Apparent Compression, Real Stability: The Intrinsic Dimension of Learning a Quantum Wavefunction
Abstract
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.
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BibTeX
@misc{wei2026apparent,
title = {{Apparent Compression, Real Stability: The Intrinsic Dimension of Learning a Quantum Wavefunction}},
author = {Lu Wei and Yufeng Wang and Chenfeng Cao and Haibin Ling},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33193},
eprint = {2609.33193},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33193},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Apparent Compression, Real Stability: The Intrinsic Dimension of Learning a Quantum Wavefunction
AU - Lu Wei
AU - Yufeng Wang
AU - Chenfeng Cao
AU - Haibin Ling
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33193
UR - https://arxiv.org/abs/2609.33193
AB - 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.
ER -