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

2026-09-27

AI Summary

This study addresses the instability of fully parameterized variational Monte Carlo (VMC) training and the unclear stability mechanisms underlying parameter-efficient approaches. By leveraging stochastic subspace projection and natural gradient optimization, this work quantifies the intrinsic dimensionality of the weight space in neural-network quantum wave functions to investigate the origins of stability in low-dimensional training. It reveals that small-dimensional subspaces confer essential stability rather than mere compression effects. Furthermore, the intrinsic dimension is found to increase at quantum phase transitions with a well-defined lower bound, effectively tracking the learning difficulty of quantum states, while sign representations are shown to critically influence dimensionality. Ultimately, this project achieves divergence-free, stable training, offering new perspectives for understanding the optimization dynamics of quantum states.

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.

Problem

• intrinsic dimension
• variational Monte Carlo
• quantum many-body systems
• training stability
• neural network wavefunction

Innovation

• Intrinsic Dimension
• Variational Monte Carlo
• Quantum Wavefunction
• Parameter-efficient Training
• Subspace Stability