# Apparent Compression, Real Stability: The Intrinsic Dimension of Learning a Quantum Wavefunction

arXiv: https://arxiv.org/abs/2609.33193 · HTML: https://mlsanity.com/paper/2609.33193/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 38 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Deep autoregressive models for the efficient variational simulation of many-body quantum systems](https://mlsanity.com/paper/1902.04057/) (2019)
- [How many degrees of freedom do we need to train deep networks: a loss landscape perspective](https://mlsanity.com/paper/2107.05802/) (2021)
- [NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems](https://mlsanity.com/paper/2112.10526/) (2021)
- [Neural network wave functions and the sign problem](https://mlsanity.com/paper/2002.04613/) (2020)
- [Geometry of learning neural quantum states](https://mlsanity.com/paper/1910.11163/) (2019)
- [Adiabatic Fine-Tuning of Neural Quantum States Enables Detection of Phase Transitions in Weight Space](https://mlsanity.com/paper/2503.17140/) (2025)
- [Parametric Quantum State Tomography with HyperRBMs](https://mlsanity.com/paper/2601.20950/) (2026)
- [Mapping Networks](https://mlsanity.com/paper/2602.19134/) (2026)
- [Low-variance estimators overcome the phase-gradient bottleneck in complex-valued neural quantum states](https://mlsanity.com/paper/2606.13912/) (2026)

## Abstract (arXiv)

How many directions in weight space does training need? The intrinsic dimension answers this with the smallest number of random directions in which training still reaches a target accuracy, and small values have motivated parameter-efficient methods such as LoRA. We measure it for variational Monte Carlo (VMC), which trains a neural network to represent the ground state of a quantum many-body system. VMC is a demanding test, because the network generates its own training samples and every gradient is noisy, and a revealing one, because the exact answer is known and every run can be scored. We train only a small latent vector that a frozen random map turns into the network's weights, with no change to the standard natural-gradient optimizer. We find that a small dimension can be misleading, while the stability it brings is real. On a magnet with a hard sign pattern, a network that cannot represent signs reaches its best energy in 8 of 28,642 directions, but only because no such network can go lower; once signs are learnable, neither the signs nor the magnitudes are cheap. The dimension rises across a quantum phase transition, so it tracks how difficult a state is at far less compute than fitting a scaling law, yet it never falls below a floor set by the random subspace itself, even where the ground state is nearly trivial. Training in the subspace, in contrast, never diverged in our experiments, whereas full-parameter training with the same settings did, and a control with matched solvers attributes the difference to the reduced dimension.