Adiabatic Fine-Tuning of Neural Quantum States Enables Detection of Phase Transitions in Weight Space

TL;DR

Problem: How do neural quantum states encode quantum phase information, and can learned weights reveal phase transitions without explicit observables? Approach: Train NQS across phase diagrams with adiabatic fine-tuning, and analyze weight trajectories using PCA, validated on TFIM and J1-J2 models. Findings: Phase transitions produce structured weight-space trajectories with PC1 minima at $|h_c/J|=1$ and $J_2/J_1=0.5$, and adiabatic fine-tuning yields smoother convergence and stronger cross-phase weight correlations. Significance: Links physical phase transitions to neural parameter geometry, enabling phase-transition detection from weights alone and suggesting connections to mode connectivity for broader ML interpretability in physics.

Abstract

Neural quantum states (NQS) have emerged as a powerful tool for approximating quantum wavefunctions using deep learning. While these models achieve remarkable accuracy, understanding how they encode physical information remains an open challenge. In this work, we introduce adiabatic fine-tuning, a scheme that trains NQS across a phase diagram, leading to strongly correlated weight representations across different models. This correlation in weight space enables the detection of phase transitions in quantum systems by analyzing the trained network weights alone. We validate our approach on the transverse field Ising model and the J1-J2 Heisenberg model, demonstrating that phase transitions manifest as distinct structures in weight space. Our results establish a connection between physical phase transitions and the geometry of neural network parameters, opening new directions for the interpretability of machine learning models in physics.

Figures (5)

• Figure 1: Results for the Ising model. From top to bottom the rows show: the energy error over the first two training iterations, the last energy error values for each value of $h$, the infidelity values, and the principal component analysis of the weights. From left to right columns show the case of independent training, and fine-tuning models starting from $h=0.0$, and $h=3.0$. The colorbar represents the value of $h$.
• Figure 2: Results for the J1-J2 model. From top to bottom the rows show: the energy error over the first two training iterations, the last energy error values for each value of $J2/J1$, the infidelity values, and the principal component analysis of the weights. From left to right columns show the case of independent training, and fine-tuning models starting from $J_2/J_1=0.0$, and $J_2/J_1=1.0$. The colorbar represents the $J_2/J_1$ ratio.
• Figure 3: Energy error as a function of training step for all values of $h$ in the TFIM model.
• Figure 4: Energy error as a function of training step for all values of $J2/J1$ in the J1-J2 model.
• Figure 5: Scatter plots and 3D visualizations of PCA projections of the weights of neural networks trained for different values of $h$ in the case of the Ising model (first row), and for different values of $J_2/J_1$ in the case of the J1-J2 model (second row). From left to right, columns show PC1 vs PC2 space, PC2 vs PC3 space, PC1 vs PC3 spaces, and the 3D PCA projection. The colorbars to the right indicate the corresponding parameter values for each model. The phase transition point for all images is represented by the divergence of the colormap, shown in white.