Hidden Unit Interpretability in RBM Quantum States:Encoding Antiferromagnetic Order in Heisenberg Spin Rings

TL;DR

This study reveals how RBMs encode antiferromagnetic order in quantum spin systems, showing that hidden units collectively capture complex correlations and that important units decrease in proportion as system size grows.

Contribution

The paper provides a detailed analysis of how RBMs encode quantum order, demonstrating the collective role of hidden units and their scaling behavior in representing antiferromagnetic states.

Findings

Hidden units specialize to capture staggered magnetization patterns.

Removing key hidden units significantly impacts energy and correlations.

The fraction of important hidden units decreases with system size, following a sublinear trend.

Abstract

We investigate how Restricted Boltzmann Machines (RBMs) encode antiferromagnetic order when trained as variational ans\"atze for one-dimensional Heisenberg spin rings with periodic boundary conditions. Through systematic hidden unit analysis and ablation studies on $N=4$ and $N=8$ spin systems, we show that individual hidden units spontaneously specialize to capture staggered magnetization patterns characteristic of antiferromagnetic ground states. Hidden units naturally segregate into two classes: those essential for ground-state energy and correlation structure, and supplementary units providing smaller corrections. Removing important units induces clear energy penalties and disrupts the staggered correlation pattern in $C_{zz}(r)$, whereas removing supplementary units has modest effects. Single-unit analysis confirms that no individual hidden unit reproduces the full antiferromagnetic correlations, indicating that quantum order emerges through collective encoding across the hidden layer. Extending this analysis to $N=8$ through $20$ with hidden unit densities $\alpha = 2$ to $5$ and ten independent seeds per configuration, we find that the fraction of important hidden units decreases with system size, consistent with sublinear growth $m' \sim N^k$ ($k \approx 0.4$). The energy-correlation impact relationship persists for small to moderate system sizes, though it weakens for the largest systems studied. These results provide a quantitative framework for RBM interpretability in quantum many-body systems.