Infinite Neural Network Quantum States: Entanglement and Training Dynamics

TL;DR

This paper introduces infinite neural network quantum states ($ abla$-NNQS), revealing their entanglement properties and simplified training dynamics, enabling exact recovery of target wavefunctions and advancing quantum many-body simulations.

Contribution

It develops a theoretical framework for $ abla$-NNQS, including neural tangent kernel analysis, and demonstrates their ability to represent volume-law entanglement and efficiently learn quantum states.

Findings

$ abla$-NNQS can exhibit volume-law entanglement.

Analytic solutions for quantum state learning are derived.

Numerical results show strong agreement with theoretical predictions.

Abstract

We study infinite limits of neural network quantum states ($\infty$-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 $\infty$-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 $\infty$-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. $\infty$-NNQS opens up new opportunities for studying entanglement and training dynamics in other physics applications, such as in finding ground states.