields a Schmidt decomposition \ref{['eq:exact-Schmidt']} of the target state $|\psi\rangle$ with the TTNS formed by tensors $\{A_j\,|\,j\in\mathcal{L}(i)\}$ providing an orthonormal basis $\{|\ell_{i,\mu}\rangle\}$ for subsystem $\mathcal{L}(i)$ -- in this case, sites 1, 2 and 3. Similarly, tensor $\Phi_i$ provides an orthonormal basis $\{|r_{i,\mu}\rangle\}$ for subsystem $\mathcal{R}(i)$ -- in this case, sites 4, 5, 6 and 7. (b) The truncated TTNS \ref{['eq:psiTrunc']} is obtained from the exact TTNS representation of the target state by inserting projection operators $p_i$ at every edge, only retaining the $M_i$-dimensional bond subspace corresponding to the largest Schmidt coefficients $\lambda_{i,1}\geq\dotsc\geq\lambda_{i,M_i}$.