# Efficient Representation of Quantum Many-body States with Deep Neural Networks

## TL;DR

This paper proves that deep neural networks can efficiently represent complex quantum many-body states, unlike shallow networks, highlighting the importance of depth in neural network representations of quantum physics.

## Contribution

The paper provides a rigorous proof demonstrating the superior representational power of deep neural networks over shallow ones for quantum states.

## Key findings

- Deep networks can efficiently represent states from polynomial quantum circuits.
- Shallow networks cannot efficiently represent these states unless a major complexity theory collapse occurs.
- Deep neural networks are fundamentally more powerful for quantum state representation.

## Abstract

The challenge of quantum many-body problems comes from the difficulty to represent large-scale quantum states, which in general requires an exponentially large number of parameters. Recently, a connection has been made between quantum many-body states and the neural network representation (\textit{arXiv:1606.02318}). An important open question is what characterizes the representational power of deep and shallow neural networks, which is of fundamental interest due to popularity of the deep learning methods. Here, we give a rigorous proof that a deep neural network can efficiently represent most physical states, including those generated by any polynomial size quantum circuits or ground states of many body Hamiltonians with polynomial-size gaps, while a shallow network through a restricted Boltzmann machine cannot efficiently represent those states unless the polynomial hierarchy in computational complexity theory collapses.