# QERNEL: a Scalable Large Electron Model

arXiv: https://arxiv.org/abs/2604.26018 · HTML: https://mlsanity.com/paper/2604.26018/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 26 references that are in the map.
- First posted on arXiv 28 Apr 2026.

## Builds on (most cited first, 10 of 10)

- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [GQA: Training Generalized Multi-Query Transformer Models from Multi-Head Checkpoints](https://mlsanity.com/paper/2305.13245/) (2023)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [A Self-Attention Ansatz for Ab-initio Quantum Chemistry](https://mlsanity.com/paper/2211.13672/) (2022)
- [Discovering Quantum Phase Transitions with Fermionic Neural Networks](https://mlsanity.com/paper/2202.05183/) (2022)
- [An ab initio foundation model of wavefunctions that accurately describes chemical bond breaking](https://mlsanity.com/paper/2506.19960/) (2025)
- [Pairing-based graph neural network for simulating quantum materials](https://mlsanity.com/paper/2311.02143/) (2023)
- [Forward Laplacian: A New Computational Framework for Neural Network-based Variational Monte Carlo](https://mlsanity.com/paper/2307.08214/) (2023)
- [Large Electron Model: A Universal Ground State Predictor](https://mlsanity.com/paper/2603.02346/) (2026)

## Built on by (most cited first, 3 of 3)

- [Low-variance estimators overcome the phase-gradient bottleneck in complex-valued neural quantum states](https://mlsanity.com/paper/2606.13912/) (2026)
- [Hamilton-Zero: A Neural Tensor-Network Foundation Model for Ground States of Arbitrary Quadratic Qubit Hamiltonians](https://mlsanity.com/paper/2608.11911/) (2026)
- [Representability-Aware Neural Networks for Reduced Density Matrices: Application to Fractional Chern Insulators](https://mlsanity.com/paper/2605.20326/) (2026)

## Abstract (arXiv)

We introduce QERNEL, a foundational neural wavefunction that variationally solves families of parameterized many-electron Hamiltonians and captures their ground states throughout parameter space within a single model. QERNEL combines FiLM-based parameter conditioning with scale-efficient architectural elements -- mixture of experts and grouped-query attention, substantially improving expressivity at low computational cost. We apply QERNEL to interacting electrons in semiconductor moir'e heterobilayers, training a single weight-shared model for systems of up to 150 electrons. By solving the many-electron Schr"odinger equation conditioned on moir'e potential depth, QERNEL captures both quantum liquid and crystal states and discovers the sharp phase transition between them, marked by abrupt changes in interaction energy and charge density. Our work establishes a foundation model for moir'e quantum materials and a scalable architecture toward a Large Electron Model for solids.