Abstract
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.
Highlights
Recognizing complex patterns is a central problem that pervades all fields of science
While short-range restricted Boltzmann machines are a subclass of entangled plaquette states, fully connected restricted Boltzmann machines are a subclass of stringbond states
We compared the power of these different classes of states and showed that, while there are cases where stringbond states require less parameters than fully connected restricted Boltzmann machines to describe the ground state of a many-body Hamiltonian, there are cases where the additional parameters in each string make string-bond states less efficient to optimize numerically
Summary
Recognizing complex patterns is a central problem that pervades all fields of science. The global wave function is taken to be the product of these tensor networks, which introduces correlations among the different clusters This construction for local clusters parametrized by a full tensor gives rise to entangled plaquette states (EPS) [28,29,30], while taking one-dimensional clusters of spins each described by a MPS leads to a string-bond states (SBS) Ansatz [31,32]. It allows for the description of states with larger local Hilbert space and has a flexible geometry.
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