# Infinite Neural Network Quantum States: Entanglement and Training Dynamics

arXiv: https://arxiv.org/abs/2112.00723 · HTML: https://mlsanity.com/paper/2112.00723/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 55 references that are in the map.
- First posted on arXiv 1 Dec 2021.

## Builds on (most cited first, 22 of 22)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [Finite Versus Infinite Neural Networks: an Empirical Study](https://mlsanity.com/paper/2007.15801/) (2020)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation](https://mlsanity.com/paper/1902.04760/) (2019)
- [Tensor Programs II: Neural Tangent Kernel for Any Architecture](https://mlsanity.com/paper/2006.14548/) (2020)
- [Deep Convolutional Networks as shallow Gaussian Processes](https://mlsanity.com/paper/1808.05587/) (2018)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Non-Gaussian processes and neural networks at finite widths](https://mlsanity.com/paper/1910.00019/) (2019)
- [Neural Networks and Quantum Field Theory](https://mlsanity.com/paper/2008.08601/) (2020)
- [Tensor Programs IIb: Architectural Universality of Neural Tangent Kernel Training Dynamics](https://mlsanity.com/paper/2105.03703/) (2021)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Gauge equivariant neural networks for quantum lattice gauge theories](https://mlsanity.com/paper/2012.05232/) (2020)
- [Neural tensor contractions and the expressive power of deep neural quantum states](https://mlsanity.com/paper/2103.10293/) (2021)
- [Quantum tangent kernel](https://mlsanity.com/paper/2111.02951/) (2021)
- [Spacetime Neural Network for High Dimensional Quantum Dynamics](https://mlsanity.com/paper/2108.02200/) (2021)
- [A quantum algorithm for training wide and deep classical neural networks](https://mlsanity.com/paper/2107.09200/) (2021)
- [Any Target Function Exists in a Neighborhood of Any Sufficiently Wide Random Network: A Geometrical Perspective](https://mlsanity.com/paper/2001.06931/) (2020)

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

- [Expressibility-induced Concentration of Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2311.04965/) (2023)
- [Quantum Lazy Training](https://mlsanity.com/paper/2202.08232/) (2022)
- [Learning ground states of gapped quantum Hamiltonians with Kernel Methods](https://mlsanity.com/paper/2303.08902/) (2023)
- [Artificial intelligence for artificial materials: moir'e atom](https://mlsanity.com/paper/2303.08162/) (2023)

## Abstract (arXiv)

We study infinite limits of neural network quantum states (∞-NNQS), which exhibit representation power through ensemble statistics, and also tractable gradient descent dynamics. Ensemble averages of Renyi entropies are expressed in terms of neural network correlators, and architectures that exhibit volume-law entanglement are presented. A general framework is developed for studying the gradient descent dynamics of neural network quantum states (NNQS), using a quantum state neural tangent kernel (QS-NTK). For ∞-NNQS the training dynamics is simplified, since the QS-NTK becomes deterministic and constant. An analytic solution is derived for quantum state supervised learning, which allows an ∞-NNQS to recover any target wavefunction. Numerical experiments on finite and infinite NNQS in the transverse field Ising model and Fermi Hubbard model demonstrate excellent agreement with theory. ∞-NNQS opens up new opportunities for studying entanglement and training dynamics in other physics applications, such as in finding ground states.