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arXiv:2201.12823 · Physical review. B./Physical review. B

Functional tensor network solving many-body Schrödinger equation

Rui Hong, Ya-Xuan Xiao, Jie Hu, An-Chun Ji, Shi-Ju Ran

Published Apr 12, 2022• 12 citations
arXivHTMLPDF

Abstract

Solving the many-body Schr\"odinger equation in continuous spaces with the presence of strong correlations is an extremely important and challenging issue in quantum physics. In this work, we propose the functional tensor network (FTN) approach to solve the many-body Schr\"odinger equation. Provided the orthonormal functional bases, we represent the coefficients of the many-body wave function as a tensor network. The observables, such as energy, can be calculated simply by tensor contractions. Simulating the ground state becomes solving a minimization problem defined by the tensor network. An efficient gradient-decent algorithm based on automatically differentiable tensors is proposed. We here take the matrix product state (MPS), whose complexity scales only linearly with the system size, as an example. We apply our approach to solve the ground state of coupled harmonic oscillators and achieve high accuracy by comparing our results with the exact solutions. Reliable results are also given in the presence of three-body interactions, where the system cannot be decoupled from isolated oscillators. Our approach is simple and, with well-controlled error, essentially different from highly nonlinear neural-network solvers. Our work extends the applications of the tensor network from quantum lattice models to systems in continuous space. The FTN can be used as a general solver of the differential equations with many variables. The MPS exemplified here can be generalized to, e.g., fermionic tensor networks to solve the electronic Schr\"odinger equation.

DOI
10.1103/physrevb.105.165116

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Solving the many-body Schr"odinger equation in continuous spaces with the presence of strong correlations is an extremely important and challenging issue in quantum physics. In this work, we propose the functional tensor network (FTN) approach to solve the many-body Schr"odinger equation. Provided the orthonormal functional bases, we represent the coefficients of the many-body wave function as a tensor network. The observables, such as energy, can be calculated simply by tensor contractions. Simulating the ground state becomes solving a minimization problem defined by the tensor network. An efficient gradient-decent algorithm based on automatically differentiable tensors is proposed. We here take the matrix product state (MPS), whose complexity scales only linearly with the system size, as an example. We apply our approach to solve the ground state of coupled harmonic oscillators and achieve high accuracy by comparing our results with the exact solutions. Reliable results are also given in the presence of three-body interactions, where the system cannot be decoupled from isolated oscillators. Our approach is simple and, with well-controlled error, essentially different from highly nonlinear neural-network solvers. Our work extends the applications of the tensor network from quantum lattice models to systems in continuous space. The FTN can be used as a general solver of the differential equations with many variables. The MPS exemplified here can be generalized to, e.g., fermionic tensor networks to solve the electronic Schr"odinger equation.
In this paper, we introduce a novel approach to solve the many-body Schrodinger equation by the tensor neural network. Based on the tensor product structure, we can do the direct numerical integration by using fixed quadrature points for the functions constructed by the tensor neural network within tolerable computational complexity. Especially, we design several types of efficient numerical methods to treat the variable-coupled Coulomb potentials with high accuracy. The corresponding machine learning method is built for solving many-body Schrodinger equation. Some numerical examples are provided to validate the accuracy and efficiency of the proposed algorithms.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
An improved version of the sum-of-Gaussians tensor neural network (SOG-TNN) architecture is presented for solving the many-electron Schrödinger equation for one-dimensional soft-Coulomb systems. Model reduction techniques are introduced to reduce the number of tensor-factorized bases under the SOG approximation of the kernel. The Slater determinant ansatz is employed so that the antisymmetric property of the wave function can be strictly preserved. Numerical results show that the SOG-TNN achieves high accuracy with remarkably small basis sizes. Robust spectral convergence with respect to the basis size is also observed, consistently characterized by a mixed algebraic-exponential model for the error decay. These findings validate that the SOG-TNN architecture provides an ultra-efficient and low-rank representation of complex multi-electron wave functions, shedding light on high-fidelity quantum calculations in larger-scale many-electron systems.
We introduce Neural Tensor Network States (νννTNS), a variational many-body wave-function ansatz that integrates deep neural networks with tensor-network architectures. In the νννTNS framework, a neural network serves as a disentangler of the wave-function, transforming the physical degrees of freedom into renormalized variables with much less entanglement. The renormalized state is then efficiently encoded by a back-flow tensor network. This construction yields a compact yet highly expressive representation of strongly correlated quantum states. Using convolutional neural networks combined with matrix product states as a concrete implementation, we obtain state-of-the-art variational energies for the spin-1/21/21/2 J1J_1J1​-J2J_2J2​ Heisenberg model on the square lattice at the highly frustrated point J2/J1=0.5J_2/J_1=0.5J2​/J1​=0.5, for systems up to 20×2020\times 2020×20 with periodic boundary conditions. Finite-size scaling of spin, dimer, and plaquette correlations exhibits power-law decay without magnetic or valence-bond long-range order, consistent with a gapless quantum spin-liquid ground state at that point.This νννTNS framework is flexible and naturally extensible to other neural and tensor-network structures, offering a general platform for investigating strongly correlated quantum many-body systems.
We present an improved version of the sum-of-Gaussians tensor neural network (SOG-TNN) architecture for solving many-electron Schrödinger equation for one-dimensional soft-Coulomb systems. Model reduction techniques are introduced to reduce the number of tensor-factorized bases under the SOG approximation of the kernel. The Slater determinant ansatz is employed so that the anti-symmetric property of the wave function can be strictly preserved. Numerical results show that the SOG-TNN achieves high accuracy with remarkably small basis sizes. Robust spectral convergence with respect to the basis size is also observed, consistently characterized by a mixed algebraic-exponential model for the error decay. These findings validate that the SOG-TNN architecture provides an ultra-efficient and low-rank representation of complex multi-electron wave functions, shedding light on high-fidelity quantum calculations in larger-scale many-electron systems.