Functional Tensor Network Solving Many-body Schrödinger Equation

Abstract

Schrödinger equation belongs to the most fundamental differential equations in quantum physics. However, the exact solutions are extremely rare, and many analytical methods are applicable only to the cases with small perturbations or weak correlations. Solving the many-body Schrödinger equation in the continuous spaces with the presence of strong correlations is an extremely important and challenging issue. In this work, we propose the functional tensor network (FTN) approach to solve the many-body Schrödinger equation. Provided the orthonormal functional bases, we represent the coefficients of the many-body wave-function as 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 the automatically differentiable tensors is proposed. We here take matrix product state (MPS) as an example, whose complexity scales only linearly with the system size. We apply our approach to solve the ground state of coupled harmonic oscillators, and achieve high accuracy by comparing with the exact solutions. Reliable results are also given with the presence of three-body interactions, where the system cannot be decoupled to isolated oscillators. Our approach is simple and with well-controlled error, superior to the highly-nonlinear neural-network solvers. Our work extends the applications of tensor network from quantum lattice models to the systems in the continuous space. 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., the fermionic tensor networks, to solve the electronic Schrödinger equation.

arXiv comment9 pages, 6 figures

References in corpus (41)

• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Area laws for the entanglement entropy - a review
• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Matrix product states represent ground states faithfully
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Criticality, the area law, and the computational power of PEPS
• Tensor networks for complex quantum systems
• Learning data driven discretizations for partial differential equations
• Ab Initio Quantum Chemistry using the Density Matrix Renormalization Group
• Deep neural network solution of the electronic Schrödinger equation
• Simulating quantum computation by contracting tensor networks
• Studying Two Dimensional Systems With the Density Matrix Renormalization Group
• Renormalization and tensor product states in spin chains and lattices
• Simulation of strongly correlated fermions in two spatial dimensions with fermionic Projected Entangled-Pair States
• PEPS as ground states: degeneracy and topology
• Tensor network decompositions in the presence of a global symmetry
• Continuous Matrix Product States for Quantum Fields
• Fermionic neural-network states for ab-initio electronic structure
• Differentiable Programming Tensor Networks
• Solving Many-Electron Schrödinger Equation Using Deep Neural Networks
• Lecture Notes of Tensor Network Contractions
• Contraction of fermionic operator circuits and the simulation of strongly correlated fermions
• A Quantum Inspired Approach to Exploit Turbulence Structures
• Simulation of fermionic lattice models in two dimensions with Projected Entangled-Pair States: Next-nearest neighbor Hamiltonians
• Calculus of continuous matrix product states
• A canonical form for Projected Entangled Pair States and applications
• Fermionic Implementation of Projected Entangled Pair States Algorithm
• Coupled Oscillator Systems Having Partial PT Symmetry
• Multigrid Renormalization
• Multigrid Algorithms for Tensor Network States
• Sliced Basis Density Matrix Renormalization Group for Electronic Structure
• Proposal for Quantum Simulation via All-Optically Generated Tensor Network States
• Continuous Tensor Network States for Quantum Fields
• Continuum tensor network field states, path integral representations and the encoding of spatial symmetries
• Quantum field tomography
• Computational Power and Correlation in Quantum Computational Tensor Network
• Field Tensor Network States

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• Time evolution of controlled many-body quantum systems with matrix product operators
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