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

Abstract

We investigate how Restricted Boltzmann Machines (RBMs) encode antiferromagnetic order when trained as variational ansätze 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 $α= 2$ 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.

Figures (12)

• Figure 1: Circular spin systems with periodic boundary conditions. Both small-scale (N=4) and larger-scale (N=8) systems are studied with antiferromagnetic coupling J=1 in the Heisenberg Hamiltonian.
• Figure 2: Energy convergence during VMC optimization demonstrating how the RBM ansatz successfully minimizes the Heisenberg Hamiltonian. Both systems show smooth convergence to nearly analytical ground state values: N=4 reaching $E/N=-1.061$ and N=8 reaching $E/N=-1.101$. The variance reaches $10^{-1}$ for the smaller system and remains at $10^{0}$ for the larger system, confirming high-quality variational states.
• Figure 3: Evolution of spin-spin correlations during training. As iterations progress, the antiferromagnetic nature emerges, with clear $(-1)^r$ oscillations established by convergence. This illustrates how the RBM network progressively learns the correlational structure of the quantum ground state.
• Figure 4: Representative hidden unit weight patterns for the N=4 system with 8 hidden units. Units 3 and 5 exhibit clear antiferromagnetic behavior with oscillating patterns and absence of strong ferromagnetic order. Unit 3 shows higher magnitude weights than unit 5, explaining its stronger impact on both energy and correlations. The worst-performing units share a common feature: three sites with ferromagnetic alignment.
• Figure 5: Energy change upon removing individual hidden units. In the N=4 system, unit 3 is critically important, causing >50% error when removed. Unit 5 also impacts energy significantly, while unit 0 has minimal effect. The N=8 system shows distributed importance across multiple units, with units 14 and 13 having the highest impact.