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arXiv:2505.05908 · Computer Physics Communications

TTNOpt: Tree tensor network package for high-rank tensor compression

Ryo Watanabe, Hidetaka Manabe, Toshiya Hikihara, Hiroshi Ueda

Published Oct 30, 2025• 2 citations
arXivHTMLPDF

Abstract

We have developed TTNOpt, a software package that utilizes tree tensor networks (TTNs) for quantum spin systems and high-dimensional data analysis. TTNOpt provides efficient and powerful TTN computations by locally optimizing the network structure, guided by the entanglement pattern of the target tensors. For quantum spin systems, TTNOpt searches for the ground state of Hamiltonians with bilinear spin interactions and magnetic fields, and computes physical properties of these states, including the variational energy, bipartite entanglement entropy (EE), single-site expectation values, and two-site correlation functions. Additionally, TTNOpt can target the lowest-energy state within a specified subspace, provided that the Hamiltonian conserves total magnetization. For high-dimensional data analysis, TTNOpt factorizes complex tensors into TTN states that maximize fidelity to the original tensors by optimizing the tensors and the network. When a TTN is provided as input, TTNOpt reconstructs the network based on the EE without referencing the fidelity of the original state. We present three demonstrations of TTNOpt:(1) Ground-state search for the hierarchical chain model with a system size of $256$. The entanglement patterns of the ground state manifest themselves in a tree structure, and TTNOpt successfully identifies the tree.(2) Factorization of a quantic tensor of the $2^{24}$ dimensions representing a three-variable function where each variant has a weak bit-wise correlation. The optimized TTN shows that its structure isolates the variables from each other.(3) Reconstruction of the matrix product network representing a $16$-variable normal distribution characterized by a tree-like correlation structure. TTNOpt can reveal hidden correlation structures of the covariance matrix.

DOI
10.1016/j.cpc.2025.109918

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We have developed TTNOpt, a software package that utilizes tree tensor networks (TTNs) for quantum spin systems and high-dimensional data analysis. TTNOpt provides efficient and powerful TTN computations by locally optimizing the network structure, guided by the entanglement pattern of the target tensors. For quantum spin systems, TTNOpt searches for the ground state of Hamiltonians with bilinear spin interactions and magnetic fields, and computes physical properties of these states, including the variational energy, bipartite entanglement entropy (EE), single-site expectation values, and two-site correlation functions. Additionally, TTNOpt can target the lowest-energy state within a specified subspace, provided that the Hamiltonian conserves total magnetization. For high-dimensional data analysis, TTNOpt factorizes complex tensors into TTN states that maximize fidelity to the original tensors by optimizing the tensors and the network. When a TTN is provided as input, TTNOpt reconstructs the network based on the EE without referencing the fidelity of the original state. We present three demonstrations of TTNOpt:(1) Ground-state search for the hierarchical chain model with a system size of 256256256. The entanglement patterns of the ground state manifest themselves in a tree structure, and TTNOpt successfully identifies the tree.(2) Factorization of a quantic tensor of the 2242^{24}224 dimensions representing a three-variable function where each variant has a weak bit-wise correlation. The optimized TTN shows that its structure isolates the variables from each other.(3) Reconstruction of the matrix product network representing a 161616-variable normal distribution characterized by a tree-like correlation structure. TTNOpt can reveal hidden correlation structures of the covariance matrix.
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.
The tree tensor network (TTN) provides an essential theoretical framework for the practical simulation of quantum many-body systems, where the network structure defined by the connectivity of the isometry tensors plays a crucial role in improving its approximation accuracy. In this paper, we propose a TTN algorithm that enables us to automatically optimize the network structure by local reconnections of isometries to suppress the bipartite entanglement entropy on their legs. The algorithm can be seamlessly implemented to such a conventional TTN approach as the density-matrix renormalization group. We apply the algorithm to the inhomogeneous antiferromagnetic Heisenberg spin chain, having a hierarchical spatial distribution of the interactions. We then demonstrate that the entanglement structure embedded in the ground state of the system can be efficiently visualized as a perfect binary tree in the optimized TTN. Possible improvements and applications of the algorithm are also discussed.
We present the Python Tree Tensor Network (pyTTN) package for the evaluation of dynamical properties of closed and open quantum systems that makes use of Tree Tensor Network (TTN) based representations of wave functions. This package includes several features allowing for easy setup of zero- and finite-temperature calculations for general Hamiltonians using single and multi-set TTN ansätze with an adaptive bond dimension using subspace expansion techniques. In addition to these core features, pyTTN provides several tools for setting up efficient simulations of open quantum system dynamics, including the use of the TTN ansatz to represent the auxiliary density operator space for the simulation of the hierarchical equation of motion method and generalized quasi-Lindblad pseudomode methods. We present a set of applications for the package, starting with the widely used benchmark case of the photo-excitation dynamics of 24 mode pyrazine, after which we consider a more challenging model describing the exciton dynamics at the interface of an n-oligothiophene donor-C60 fullerene acceptor system. Finally, we consider applications to open quantum systems, including the spin-boson model, a set of extended dissipative spin models, and an Anderson impurity model. By combining ease of use and an efficient implementation, along with an extendable design that allows for the addition of future extensions, pyTTN can be integrated into a wide range of computational modeling software.
Abstract. Compactly representing and efficently applying linear operators are fundamental ingredients in tensor network methods for simulating quantum many-body problems and solving high-dimensional problems in scientific computing. In this work, we study such representations for tree tensor networks, the so-called tree tensor network operators (TTNOs), paying particular attention to Hamiltonian operators that involve long-range pairwise interactions between particles. Generalizing the work by Lin, Tong, and others on matrix product operators, we establish a direct connection between the hierarchical low-rank structure of the interaction matrix and the TTNO property. This connection allows us to arrive at very compact TTNO representations by compressing the interaction matrix into a hierarchically semiseparable matrix. Numerical experiments for different quantum spin systems validate our results and highlight the potential advantages of TTNOs over matrix product operators.
The quantum state preparation of probability distributions is an important subroutine for many quantum algorithms. When embedding D -dimensional multivariate probability distributions by discretizing each dimension into 2n points, we need a state preparation circuit comprising a total of nD qubits, which is often difficult to compile. In this study, we propose a scalable method to generate state preparation circuits for D -dimensional multivariate normal distributions, utilizing tree tensor networks (TTN). We establish theoretical guarantees that multivariate normal distributions with 1D correlation structures can be efficiently represented using TTN. Based on these analyses, we propose a compilation method that uses automatic structural optimization to find the most efficient network structure and compact circuit. We apply our method to state preparation circuits for various high-dimensional random multivariate normal distributions. The numerical results suggest that our method can dramatically reduce the circuit depth and CNOT count while maintaining fidelity compared to existing approaches.
We present the Python Tree Tensor Network package (pyTTN) for the evaluation of dynamical properties of closed and open quantum systems that makes use of Tree Tensor Network (TTN), or equivalently the multi-layer multiconfiguration time-dependent Hartree (ML-MCTDH), based representations of wavefunctions. This package includes several features allowing for easy setup of zero- and finite-temperature calculations for general Hamiltonians using single and multi-set TTN ansätze with an adaptive bond dimension through the use of subspace expansion techniques. All core features are implemented in C++ with Python bindings provided to simplify the use of this package. In addition to these core features, pyTTN provides several tools for setting up efficient simulation of open quantum system dynamics, including the use of the TTN ansatz to represent the auxiliary density operator space for the simulation of the Hierarchical Equation of Motion (HEOM) method and generalised pseudomode methods; furthermore we demonstrate that the two approaches are equivalent up to a non-unitary normal mode transformation acting on the pseudomode degrees of freedom. We present a set of applications of the package, starting with the widely used benchmark case of the photo-excitation dynamics of 24 mode pyrazine, following which we consider a more challenging model describing the exciton dynamics at the interface of a nnn-oligothiophene donor-C60_{60}60​ fullerene acceptor system. Finally, we consider applications to open quantum systems, including the spin-boson model, a set of extended dissipative spin models, and an Anderson impurity model. By combining ease of use, an efficient implementation, as well as an extendable design allowing for the addition of future extensions, pyTTN can be integrated in a wide range of computational modelling software.