# Scaling of neural-network quantum states for time evolution

arXiv: https://arxiv.org/abs/2104.10696 · HTML: https://mlsanity.com/paper/2104.10696/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 104 references that are in the map.
- First posted on arXiv 21 Apr 2021.

## Builds on (most cited first, 16 of 16)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Aggregated Residual Transformations for Deep Neural Networks](https://mlsanity.com/paper/1611.05431/) (2016)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Neural Autoregressive Distribution Estimation](https://mlsanity.com/paper/1605.02226/) (2016)
- [Deep autoregressive models for the efficient variational simulation of many-body quantum systems](https://mlsanity.com/paper/1902.04057/) (2019)
- [Quantum Entanglement in Deep Learning Architectures](https://mlsanity.com/paper/1803.09780/) (2018)
- [Neural-Network Quantum States, String-Bond States, and Chiral Topological States](https://mlsanity.com/paper/1710.04045/) (2017)
- [The universal approximation theorem for complex-valued neural networks](https://mlsanity.com/paper/2012.03351/) (2020)
- [Neural network wave functions and the sign problem](https://mlsanity.com/paper/2002.04613/) (2020)

## Built on by (most cited first, 4 of 4)

- [Variational optimization of the amplitude of neural-network quantum many-body ground states](https://mlsanity.com/paper/2308.09664/) (2023)
- [Large Language Model Scaling Laws for Neural Quantum States in Quantum Chemistry](https://mlsanity.com/paper/2509.12679/) (2025)
- [An Empirical Study of Quantum Dynamics as a Ground State Problem with Neural Quantum States](https://mlsanity.com/paper/2206.09241/) (2022)
- [Capturing reduced-order quantum many-body dynamics out of equilibrium via neural ordinary differential equations](https://mlsanity.com/paper/2512.13913/) (2025)

## Abstract (arXiv)

Simulating quantum many-body dynamics on classical computers is a challenging problem due to the exponential growth of the Hilbert space. Artificial neural networks have recently been introduced as a new tool to approximate quantum-many body states. We benchmark the variational power of the restricted Boltzmann machine quantum states and different shallow and deep neural autoregressive quantum states to simulate global quench dynamics of a non-integrable quantum Ising chain. We find that the number of parameters required to represent the quantum state at a given accuracy increases exponentially in time. The growth rate is only slightly affected by the network architecture over a wide range of different design choices: shallow and deep networks, small and large filter sizes, dilated and normal convolutions, with and without shortcut connections.