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Neural-Network Quantum States have been recently introduced as an Ansatz fordescribing the wave function of quantum many-body systems. We show that thereare strong connections between Neural-Network Quantum States in the form ofRestricted Boltzmann Machines and some classes of Tensor-Network states inarbitrary dimensions. In particular we demonstrate that short-range RestrictedBoltzmann Machines are Entangled Plaquette States, while fully connectedRestricted Boltzmann Machines are String-Bond States with a nonlocal geometryand low bond dimension. These results shed light on the underlying architectureof Restricted Boltzmann Machines and their efficiency at representing many-bodyquantum states. String-Bond States also provide a generic way of enhancing thepower of Neural-Network Quantum States and a natural generalization to systemswith larger local Hilbert space. We compare the advantages and drawbacks ofthese different classes of states and present a method to combine themtogether. This allows us to benefit from both the entanglement structure ofTensor Networks and the efficiency of Neural-Network Quantum States into asingle Ansatz capable of targeting the wave function of strongly correlatedsystems. While it remains a challenge to describe states with chiraltopological order using traditional Tensor Networks, we show thatNeural-Network Quantum States and their String-Bond States extension candescribe a lattice Fractional Quantum Hall state exactly. In addition, weprovide numerical evidence that Neural-Network Quantum States can approximate achiral spin liquid with better accuracy than Entangled Plaquette States andlocal String-Bond States. Our results demonstrate the efficiency of neuralnetworks to describe complex quantum wave functions and pave the way towardsthe use of String-Bond States as a tool in more traditional machine-learningapplications.

Neural-Network Quantum States, String-Bond States, and Chiral Topological States