# Excited Pfaffians: Generalized Neural Wave Functions Across Structure and State

arXiv: https://arxiv.org/abs/2603.14515 · HTML: https://mlsanity.com/paper/2603.14515/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 69 references that are in the map.
- First posted on arXiv 15 Mar 2026.

## Builds on (most cited first, 21 of 21)

- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [A Self-Attention Ansatz for Ab-initio Quantum Chemistry](https://mlsanity.com/paper/2211.13672/) (2022)
- [Ab-Initio Potential Energy Surfaces by Pairing GNNs with Neural Wave Functions](https://mlsanity.com/paper/2110.05064/) (2021)
- [Ab-initio quantum chemistry with neural-network wavefunctions](https://mlsanity.com/paper/2208.12590/) (2022)
- [Better, Faster Fermionic Neural Networks](https://mlsanity.com/paper/2011.07125/) (2020)
- [Generalizing Neural Wave Functions](https://mlsanity.com/paper/2302.04168/) (2023)
- [Gold-standard solutions to the Schr"odinger equation using deep learning: How much physics do we need?](https://mlsanity.com/paper/2205.09438/) (2022)
- [A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions](https://mlsanity.com/paper/2401.10190/) (2024)
- [Electronic excited states in deep variational Monte Carlo](https://mlsanity.com/paper/2203.09472/) (2022)
- [Sampling-free Inference for Ab-Initio Potential Energy Surface Networks](https://mlsanity.com/paper/2205.14962/) (2022)
- [Accurate Computation of Quantum Excited States with Neural Networks](https://mlsanity.com/paper/2308.16848/) (2023)
- [An ab initio foundation model of wavefunctions that accurately describes chemical bond breaking](https://mlsanity.com/paper/2506.19960/) (2025)
- [Neural Wave Functions for Superfluids](https://mlsanity.com/paper/2305.06989/) (2023)
- [Variational Monte Carlo on a Budget -- Fine-tuning pre-trained Neural Wavefunctions](https://mlsanity.com/paper/2307.09337/) (2023)
- [Neural Pfaffians: Solving Many Many-Electron Schr"odinger Equations](https://mlsanity.com/paper/2405.14762/) (2024)
- [Accurate Ab-initio Neural-network Solutions to Large-Scale Electronic Structure Problems](https://mlsanity.com/paper/2504.06087/) (2025)
- [Accurate and scalable exchange-correlation with deep learning](https://mlsanity.com/paper/2506.14665/) (2025)
- [Transferable Neural Wavefunctions for Solids](https://mlsanity.com/paper/2405.07599/) (2024)
- [Highly Accurate Real-space Electron Densities with Neural Networks](https://mlsanity.com/paper/2409.01306/) (2024)
- [Determinant-free fermionic wave function using feed-forward neural networks](https://mlsanity.com/paper/2108.08631/) (2021)

## Built on by (most cited first, 1 of 1)

- [Position: The Inevitable Transition to Machine Learning in Quantum Chemistry](https://mlsanity.com/paper/2607.18281/) (2026)

## Abstract (arXiv)

Neural-network wave functions in Variational Monte Carlo (VMC) have achieved great success in accurately representing both ground and excited states. However, achieving sufficient numerical accuracy in state overlaps requires increasing the number of Monte Carlo samples, and consequently the computational cost, with the number of states. We present a nearly constant sample-size approach, Multi-State Importance Sampling (MSIS), that leverages samples from all states to estimate pairwise overlap. To efficiently evaluate all states for all samples, we introduce Excited Pfaffians. Inspired by Hartree-Fock, this architecture represents many states within a single neural network. Excited Pfaffians also serve as generalized wave functions, allowing a single model to represent multi-state potential energy surfaces. On the carbon dimer, we match the O(Ns4)-scaling natural excited states while training >200× faster and modeling 50% more states. Our favorable scaling enables us to be the first to use neural networks to find all distinct energy levels of the beryllium atom. Finally, we demonstrate that a single wave function can represent excited states across various molecules.