Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2512.20215 · Journal of Statistical Mechanics Theory and Experiment

Tree tensor network states represent low-energy states faithfully

Thomas Barthel

quant-phcond-mat.stat-mechcond-mat.str-elphysics.comp-phSubmitted Dec 23, 2025• Announced Dec 24, 2025• 0 citations
arXivHTMLPDF

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.

Comments
9 pages, 2 figures

Related papers

Cost scaling of MPS and TTNS simulations for 2D and 3D systems with area-law entanglement

Thomas Barthel

Tensor network states are an indispensable tool for the simulation of strongly correlated quantum many-body systems. In recent years, tree tensor network states (TTNS) have been successfully used for two-dimensional systems and to benchmark quantum simulation approaches for condensed matter, nuclear, and particle physics. In comparison to the more traditional approach based on matrix product states (MPS), the graph distance of physical degrees of freedom can be drastically reduced in TTNS. Surprisingly, it turns out that, for large systems in $D>1$ spatial dimensions, MPS simulations of low-energy states are nevertheless more efficient than TTNS simulations. With a focus on $D=2$ and 3, the scaling of computational costs for different boundary conditions is determined under the assumption that the system obeys an entanglement (log-)area law, implying that bond dimensions scale exponentially in the surface area of the associated subsystems.

quant-pharXiv:2601.08132•Jan 14, 2026•0 citations

Nearly Tight Bounds for Testing Tree Tensor Network States

Benjamin Lovitz, Angus Lowe

Tree tensor network states (TTNS) generalize the notion of having low Schmidt-rank to multipartite quantum states, through a parameter known as the bond dimension. This leads to succinct representations of quantum many-body systems with a tree-like entanglement structure. In this work, we study the task of testing whether an unknown pure state is a TTNS onnqudits with bond dimension at mostr, or is far in trace distance from any such state. We first establish that, independent of the dimension of the state,O(nr2) copies suffice to accomplish this task with onesided error. We then prove that Ω(nr2/ logn) copies are necessary for any test with one-sided error wheneverr≥ 2 + logn. In particular, this closes a roughly quadratic gap in the previous bounds for testing matrix product states in this setting. On the other hand, whenr= 2 we show that Θ(√n) copies are both necessary and sufficient for the related task of testing whether a state is a product ofnbipartite states having Schmidt-rank at mostr, for some choice of the qudit dimensions. We also study the performance of tests using measurements performed on a small number of copies at a time. Here, we obtain new bounds for testing rank, Schmidt-rank, and TTNS when the tester is restricted to making measurements onr+ 1 copies of the state.

arXiv:2410.21417•IEEE Transactions on Information Theory•Mar 4, 2026•0 citations

Tensor Network States with Low-Rank Tensors

Hao Chen, Thomas Barthel

Tensor networks are used to efficiently approximate states of strongly-correlated quantum many-body systems. More generally, tensor network approximations may allow to reduce the costs for operating on an order-$N$ tensor from exponential to polynomial in $N$, and this has become a popular approach for machine learning. We introduce the idea of imposing low-rank constraints on the tensors that compose the tensor network. With this modification, the time and space complexities for the network optimization can be substantially reduced while maintaining high accuracy. We detail this idea for tree tensor network states (TTNS) and projected entangled-pair states. Simulations of spin models on Cayley trees with low-rank TTNS exemplify the effect of rank constraints on the expressive power. We find that choosing the tensor rank $r$ to be on the order of the bond dimension $m$, is sufficient to obtain high-accuracy groundstate approximations and to substantially outperform standard TTNS computations. Thus low-rank tensor networks are a promising route for the simulation of quantum matter and machine learning on large data sets.

arXiv:2205.15296•May 30, 2022•1 citation

Efficient tree tensor network states (TTNS) for quantum chemistry: Generalizations of the density matrix renormalization group algorithm

Naoki Nakatani, Garnet Kin-Lic Chan

We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general molecules. We describe an optimal tree tensor network state algorithm for quantum chemistry. We introduce the concept of half-renormalization which greatly improves the efficiency of the calculations. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states versus matrix product states. We carry out benchmark calculations both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree molecules, whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling associated with tree algorithms. In tree like molecules, tree network states are easily superior to matrix product states. As an illustration, our largest dendrimer calculation with tree tensor network states correlates 110 electrons in 110 active orbitals.

arXiv:1302.2298•The Journal of Chemical Physics•Apr 4, 2013•184 citations

Proper Agnostic Learning of Matrix Product States and Tree Tensor Networks

Constantin Cedillo Vayson de Pradenne, Jordan Cotler

We establish proper agnostic learning of matrix product states and tree tensor networks. Given copies of an arbitrary quantum state $ρ$, our algorithms return a state $|ψ\rangle$ of chosen bond dimension such that $\langleψ| ρ|ψ\rangle$ is within $\varepsilon$ of the optimum over the model class, without assuming that $ρ$ itself belongs to or is well approximated by that class. The main idea is to use improper learning to compress the mixed-state objective, reducing proper agnostic learning to optimization against a finite collection of explicitly specified pure states. We then introduce a comparator-dual compression procedure that reduces the bond dimension of these targets while uniformly preserving their overlaps with all bounded-bond comparators, with an error independent of system size. For matrix product states, this gives polynomial copy complexity in the system size, local dimension, bond dimension, and $1/\varepsilon$, together with polynomial runtime in the system size for fixed local dimension, bond dimension, and accuracy. The same framework yields proper agnostic learning of tree tensor networks on bounded-degree trees, with both copy complexity and runtime polynomial in the system size when the remaining parameters are fixed.

quant-pharXiv:2609.30148•Sep 25, 2026•0 citations

Entanglement bipartitioning and tree tensor networks

Kouichi Okunishi, Hiroshi Ueda, Tomotoshi Nishino

Abstract We propose the entanglement bipartitioning approach to design an optimal network structure of the tree tensor network (TTN) for quantum many-body systems. Given an exact ground-state wavefunction, we perform sequential bipartitioning of spin-cluster nodes so as to minimize the mutual information or the maximum loss of the entanglement entropy associated with the branch to be bipartitioned. We demonstrate that entanglement bipartitioning of up to 16 sites gives rise to nontrivial tree network structures for S = 1/2 Heisenberg models in one and two dimensions. The resulting TTNs enable us to obtain better variational energies, compared with standard TTNs such as the uniform matrix product state and perfect binary tree tensor network.

arXiv:2210.11741•Progress of Theoretical and Experimental Physics•Jan 28, 2023•15 citations
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<1α<1α<1 Rényi entanglement entropies for single-branch cuts obey an area law, as in ground and low-energy states of certain gapped systems.
Tensor network states are an indispensable tool for the simulation of strongly correlated quantum many-body systems. In recent years, tree tensor network states (TTNS) have been successfully used for two-dimensional systems and to benchmark quantum simulation approaches for condensed matter, nuclear, and particle physics. In comparison to the more traditional approach based on matrix product states (MPS), the graph distance of physical degrees of freedom can be drastically reduced in TTNS. Surprisingly, it turns out that, for large systems in D>1D>1D>1 spatial dimensions, MPS simulations of low-energy states are nevertheless more efficient than TTNS simulations. With a focus on D=2D=2D=2 and 3, the scaling of computational costs for different boundary conditions is determined under the assumption that the system obeys an entanglement (log-)area law, implying that bond dimensions scale exponentially in the surface area of the associated subsystems.
Tree tensor network states (TTNS) generalize the notion of having low Schmidt-rank to multipartite quantum states, through a parameter known as the bond dimension. This leads to succinct representations of quantum many-body systems with a tree-like entanglement structure. In this work, we study the task of testing whether an unknown pure state is a TTNS onnqudits with bond dimension at mostr, or is far in trace distance from any such state. We first establish that, independent of the dimension of the state,O(nr2) copies suffice to accomplish this task with onesided error. We then prove that Ω(nr2/ logn) copies are necessary for any test with one-sided error wheneverr≥ 2 + logn. In particular, this closes a roughly quadratic gap in the previous bounds for testing matrix product states in this setting. On the other hand, whenr= 2 we show that Θ(√n) copies are both necessary and sufficient for the related task of testing whether a state is a product ofnbipartite states having Schmidt-rank at mostr, for some choice of the qudit dimensions. We also study the performance of tests using measurements performed on a small number of copies at a time. Here, we obtain new bounds for testing rank, Schmidt-rank, and TTNS when the tester is restricted to making measurements onr+ 1 copies of the state.
Tensor networks are used to efficiently approximate states of strongly-correlated quantum many-body systems. More generally, tensor network approximations may allow to reduce the costs for operating on an order-NNN tensor from exponential to polynomial in NNN, and this has become a popular approach for machine learning. We introduce the idea of imposing low-rank constraints on the tensors that compose the tensor network. With this modification, the time and space complexities for the network optimization can be substantially reduced while maintaining high accuracy. We detail this idea for tree tensor network states (TTNS) and projected entangled-pair states. Simulations of spin models on Cayley trees with low-rank TTNS exemplify the effect of rank constraints on the expressive power. We find that choosing the tensor rank rrr to be on the order of the bond dimension mmm, is sufficient to obtain high-accuracy groundstate approximations and to substantially outperform standard TTNS computations. Thus low-rank tensor networks are a promising route for the simulation of quantum matter and machine learning on large data sets.
We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general molecules. We describe an optimal tree tensor network state algorithm for quantum chemistry. We introduce the concept of half-renormalization which greatly improves the efficiency of the calculations. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states versus matrix product states. We carry out benchmark calculations both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree molecules, whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling associated with tree algorithms. In tree like molecules, tree network states are easily superior to matrix product states. As an illustration, our largest dendrimer calculation with tree tensor network states correlates 110 electrons in 110 active orbitals.
We establish proper agnostic learning of matrix product states and tree tensor networks. Given copies of an arbitrary quantum state ρρρ, our algorithms return a state ∣ψ⟩|ψ\rangle∣ψ⟩ of chosen bond dimension such that ⟨ψ∣ρ∣ψ⟩\langleψ| ρ|ψ\rangle⟨ψ∣ρ∣ψ⟩ is within ε\varepsilonε of the optimum over the model class, without assuming that ρρρ itself belongs to or is well approximated by that class. The main idea is to use improper learning to compress the mixed-state objective, reducing proper agnostic learning to optimization against a finite collection of explicitly specified pure states. We then introduce a comparator-dual compression procedure that reduces the bond dimension of these targets while uniformly preserving their overlaps with all bounded-bond comparators, with an error independent of system size. For matrix product states, this gives polynomial copy complexity in the system size, local dimension, bond dimension, and 1/ε1/\varepsilon1/ε, together with polynomial runtime in the system size for fixed local dimension, bond dimension, and accuracy. The same framework yields proper agnostic learning of tree tensor networks on bounded-degree trees, with both copy complexity and runtime polynomial in the system size when the remaining parameters are fixed.
Abstract We propose the entanglement bipartitioning approach to design an optimal network structure of the tree tensor network (TTN) for quantum many-body systems. Given an exact ground-state wavefunction, we perform sequential bipartitioning of spin-cluster nodes so as to minimize the mutual information or the maximum loss of the entanglement entropy associated with the branch to be bipartitioned. We demonstrate that entanglement bipartitioning of up to 16 sites gives rise to nontrivial tree network structures for S = 1/2 Heisenberg models in one and two dimensions. The resulting TTNs enable us to obtain better variational energies, compared with standard TTNs such as the uniform matrix product state and perfect binary tree tensor network.