  Neural-Network Quantum States have been recently introduced as an Ansatz for
describing the wave function of quantum many-body systems. We show that there
are strong connections between Neural-Network Quantum States in the form of
Restricted Boltzmann Machines and some classes of Tensor-Network states in
arbitrary dimensions. In particular we demonstrate that short-range Restricted
Boltzmann Machines are Entangled Plaquette States, while fully connected
Restricted Boltzmann Machines are String-Bond States with a nonlocal geometry
and low bond dimension. These results shed light on the underlying architecture
of Restricted Boltzmann Machines and their efficiency at representing many-body
quantum states. String-Bond States also provide a generic way of enhancing the
power of Neural-Network Quantum States and a natural generalization to systems
with larger local Hilbert space. We compare the advantages and drawbacks of
these different classes of states and present a method to combine them
together. This allows us to benefit from both the entanglement structure of
Tensor Networks and the efficiency of Neural-Network Quantum States into a
single Ansatz capable of targeting the wave function of strongly correlated
systems. While it remains a challenge to describe states with chiral
topological order using traditional Tensor Networks, we show that
Neural-Network Quantum States and their String-Bond States extension can
describe a lattice Fractional Quantum Hall state exactly. In addition, we
provide numerical evidence that Neural-Network Quantum States can approximate a
chiral spin liquid with better accuracy than Entangled Plaquette States and
local String-Bond States. Our results demonstrate the efficiency of neural
networks to describe complex quantum wave functions and pave the way towards
the use of String-Bond States as a tool in more traditional machine-learning
applications.
