# Neural-Network Quantum States, String-Bond States, and Chiral Topological States. (arXiv:1710.04045v3 [quant-ph] UPDATED)

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.