Quantum Entanglement in Neural Network States

Abstract

Machine learning, one of today's most rapidly growing interdisciplinary fields, promises an unprecedented perspective for solving intricate quantum many-body problems. Understanding the physical aspects of the representative artificial neural-network states is recently becoming highly desirable in the applications of machine learning techniques to quantum many-body physics. Here, we study the quantum entanglement properties of neural-network states, with a focus on the restricted-Boltzmann-machine (RBM) architecture. We prove that the entanglement of all short-range RBM states satisfies an area law for arbitrary dimensions and bipartition geometry. For long-range RBM states we show by using an exact construction that such states could exhibit volume-law entanglement, implying a notable capability of RBM in representing efficiently quantum states with massive entanglement. We further examine generic RBM states with random weight parameters. We find that their averaged entanglement entropy obeys volume-law scaling and meantime strongly deviates from the Page-entropy of the completely random pure states. We show that their entanglement spectrum has no universal part associated with random matrix theory and bears a Poisson-type level statistics. Using reinforcement learning, we demonstrate that RBM is capable of finding the ground state (with power-law entanglement) of a model Hamiltonian with long-range interaction. In addition, we show, through a concrete example of the one-dimensional symmetry-protected topological cluster states, that the RBM representation may also be used as a tool to analytically compute the entanglement spectrum. Our results uncover the unparalleled power of artificial neural networks in representing quantum many-body states, which paves a novel way to bridge computer science based machine learning techniques to outstanding quantum condensed matter physics problems.

17 pages, 8 figures. Version published in Phys. Rev. X

References in corpus (31)

• The density-matrix renormalization group in the age of matrix product states
• Many body localization and thermalization in quantum statistical mechanics
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Multi-party entanglement in graph states
• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• A class of quantum many-body states that can be efficiently simulated
• Learning phase transitions by confusion
• Matrix product states represent ground states faithfully
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Entropy scaling and simulability by Matrix Product States
• Measuring entanglement growth in quench dynamics of bosons in an optical lattice
• Entanglement spectrum in one-dimensional systems
• Bipartite entanglement and entropic boundary law in lattice spin systems
• Ground state entanglement and geometric entropy in the Kitaev's model
• Self-Learning Monte Carlo Method
• Measuring entanglement entropy of a generic many-body system with a quantum switch
• Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines
• Weak binding between two aromatic rings: feeling the van der Waals attraction by quantum Monte Carlo methods
• An exact mapping between the Variational Renormalization Group and Deep Learning
• Many-body localisation implies that eigenvectors are matrix-product states
• Stochastic Error Cancellation in Analog Quantum Simulation
• Entanglement Across a Transition to Quantum Chaos
• Deep learning and the renormalization group
• Restricted Boltzmann Machines for the Long Range Ising Models
• Machine learning for many-body physics: efficient solution of dynamical mean-field theory
• Statistical Bubble Localization with Random Interactions
• Area law in one dimension: Degenerate ground states and Renyi entanglement entropy
• Detecting intermediate mass black holes in globular clusters with machine learning
• Measuring quantum entanglement, machine learning and wave function tomography: Bridging theory and experiment with the quantum gas microscope

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