Variational optimization of the amplitude of neural-network quantum many-body ground states

TL;DR

This work proposes a CNN-based amplitude network (aCNN) to variationally optimize the amplitude of a neural-network quantum state while fixing the sign structure, yielding competitive ground-state energies for unfrustrated and frustrated spin models. By expressing the wave function as $\\psi_{\\theta}(\\sigma)=s(\\sigma)\\exp[a_{\\theta}(\\sigma)]$ and enforcing symmetries through data augmentation and convolutional design, the approach achieves energies better than VMC and comparable to DMRG and QMC benchmarks in several cases, with particularly strong performance against a complex-valued CNN in the frustrated $J_1$-$J_2$ model. The study highlights that optimizing the amplitude alone can be more effective than jointly optimizing sign and amplitude for certain problems, and it identifies sign-structure optimization as a key future direction, especially for strongly frustrated systems. The results advocate using specialized amplitude networks with symmetry-aware architectures as a practical path for accurate ground-state predictions in quantum many-body systems. The authors also suggest extending to phase networks to handle complex wave functions and sign problems in broader contexts, with code and data available for reproducibility.

Abstract

Neural-network quantum states (NQSs), variationally optimized by combining traditional methods and deep learning techniques, is a new way to find quantum many-body ground states and gradually becomes a competitor of traditional variational methods. However, there are still some difficulties in the optimization of NQSs, such as local minima, slow convergence, and sign structure optimization. Here, we split a quantum many-body variational wave function into a multiplication of a real-valued amplitude neural network and a sign structure, and focus on the optimization of the amplitude network while keeping the sign structure fixed. The amplitude network is a convolutional neural network (CNN) with residual blocks, namely a ResNet. Our method is tested on three typical quantum many-body systems. The obtained ground state energies are lower than or comparable to those from traditional variational Monte Carlo (VMC) methods and density matrix renormalization group (DMRG). Surprisingly, for the frustrated Heisenberg $J_1$-$J_2$ model, our results are better than those of the complex-valued CNN in the literature, implying that the sign structure of the complex-valued NQS is difficult to be optimized. We will study the optimization of the sign structure of NQSs in the future.

Figures (3)

• Figure 1: The architecture of the aCNN. The input is a spin configuration $\sigma$ of the system, while the output is $a(\sigma)$[Eq. \ref{['eq:amp_out']}]. (a) ${\cal C}_{\rm 4v}$ symmetrization of the aCNN as a data augmentation. $v_0$ and $v_{\mathrm{out}}$ denote the input and output layers, respectively. (b) A single spin configuration $\sigma_i$ propagation in the aCNN, where $f$ is an activation function. (c) Residual block, where $k_l$ is the convolution kernel of layer $l$. The shortcut connection achieves cross-layer information transmission.
• Figure 2: Spin structure factor $S^2(\vec{k})$ as defined in Eq. \ref{['eq:ssf']}. The ${\boldsymbol{\mathrm{k}}}$-space distribution of $S^2(\vec{k})$ of $10\times 10$ square lattice of the Heisenberg model, which peaks at $\vec{k}=(\pi,\pi)$. $k_x$ and $k_y$ are the two components of $\vec{k}$.
• Figure 3: Ground state energies of the Heisenberg $J_1$-$J_2$ model with $L=10$. The aCNN is optimized with different Marshall sign structures, the checkerboard-patterned (exact for $J_2=0$) and the stripe-patterned (exact for $J_1=0$). The VMC($p=0$) results are from Ref. Choo2019_CNN_ComplexValued. VMC($p=\infty$) denotes the variance extrapolated results from Ref. Sorella2013_VMC, which can be regarded as the exact ground state energies. The inset shows $E - E(\mathrm{VMC}(p=0))$, the relative energies with respect to the VMC($p=0$) energies, for an enlarged view.