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arXiv:2210.11741 · Progress of Theoretical and Experimental Physics

Entanglement bipartitioning and tree tensor networks

Kouichi Okunishi, Hiroshi Ueda, Tomotoshi Nishino

Published Jan 28, 2023• 15 citations
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Abstract

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.

DOI
10.1093/ptep/ptad018

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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 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.
We investigate the disordered spin-12\frac1221​Heisenberg model in two dimensions and employ tree tensor networks (TTNs) with a physics-informed structural optimization of the tree layout, to simulate dynamics in the many-body localization problem. We find that TTNs are able to capture two-dimensional entanglement patterns more effectively than matrix product states (MPS) while being more efficient to contract than projected entangled pair states (PEPS) to probe larger systems and longer times. Structural optimization of the trees based on time evolution of the entanglement in the system allows to keep the necessary bond dimensions low and to maximally exploit the increased expressiveness of TTNs over MPS. In this way, we achieve more accurate results in all considered parameter regimes both below and above the ergodicity-to-localization crossover at a comparable compute-time cost.
Tensor networks provide a powerful and elegant approach to quantum manybody simulation. The simplest example is the density matrix renormalisation group (DMRG), which is based on the variational update of a matrix product state (MPS). It has proved to be the most accurate approach for the numerical study of strongly correlated one dimensional systems. We use DMRG to study the one dimensional disordered Bose-Hubbard model at fillings N=L = 1=2, 1 and 2 and show that the whole phase diagram for each can be successfully obtained by analysing entanglement properties alone. We �nd that the average entanglement is insufficient to accurately locate all of the phases, however using the standard error on the mean we are able to construct a phase diagram that is consistent with previous studies. It has recently been shown that there is a connection between the geometry of tensor networks and the entanglement and correlation properties that it can encode, which is a generalisation of the so called area law for entanglement entropy. This suggests that whilst gapped quantum systems can be accurately modeled using an MPS, a tensor network with a holographic geometry is natural to capture the logarithmic entanglement scaling and power law decaying correlation functions of critical systems. We create an algorithm for the disordered Heisenberg Hamiltonian that self assembles a tensor network based on the disorder in the couplings. The geometry created is that of a disordered tree tensor network (TTN) that when averaged has the holographic properties characteristic of critical systems. We continue the analysis of holographic tensor network geometry by considering the average length of leaf-to-leaf paths in various tree graphs, which is related to two-point correlation functions in tensor networks. For regular, complete trees we analytically calculate the average path length and all statistical moments, and generalise it for any splitting number. We then turn to the Catalan trees, which is the set of unique binary trees with n vertices, as it has a similar geometry to the disordered TTNs. We calculate the average depth of a leaf and show that it is equal to the average path length. We compare these analytic results with the structures found in the TTN and randomly constructed trees to show that the renormalisation involved in the TTN algorithm is crucial in the selection of the tree structure.
Tensor network states (TNSs) offer an efficient representation for the ground state of quantum many-body systems and play an important role in the simulations of them. Numerous TNSs have been proposed in the past few decades. However, due to the high cost of TNS for two-dimensional systems, a balance between the encoded entanglement and computational complexity of TNS is yet to be reached. In this work we introduce a new tree tensor network-based (TTN-based) TNS dubbed a ``fully augmented tree tensor network (FATTN) by releasing the constraint in augmented tree tensor network (ATTN). When disentanglers are augmented in the physical layer of TTN, FATTN can provide more entanglement than TTN and ATTN. At the same time, FATTN maintains the scaling of computational cost with bond dimension in TTN and ATTN. Benchmark results on the ground-state energy for the transverse Ising model are provided to demonstrate the improvement of accuracy of FATTN over TTN and ATTN. Moreover, FATTN is quite flexible, which can be constructed as an interpolation between TTN and the multiscale entanglement renormalization ansatz to reach a balance between the encoded entanglement and the computational cost.
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.