On the descriptive power of Neural-Networks as constrained Tensor Networks with exponentially large bond dimension

Abstract

In many cases, Neural networks can be mapped into tensor networks with an exponentially large bond dimension. Here, we compare different sub-classes of neural network states, with their mapped tensor network counterpart for studying the ground state of short-range Hamiltonians. We show that when mapping a neural network, the resulting tensor network is highly constrained and thus the neural network states do in general not deliver the naive expected drastic improvement against the state-of-the-art tensor network methods. We explicitly show this result in two paradigmatic examples, the 1D ferromagnetic Ising model and the 2D antiferromagnetic Heisenberg model, addressing the lack of a detailed comparison of the expressiveness of these increasingly popular, variational ansätze.

arXiv comment18 pages, 8 figures

References in corpus (38)

• The density-matrix renormalization group in the age of matrix product states
• Area laws for the entanglement entropy - a review
• Machine learning and the physical sciences
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• 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
• Efficient simulation of one-dimensional quantum many-body systems
• An Area Law for One Dimensional Quantum Systems
• Classical simulation of quantum many-body systems with a tree tensor network
• Finite-Size Scaling of the Ground State Parameters of the Two-Dimensional Heisenberg Model
• Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• Quantum Entanglement in Neural Network States
• Why does deep and cheap learning work so well?
• Renormalization and tensor product states in spin chains and lattices
• Restricted-Boltzmann-Machine Learning for Solving Strongly Correlated Quantum Systems
• Towards Quantum Machine Learning with Tensor Networks
• Machine Learning Topological States
• Tensor network states and geometry
• Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
• Learning Thermodynamics with Boltzmann Machines
• Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks
• Self-Learning Monte Carlo Method
• Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines
• Equivalence of restricted Boltzmann machines and tensor network states
• Study of the Two-Dimensional Frustrated J1-J2 Model with Neural Network Quantum States
• Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
• Neural-Network Quantum States, String-Bond States, and Chiral Topological States
• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
• Density Matrix Renormalization Group in the Heisenberg Picture
• Fractional quantum Hall effect in the interacting Hofstadter model via tensor networks
• Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks
• Investigating ultrafast quantum magnetism with machine learning
• Generalized Transfer Matrix States from Artificial Neural Networks
• Projective quantum Monte Carlo simulations guided by unrestricted neural network states
• Self-learning projective quantum Monte Carlo simulations guided by restricted Boltzmann machines
• Tensor networks-a new tool for old problems

Cited by in corpus (7)

• Tensor networks for quantum machine learning
• Intrinsic dimension of path integrals: data mining quantum criticality and emergent simplicity
• Ground state search by local and sequential updates of neural network quantum states
• Matrix Product States with Backflow correlations
• Neural-network Quantum States for Spin-1 systems: spin-basis and parameterization effects on compactness of representations
• Transfer entropy and O-information to detect grokking in tensor network multi-class classification problems
• Basis dependence of Neural Quantum States for the Transverse Field Ising Model