Quantum Entanglement in Neural Network States

By Dong-Ling Deng, Xiaopeng Li. 2017

431 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4ea909c9509aa556786687fb044f2ce86100e4ec

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Citations in the corpus

A high-bias, low-variance introduction to Machine Learning for physicists (2018, positive): “states (Deng et al., 2017) and tensor product based generative models have been used to understand MNIST and other ML datasets (Stoudenmire and Schwab, 2016).”
Machine learning \& artificial intelligence in the quantum domain (2017, background): “In subsequent works, the authors have also analyzed the structure of entanglement of NQS states (Deng et al., 2017), and have provided analytic proofs of the representation power of deep restricted BMs, proving they can e.”
Deep autoregressive models for the efficient variational simulation of many-body quantum systems (2019, background): “[8] Dong-Ling Deng, Xiaopeng Li, and S Das Sarma, “Quan-”
Thermodynamics (2021, neutral): “From the panel showing the entanglement entropy, it is clear that while M = 8 is sufficient to represent certain states with volume-law entanglement [41], it does not capture the full entanglement structure of the time evolved state for times Jt ≳ 2 .”
Reconstructing quantum states with generative models (2018, background): “Notably, due to their intrinsically nonlocal structure, RBMs can represent highly entangled many-body states using a small number of parameters [27–30].”
Neural-Network Quantum States, String-Bond States, and Chiral Topological States (2017, methodology): “It was recently shown that while fully connected RBM can exhibit volume-law entanglement, contrary to local tensor networks, all short-range RBM satisfy an area law[72].”