Tree tensor network states represent low-energy states faithfully

TL;DR

This paper demonstrates that tree tensor network states can accurately approximate low-energy quantum states, with bounds on error and bond dimensions linked to entanglement properties, especially for systems obeying an area law.

Contribution

It extends error bounds and bond dimension estimates for tree tensor network states based on entanglement spectra, generalizing previous matrix product state results.

Findings

Error bounds depend on Schmidt spectra and Rényi entropies.

Efficient TTNS approximations exist for states obeying an area law.

Applicable to ground and low-energy states of gapped 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 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. For tree lattices, 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.