# On the descriptive power of Neural-Networks as constrained Tensor Networks with exponentially large bond dimension

## TL;DR

This paper compares neural network states and their tensor network counterparts, revealing that neural networks are highly constrained and do not significantly outperform traditional tensor network methods in modeling ground states of certain Hamiltonians.

## Contribution

It provides a detailed comparison showing that neural networks mapped to tensor networks are limited in expressiveness compared to unconstrained tensor network approaches.

## Key findings

- Neural network states do not drastically outperform tensor networks in the studied models.
- Mapped tensor networks are highly constrained, limiting their expressiveness.
- Explicit comparison in 1D Ising and 2D Heisenberg models demonstrates these limitations.

## Abstract

In many cases, Neural networks can be mapped into tensor networks with an exponentially large bond dimension. Here, we compare different sub-classes of neural network states, with their mapped tensor network counterpart for studying the ground state of short-range Hamiltonians. We show that when mapping a neural network, the resulting tensor network is highly constrained and thus the neural network states do in general not deliver the naive expected drastic improvement against the state-of-the-art tensor network methods. We explicitly show this result in two paradigmatic examples, the 1D ferromagnetic Ising model and the 2D antiferromagnetic Heisenberg model, addressing the lack of a detailed comparison of the expressiveness of these increasingly popular, variational ans\"atze.