# Emergent Quantumness in Neural Networks

arXiv: https://arxiv.org/abs/2012.05082 · HTML: https://mlsanity.com/paper/2012.05082/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Dec 2020.

## Built on by (most cited first, 10 of 10)

- [Towards a Theory of Evolution as Multilevel Learning](https://mlsanity.com/paper/2110.14602/) (2021)
- [Towards a theory of quantum gravity from neural networks](https://mlsanity.com/paper/2111.00903/) (2021)
- [Thermodynamics of Evolution and the Origin of Life](https://mlsanity.com/paper/2110.15066/) (2021)
- [Dataset-learning duality and emergent criticality](https://mlsanity.com/paper/2405.17391/) (2024)
- [Emergent field theories from neural networks](https://mlsanity.com/paper/2411.08138/) (2024)
- [Autonomous particles](https://mlsanity.com/paper/2301.10077/) (2023)
- [Geometric Learning Dynamics](https://mlsanity.com/paper/2504.14728/) (2025)
- [Self-organized criticality in neural networks](https://mlsanity.com/paper/2107.03402/) (2021)
- [Bio-inspired Machine Learning: programmed death and replication](https://mlsanity.com/paper/2207.04886/) (2022)
- [Experimental verification of the quantum nature of a neural network](https://mlsanity.com/paper/2209.07577/) (2022)

## Abstract (arXiv)

It was recently shown that the Madelung equations, that is, a hydrodynamic form of the Schr"odinger equation, can be derived from a canonical ensemble of neural networks where the quantum phase was identified with the free energy of hidden variables. We consider instead a grand canonical ensemble of neural networks, by allowing an exchange of neurons with an auxiliary subsystem, to show that the free energy must also be multivalued. By imposing the multivaluedness condition on the free energy we derive the Schr"odinger equation with "Planck's constant" determined by the chemical potential of hidden variables. This shows that quantum mechanics provides a correct statistical description of the dynamics of the grand canonical ensemble of neural networks at the learning equilibrium. We also discuss implications of the results for machine learning, fundamental physics and, in a more speculative way, evolutionary biology.