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arXiv:2002.07459 · SIAM Journal on Scientific Computing

Inversion Symmetry of Singular Values and a New Orbital Ordering Method in Tensor Train Approximations for Quantum Chemistry

Mi-Song Dupuy, Gero Friesecke

Published Jan 1, 2021• 6 citations
arXivHTMLPDF

Abstract

The tensor train (TT) approximation of electronic wave functions lies at the core of the quantum chemistry density matrix renormalization group (QC-DMRG) method, a recent state-of-the-art method for numerically solving the $N$-electron Schrödinger equation. It is well known that the accuracy of TT approximations is governed by the decay of the associated singular values, which in turn strongly depends on the ordering of the one-body basis. Here we find that the singular values $s_1\ge s_2\ge \cdots \ge s_d$ of tensors representing ground states of noninteracting Hamiltonians possess a surprising inversion symmetry, $s_1s_d=s_2s_{d-1}$$=s_3s_{d-2}=\ldots,$ thus reducing the tail behavior to the leading singular value and a single hidden invariant, which moreover depends explicitly on the ordering of the basis. For correlated wave functions, we find that the tail is upper bounded by a suitable superposition of the invariants. Optimizing the invariants or their superposition thus provides a new ordering scheme for QC-DMRG. Numerical tests on simple examples, i.e., linear combinations of a few Slater determinants, show that the new scheme reduces the tail of the singular values by several orders of magnitude over existing methods, including the widely used Fiedler order.

DOI
10.1137/20m1320122

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The tensor train (TT) approximation of electronic wave functions lies at the core of the quantum chemistry density matrix renormalization group (QC-DMRG) method, a recent state-of-the-art method for numerically solving the NNN-electron Schrödinger equation. It is well known that the accuracy of TT approximations is governed by the decay of the associated singular values, which in turn strongly depends on the ordering of the one-body basis. Here we find that the singular values s1≥s2≥⋯≥sds_1\ge s_2\ge \cdots \ge s_ds1​≥s2​≥⋯≥sd​ of tensors representing ground states of noninteracting Hamiltonians possess a surprising inversion symmetry, s_1s_d=s_2s_{d-1}$$=s_3s_{d-2}=\ldots, thus reducing the tail behavior to the leading singular value and a single hidden invariant, which moreover depends explicitly on the ordering of the basis. For correlated wave functions, we find that the tail is upper bounded by a suitable superposition of the invariants. Optimizing the invariants or their superposition thus provides a new ordering scheme for QC-DMRG. Numerical tests on simple examples, i.e., linear combinations of a few Slater determinants, show that the new scheme reduces the tail of the singular values by several orders of magnitude over existing methods, including the widely used Fiedler order.
The tensor train (TT) is a popular way of representing high-dimensional hyper-rectangular data structures called tensors. It is widely used, for example, in quantum chemistry under the name “matrix product state”. The complexity of the TT model mainly depends on the bond dimensions that connect TT cores, constituting the model. Unlike canonical polyadic decomposition, the TT model complexity may depend on the order of the modes/indices in the data structures or the order of the core tensors in the TT, in general. This letter aims to provide methods for optimizing the order of the modes to reduce the bond dimensions. Since the number of possible orderings of the cores is exponentially high, we propose a greedy algorithm that provides a suboptimal solution. We consider three problem setups, i.e., specifications of the tensor: tensor given by a list of all its elements, tensor described by a TT model with some default order of the modes, and tensor obtained by sampling a multivariate function.
This article compares the tensor method density matrix renormalization group (DMRG) with two neural network based methods -namely FermiNet and PauliNet) for determining the ground state wavefunction of the many-body electronic Schrödinger problem. We provide numerical simulations illustrating the main features of the methods and showing convergence with respect to some parameter, such as the rank for DMRG, and number of pretraining iterations for neural networks. We then compare the obtained energy with the methods for a few atoms and molecules, for some of which the exact value of the energy is known for the sake of comparison. In the last part of the article, we propose a new kind of neural network to solve the Schrödinger problem based on the training of the wavefunction on a simplex, and an explicit permutation for evaluating the wavefunction on the whole space. We provide numerical results on a toy problem for the sake of illustration.
This article compares the tensor method density matrix renormalization group (DMRG) with two neural network based methods -namely FermiNet and PauliNet) for determining the ground state wavefunction of the many-body electronic Schrödinger problem. We provide numerical simulations illustrating the main features of the methods and showing convergence with respect to some parameter, such as the rank for DMRG, and number of pretraining iterations for neural networks. We then compare the obtained energy with the methods for a few atoms and molecules, for some of which the exact value of the energy is known for the sake of comparison. In the last part of the article, we propose a new kind of neural network to solve the Schrödinger problem based on the training of the wavefunction on a simplex, and an explicit permutation for evaluating the wavefunction on the whole space. We provide numerical results on a toy problem for the sake of illustration.
Strongly correlated systems exhibit phenomena -- such as high-T_c superconductivity or the fractional quantum Hall effect -- that are not explicable by classical and semi-classical methods. Moreover, due to the exponential scaling of the associated Hilbert space, solving the proposed model Hamiltonians by brute-force numerical methods is bound to fail. Thus, it is important to develop novel numerical and analytical methods that can explain the physics in this regime. Tensor Network states are quantum many-body states that help to overcome some of these difficulties by defining a family of states that depend only on a small number of parameters. Their use is twofold: they are used as variational ansatzes in numerical algorithms as well as providing a framework to represent a large class of exactly solvable models that are believed to represent all possible phases of matter. The present thesis investigates mathematical properties of these states thus deepening the understanding of how and why Tensor Networks are suitable for the description of quantum many-body systems. It is believed that tensor networks can represent ground states of local Hamiltonians, but how good is this representation? This question is of fundamental importance as variational algorithms based on tensor networks can only perform well if any ground state can be approximated efficiently in such a way. While any state can be written as a tensor network state, the number of parameters needed for the description might be too large. This is not the case for one-dimensional systems: only a few parameters are required to have a good approximation of their ground states; that, in turn, allows for numerical algorithms based on tensor networks performing well. The situation in two dimensions is somewhat more complicated, but it is known that ground states of local Hamiltonians can be expressed as tensor networks with sub-exponentially many parameters. In the present thesis, we improve on these existing bounds strengthening the claim that the language of tensor networks is suitable to describe many-body systems. Another central question is how symmetries of the system such as translational invariance, time-reversal symmetry or local unitary symmetry can be reflected in tensor networks. This question is important as systems appearing in nature might intrinsically possess certain symmetries; on one hand, understanding these symmetries simplifies the description of these systems. On the other hand, the presence of symmetries leads to the appearance of novel phases -- symmetry-protected topological (SPT) order, -- and tensor networks provide the right language to classify these phases. In one dimension and for certain classes of two-dimensional tensor networks (states generated by so-called injective tensors) it is well understood how symmetries of the state can be described. A general framework, however, has yet to be developed. In the present thesis, we contribute to the development of the theory in two ways. We first investigate the question for injective tensors, and generalize the existing proof for any geometry including the hyperbolic geometry used in the AdS/CFT correspondence. Second, we introduce a class of tensor network states that include previously known examples of states exhibiting SPT order. We show how symmetries are reflected in these states thus deepening the understanding of SPT order in two dimensions.
Tensors arise naturally in high-dimensional problems in chemistry, financial mathematics, and many other areas. The numerical treatment of such problems is difficult due to the curse of dimensionality: the number of unknowns and the computational complexity grow exponentially with the dimension of the problem. To break the curse of dimensionality, low-parametric representations, or formats, have to be used. In this paper we make use of the TT-format (tensor-train format) which is one of the most effective stable representations of high-dimensional tensors. Basic linear algebra operations in the TT-format are now well developed. Our goal is to provide a “black-box” type of solver for linear systems where both the matrix and the right-hand side are in the TT-format. An efficient DMRG (density matrix renormalization group) method is proposed, and several tricks are employed to make it work. The numerical experiments confirm the effectiveness of our approach.
Abstract We present a fully numerical framework for the optimization of molecule-specific quantum chemical basis functions within the quantics tensor train format using a finite-difference scheme. The optimization is driven by solving the Hartree–Fock equations (HF) with the density-matrix renormalization group algorithm on Cartesian grids that are iteratively refined. In contrast to the standard way of tackling the mean-field problem by expressing the molecular orbitals as linear combinations of atomic orbitals (LCAO) our method only requires as much basis functions as there are electrons within the system. Benchmark calculations for atoms and molecules with up to ten electrons show excellent agreement with LCAO calculations with large basis sets supporting the validity of the tensor network approach. Our work therefore offers a promising alternative to well-established HF-solvers and could pave the way to define highly accurate, fully numerical, molecule-adaptive basis sets, which, in the future, could lead to benefits for post-HF calculations.