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arXiv:1801.09998 · Journal of Chemical Theory and Computation

T3NS: Three-Legged Tree Tensor Network States

Klaas Gunst, Frank Verstraete, Sebastian Wouters, Örs Legeza, Dimitri Van Neck

Published Feb 26, 2018• 64 citations
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Abstract

Abstract We present a new variational tree tensor network state (TTNS) ansatz, the three-legged tree tensor network state (T3NS). Physical tensors are interspersed with branching tensors. Physical tensors have one physical index and at most two virtual indices, as in the matrix product state (MPS) ansatz of the density matrix renormalization group (DMRG). Branching tensors have no physical index, but up to three virtual indices. In this way, advantages of DMRG, in particular a low computational cost and a simple implementation of symmetries, are combined with advantages of TTNS, namely incorporating more entanglement. Our code is capable of simulating quantum chemical Hamiltonians, and we present several proof-of-principle calculations on LiF, N2, and the bis(μ-oxo) and μ–η2:η2 peroxo isomers of [Cu2O2]2+.

DOI
10.1021/acs.jctc.8b00098

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Abstract We present a new variational tree tensor network state (TTNS) ansatz, the three-legged tree tensor network state (T3NS). Physical tensors are interspersed with branching tensors. Physical tensors have one physical index and at most two virtual indices, as in the matrix product state (MPS) ansatz of the density matrix renormalization group (DMRG). Branching tensors have no physical index, but up to three virtual indices. In this way, advantages of DMRG, in particular a low computational cost and a simple implementation of symmetries, are combined with advantages of TTNS, namely incorporating more entanglement. Our code is capable of simulating quantum chemical Hamiltonians, and we present several proof-of-principle calculations on LiF, N2, and the bis(μ-oxo) and μ–η2:η2 peroxo isomers of [Cu2O2]2+.
We extend the three-legged tree tensor network state (T3NS) [ J. Chem. Theory Comput. 2018, 14, 2026−2033] by including spin and the real abelian point group symmetries. T3NS intersperses physical tensors with branching tensors. Physical tensors have one physical index and at most two virtual indices. Branching tensors have up to three virtual indices and no physical index. In this way, T3NS combines the low computational cost of matrix product states and their simplicity for implementing symmetries, with the better entanglement representation of tree tensor networks. By including spin and point group symmetries, more accurate calculations can be obtained with lower computational effort. We illustrate this by presenting calculations on the bis(μ-oxo) and μ–η 2:η 2 peroxo isomers of [Cu 2 O 2 ] 2+ . The used implementation is available on github .
We study the tree-tensor-network-state (TTNS) method with variable tensor orders for quantum chemistry. TTNS is a variational method to efficiently approximate complete active space (CAS) configuration interaction (CI) wave functions in a tensor product form. TTNS can be considered as a higher order generalization of the matrix product state (MPS) method. The MPS wave function is formulated as products of matrices in a multiparticle basis spanning a truncated Hilbert space of the original CAS-CI problem. These matrices belong to active orbitals organized in a one-dimensional array, while tensors in TTNS are defined upon a tree-like arrangement of the same orbitals. The tree-structure is advantageous since the distance between two arbitrary orbitals in the tree scales only logarithmically with the number of orbitals N, whereas the scaling is linear in the MPS array. It is found to be beneficial from the computational costs point of view to keep strongly correlated orbitals in close vicinity in both arrangements; therefore, the TTNS ansatz is better suited for multireference problems with numerous highly correlated orbitals. To exploit the advantages of TTNS a novel algorithm is designed to optimize the tree tensor network topology based on quantum information theory and entanglement. The superior performance of the TTNS method is illustrated on the ionic-neutral avoided crossing of LiF. It is also shown that the avoided crossing of LiF can be localized using only ground state properties, namely one-orbital entanglement.
We introduce a class of tensor-network states in which physical legs are shared among local tensors, termed leg-tied tensor ansätze (LETTA). Physical leg ties encode long-range correlations directly, while a virtual matrix product state (MPS) backbone retains short-range multipartite entanglement. The linear virtual backbone allows us to develop a deterministic density matrix renormalization group-like variational optimization algorithm using exact contractions over the active tie-boundary sets and local minimization. We demonstrate the advantages of LETTA for the two-dimensional frustrated J1J_1J1​--J2J_2J2​ Heisenberg model and the three-dimensional transverse-field Ising model. Our results show that LETTA is substantially more accurate than same-bond-dimension MPS calculations and can typically reach the accuracy of much larger MPS calculations using one order of magnitude fewer variational parameters. LETTA thus opens the door for explicitly correlated tensor-network states that can encode long-range correlation beyond virtual bonds.
We present how to compute vibrational eigenstates with tree tensor network states (TTNSs), the underlying ansatz behind the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method. The eigenstates are computed with an algorithm that is based on the density matrix renormalization group (DMRG). We apply this to compute the vibrational spectrum of acetonitrile (CH3CN) to high accuracy and compare TTNSs with matrix product states (MPSs), the ansatz behind the DMRG. The presented optimization scheme converges much faster than ML-MCTDH-based optimization. For this particular system, we found no major advantage of the more general TTNS over MPS. We highlight that for both TTNS and MPS, the usage of an adaptive bond dimension significantly reduces the amount of required parameters. We furthermore propose a procedure to find good trees.
We develop and employ general Tree Tensor Networks (TTNs) to compute the vibrational spectra for two model systems: a set of 64-dimensional coupled oscillators and acetonitrile. We explore various tree architectures, ranging from the simple linear structure of Matrix Product States (MPS), to trees where only the leaf nodes carry a physical leg -- as commonly seen in the underlying ansatz of the Multilayer Multiconfiguration Time-Dependent Hartree (ML-MCTDH) method -- and further to more general trees in which all nodes are allowed to possess a physical leg. In addition, we implement Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) methods and Inverse Iteration methods as eigensolvers. Benchmarking runtime and accuracy shows that all tested topologies can reach high accuracy. For acetonitrile, inverse-iteration refinement brings all 84 computed states below 1~cm−1^{-1}−1 error, while the fork-4 tree, a comb-like tree with four backbone nodes, provides the best overall balance between accuracy and cost. MPS remains computationally attractive, whereas more connected trees generally improve accuracy at fixed bond dimension. All numerical simulations were performed using PyTreeNet, a Python package designed for flexible tensor network computations.
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.