# Geometry of learning neural quantum states

arXiv: https://arxiv.org/abs/1910.11163 · HTML: https://mlsanity.com/paper/1910.11163/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Oct 2019.

## Built on by (most cited first, 14 of 14)

- [NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems](https://mlsanity.com/paper/2112.10526/) (2021)
- [A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions](https://mlsanity.com/paper/2401.10190/) (2024)
- [Neural Error Mitigation of Near-Term Quantum Simulations](https://mlsanity.com/paper/2105.08086/) (2021)
- [Exponentially improved efficient machine learning for quantum many-body states with provable guarantees](https://mlsanity.com/paper/2304.04353/) (2023)
- [Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo](https://mlsanity.com/paper/2106.10558/) (2021)
- [Are queries and keys always relevant? A case study on Transformer wave functions](https://mlsanity.com/paper/2405.18874/) (2024)
- [Worth Their Weight: Randomized and Regularized Block Kaczmarz Algorithms without Preprocessing](https://mlsanity.com/paper/2502.00882/) (2025)
- [A Sketch-and-Project Analysis of Subsampled Natural Gradient Algorithms](https://mlsanity.com/paper/2508.21022/) (2025)
- [Quantum Speedup of Natural Gradient for Variational Bayes](https://mlsanity.com/paper/2106.05807/) (2021)
- [An Introduction to Quantum Computing for Statisticians and Data Scientists](https://mlsanity.com/paper/2112.06587/) (2021)
- [Understanding and eliminating spurious modes in variational Monte Carlo using collective variables](https://mlsanity.com/paper/2211.09767/) (2022)
- [Reinforcement learning to learn quantum states for Heisenberg scaling accuracy](https://mlsanity.com/paper/2412.02334/) (2024)
- [Learning topological defects formation with neural networks in a quantum phase transition](https://mlsanity.com/paper/2204.06769/) (2022)
- [Apparent Compression, Real Stability: The Intrinsic Dimension of Learning a Quantum Wavefunction](https://mlsanity.com/paper/2609.33193/) (2026)

## Abstract (arXiv)

Combining insights from machine learning and quantum Monte Carlo, the stochastic reconfiguration method with neural network Ansatz states is a promising new direction for high-precision ground state estimation of quantum many-body problems. Even though this method works well in practice, little is known about the learning dynamics. In this paper, we bring to light several hidden details of the algorithm by analyzing the learning landscape. In particular, the spectrum of the quantum Fisher matrix of complex restricted Boltzmann machine states exhibits a universal initial dynamics, but the converged spectrum can dramatically change across a phase transition. In contrast to the spectral properties of the quantum Fisher matrix, the actual weights of the network at convergence do not reveal much information about the system or the dynamics. Furthermore, we identify a new measure of correlation in the state by analyzing entanglement in eigenvectors. We show that, generically, the learning landscape modes with least entanglement have largest eigenvalue, suggesting that correlations are encoded in large flat valleys of the learning landscape, favoring stable representations of the ground state.