# Quantum Mechanics and Neural Networks

arXiv: https://arxiv.org/abs/2504.05462 · HTML: https://mlsanity.com/paper/2504.05462/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Apr 2025.

## Built on by (most cited first, 7 of 7)

- [Fermions and Supersymmetry in Neural Network Field Theories](https://mlsanity.com/paper/2511.16741/) (2025)
- [Gauge-covariant stochastic neural fields: Stability and finite-width effects](https://mlsanity.com/paper/2508.18948/) (2025)
- [Viability of perturbative expansion for quantum field theories on neurons](https://mlsanity.com/paper/2508.03810/) (2025)
- [Topological Effects in Neural Network Field Theory](https://mlsanity.com/paper/2604.02313/) (2026)
- [Spinning Conformal Correlators from Neural Networks](https://mlsanity.com/paper/2608.15001/) (2026)
- [What Neural Network Field Theory Can and Cannot Realise on a Computer](https://mlsanity.com/paper/2608.21523/) (2026)
- [Neural Network Field Theory at Finite Width](https://mlsanity.com/paper/2608.21588/) (2026)

## Abstract (arXiv)

We demonstrate that any Euclidean-time quantum mechanical theory may be represented as a neural network, ensured by the Kosambi-Karhunen-Lo`eve theorem, mean-square path continuity, and finite two-point functions. The additional constraint of reflection positivity, which is related to unitarity, may be achieved by a number of mechanisms, such as imposing neural network parameter space splitting or the Markov property. Non-differentiability of the networks is related to the appearance of non-trivial commutators. Neural networks acting on Markov processes are no longer Markov, but still reflection positive, which facilitates the definition of deep neural network quantum systems. We illustrate these principles in several examples using numerical implementations, recovering classic quantum mechanical results such as Heisenberg uncertainty, non-trivial commutators, and the spectrum.