Explorative Curriculum Learning for Strongly Correlated Electron Systems

TL;DR

This work addresses the high computational cost of obtaining quantum states for strongly correlated electron systems by introducing an explorative curriculum-learning framework that uses transfer learning to efficiently traverse large parameter spaces defined by $\bm{\lambda}=(\tilde{N},\tilde{U})$ within the Hubbard model. The approach connects transfer learning to perturbation theory, deriving a first-order energy correction for zero-shot learning and using it to guide the curriculum order of parameter changes. It then introduces Pairing-Net, a transferable NQS architecture combining a neural pairing product (NPP) state and a neural correlation factor (NCF) to satisfy antisymmetry while capturing complex electron correlations, enabling robust transfer across parameter regimes. Experimental results on 2D Hubbard models up to $M=6\times6$ and on frustrated $J_1$-$J_2$ systems show substantial improvements in convergence speed and stability, with reported speedups up to ~200-fold and improved generalization across parameter variations. These findings offer a scalable path for efficient exploration of quantum phases in strongly correlated electron systems and motivate extensions to multiorbital and more complex lattice models.

Abstract

Recent advances in neural network quantum states (NQS) have enabled high-accuracy predictions for complex quantum many-body systems such as strongly correlated electron systems. However, the computational cost remains prohibitive, making exploration of the diverse parameters of interaction strengths and other physical parameters inefficient. While transfer learning has been proposed to mitigate this challenge, achieving generalization to large-scale systems and diverse parameter regimes remains difficult. To address this limitation, we propose a novel curriculum learning framework based on transfer learning for NQS. This facilitates efficient and stable exploration across a vast parameter space of quantum many-body systems. In addition, by interpreting NQS transfer learning through a perturbative lens, we demonstrate how prior physical knowledge can be flexibly incorporated into the curriculum learning process. We also propose Pairing-Net, an architecture to practically implement this strategy for strongly correlated electron systems, and empirically verify its effectiveness. Our results show an approximately 200-fold speedup in computation and a marked improvement in optimization stability compared to conventional methods.

Figures (10)

• Figure 1: Illustration of the parameter order in curriculum learning. Circles and the filled one represent possible targets and the pre-training task, respectively, in the parameter space of $\bm{\lambda}$. This illustrates that iteratively performing transfer learning to the nearest neighboring parameter in the space is crucial for efficient and stable curriculum learning.
• Figure 2: Pairing-Net architecture. The input to the network is generated from a single sampled configuration, and the output is the wave function amplitude $\Psi_{\theta_{k},\varphi_{k}}(x)$, parameterized by neural networks $\theta_k$ and $\varphi_k$. Pairing-Net decomposes the wave function into two neural networks: A neural pair-product (NPP) wave function $\Psi^{(\mathrm{NPP})}_{\theta_{k}}(x)$ and a neural correlation factor (NCF) $C_{\varphi_{k}}(x)$. The input to the NPP wave function consists of the information $\bm{\xi}_{i}$ for each sampled configuration $x$, the information $\delta\bm{\xi}_{ij}$ for electron pairs, and the electron correlation parameters $\bm{\lambda}_{k}$ characterizing the system. The input to the NCF is a sampled configuration of all electrons $\bm{v}$.
• Figure 3: The loss curves for $M=4\times4$ over 100 epochs applied to $(\tilde{N}_{2},\tilde{U}_{2})=(16/16,4)$ using the various pre-trained NQS.
• Figure 4: The dependence of $\Delta E^{(0)}$, acquired through pre-training, on changes in the target parameter $\tilde{U}_2$. Pre-trained NQS with $\tilde{U}_1 = 2, 4, 6, 8$ were used.
• Figure 5: Schematic representation of the two-dimensional square lattice Hubbard model. The arrows indicate electrons on the lattice with spin-up or spin-down. $t$ denotes the electron hopping between nearest-neighbor lattice sites, and $U$ represents the on-site Coulomb repulsion.