learning, one of today�s most rapidly growing interdisciplinary fields, promises an\nunprecedented perspective for solving intricate quantum many-body problems. Understanding the physical\naspects of the representative artificial neural-network states has recently become highly desirable in the\napplications of machine-learning techniques to quantum many-body physics. In this paper, we explore the\ndata structures that encode the physical features in the network states by studying the quantum\nentanglement properties, with a focus on the restricted-Boltzmann-machine (RBM) architecture. We\nprove that the entanglement entropy of all short-range RBM states satisfies an area law for arbitrary\ndimensions and bipartition geometry. For long-range RBM states, we show by using an exact construction\nthat such states could exhibit volume-law entanglement, implying a notable capability of RBM in\nrepresenting quantum states with massive entanglement. Strikingly, the neural-network representation for\nthese states is remarkably efficient, in the sense that the number of nonzero parameters scales only linearly\nwith the system size. We further examine the entanglement properties of generic RBM states by randomly\nsampling the weight parameters of the RBM. We find that their averaged entanglement entropy obeys\nvolume-law scaling, and the meantime strongly deviates from the Page entropy of the completely random\npure states.We show that their entanglement spectrum has no universal part associated with random matrix\ntheory and bears a Poisson-type level statistics. Using reinforcement learning, we demonstrate that RBM is\ncapable of finding the ground state (with power-law entanglement) of a model Hamiltonian with a longrange\ninteraction. In addition, we show, through a concrete example of the one-dimensional symmetryprotected\ntopological cluster states, that the RBM representation may also be used as a tool to analytically\ncompute the entanglement spectrum. Our results uncover the unparalleled power of artificial neural\nnetworks in representing quantum many-body states regardless of how much entanglement they possess,\nwhich paves a novel way to bridge computer-science-based machine-learning techniques to outstanding\nquantum condensed-matter physics problems
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