用改进的李兹方法提升量子多体系统神经网络态的精度
Generalized Lanczos method for systematic optimization of neural-network quantum states
- 结合监督学习与变分蒙特卡洛，用李兹法系统优化神经网络量子态
- 在二维自旋模型中显著降低能量，在强阻挫区域表现更优
- 计算成本线性增长，适合大规模量子系统模拟
近年来，人工智能在自然科学领域取得显著进展。在量子多体计算中，基于神经网络量子态（NQS）的基态求解器已达到与变分蒙特卡洛、密度矩阵重整化群和量子蒙特卡洛等传统方法相当甚至更优的基态能量精度。本文提出一种结合监督学习、变分蒙特卡洛（VMC）与李兹法的系统性优化方法，称为NQS李兹法。该算法分为两部分：监督学习部分用神经网络表示李兹态，变分蒙特卡洛部分进一步优化神经网络。我们分析了欠拟合问题，并证明该方法在二维Heisenberg $J_1$-$J_2$模型的强阻挫区域能系统性降低能量。相较于现有结合李兹法与受限玻尔兹曼机的方法，本方法计算成本呈线性增长，具有更好可扩展性。
原文摘要 · Abstract (English)
Recently, artificial intelligence for science has made significant inroads into various fields of natural science research. In the field of quantum many-body computation, researchers have developed numerous ground state solvers based on neural-network quantum states (NQSs), achieving ground state energies with accuracy comparable to or surpassing traditional methods such as variational Monte Carlo methods, density matrix renormalization group, and quantum Monte Carlo methods. Here, we combine supervised learning, variational Monte Carlo (VMC), and the Lanczos method to develop a systematic approach to improving the NQSs of many-body systems, which we refer to as the NQS Lanczos method. The algorithm mainly consists of two parts: the supervised learning part and the VMC optimization part. Through supervised learning, the Lanczos states are represented by the NQSs. Through VMC, the NQSs are further optimized. We analyze the reasons for the underfitting problem and demonstrate how the NQS Lanczos method systematically improves the energy in the highly frustrated regime of the two-dimensional Heisenberg $J_1$-$J_2$ model. Compared to the existing method that combines the Lanczos method with the restricted Boltzmann machine, the primary advantage of the NQS Lanczos method is its linearly increasing computational cost.
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