许多用于研究量子多体模型的数值算法在收敛速度和收敛精度上非常依赖于试探波函数。试探波函数的构造通常很大程度上依赖于研究者的经验和直觉，要求研究者在研究哈密顿量时对其本征态波函数的性质有先验的了解。在本文中，我们利用神经网络能够提取和编码特征的能力，将神经网络应用于对qubit格点模型基态波函数的性质的研究。我们主要关注一类实哈密顿量，它们的基态可以表示为实值向量，基态波函数的性质反映在展开系数的符号和幅值上。我们利用神经网络对基态波函数展开系数的符号和幅值进行了研究和分析，揭示了基态波函数的一些有趣的性质，这些性质启发我们可以利用数据驱动的方式来构造试探波函数，以减少对经验和直觉的依赖。

       我们利用神经网络研究了基态波函数展开系数的符号特性，揭示了波函数的符号规则与自旋序/电荷序之间的对应关系。其中，波函数的符号规则表示的是波函数展开系数的符号分布所遵循的函数形式。首先，我们在Gutzwiller平均场框架下研究了自旋1/2系统中波函数的符号规则，发现它能够反映出自旋序的信息，我们把这样的符号规则统称为首阶符号规则。其次，我们基于Gutzwiller平均场理论设计了一种名为单隐藏层单神经元前馈神经网络(shn-FNN)的结构。Shn-FNN具有很强的可解释性，能够学习和理解有序态波函数中的首阶符号规则，并将序的信息可视化地反映在它的编码参数之中。然后，我们利用shn-FNN研究了一系列qubit格点模型的基态波函数的符号规则，包括广义Ising环，自旋-1/2XY链，（阻挫）海森堡环，定义在三角格子上的反铁磁XY模型以及任意填充的Fermi-Hubbard环。我们的研究结果表明：（1）在所有qubit格点模型中，基态波函数的首阶符号规则和它的自旋序/电荷序密切相关；（2）对于无阻挫自旋-1/2模型，首阶符号规则能够精确地描述基态波函数的符号分布；（3）对于阻挫自旋模型和强关联费米子模型，首阶符号规则能够部分有效地描述基态波函数的符号分布，量子涨落只是导致首阶符号规则准确度的下降而不是使其完全失效。对于所有的qubit格点模型，我们都能够利用shn-FNN总结出首阶符号规则的通用形式，这为后续对基态波函数的推断提供了精确或近似精确的符号信息。

我们聚焦于具有U(1)对称性的qubit格点模型，利用神经网络研究了基态波函数展开系数的幅值特性。当一个qubit系统具有U(1)对称性时，希尔伯特空间可以根据守恒荷划分为不同的扇区，我们所关注的qubit格点模型的基态常常位于维度最大的扇区中。我们发现，利用神经网络可以从小扇区中的最低能量子态的波函数展开系数的幅值出发，推断基态波函数展开系数的的幅值。我们讨论了若干个自旋-1/2和费米子模型，包括无自旋自由费米子环，一维和方形晶格上的反铁磁海森堡模型，阻挫海森堡环以及Fermi-Hubbard环。对于这些模型，神经网络均能有效地实现对基态波函数展开系数的幅值的推断。此外我们还讨论了影响推断精度的因素，如集成学习、数据标准化、数据集的构造和神经网络的结构等等。最后，通过结合推断的幅值和首阶符号规则，我们为qubit格点模型构造出了十分接近真实基态波函数的试探波函数。

The convergence speed and precision of many numerical algorithms used for studying quantum many-body models depend significantly on trial wave functions. The construction of trial wave functions often heavily relies on the researcher’s experience and intuition, requiring researchers to have a certain level of prior knowledge about the properties of the eigenstate wave functions when studying the Hamiltonian. In this paper, we utilize the capability of neural networks to extract and encode features, applying neural networks to study the properties of the ground-state wave functions in qubit lattice models. We primarily focus on a class of real Hamiltonians, whose ground states can be represented as real-valued vectors. The properties of the ground-state wave function are reflected in the signs and amplitudes of the expansion coefficients.We employ the neural networks to investigate and analyze the signs and amplitudes of the expansion coefficients in the ground-state wave function. This exploration reveals intriguing properties of the ground-state wave functions, inspiring us to employ data-driven approaches for constructing trial wave functions, thus reducing the reliance on experience and intuition.

We employed the neural network to study the properties of the signs of the expansion coefficients in the ground-state wave functions, and revealed the correspondence between the sign rules of the wave functions and spin/charge orders. The sign rule of the wave function represent the functional form followed by the sign distribution of the expansion coefficients. First, within the Gutzwiller mean-field framework, we investigated the sign rules of wave functions in the spin 1/2 systems, we found that these sign rules could reflect information about the spin orders, we call these sign rules as leading-order sign rules. Secondly, we designed a
neural network structure, which is called single-hidden-neuron feed-forward neural network (shn-FNN), based on the Gutzwiller mean-field theory. Shn-FNN has strong interpretability, it can learn and understand the leading-order sign rules in the wave functions of ordered states, information of orders can be visualized by its encoding parameters. Thirdly, we employed shn-FNN to study the sign rules of ground-state wave functions for a range of qubitlattice models, including the generalized Ising ring, spin-1/2 XY chains, (frustrated) Heisenberg rings, antiferromagnetic XY model on triangular lattices and Fermi-Hubbard ring with
arbitrary fillings. Our results indicate that: (1)For qubit lattice models， the leading-order sign rules of the ground-state wave functions are closely related to spin/charge orders. (2)For unfrustrated spin-1/2 models, leading-order sign rules can precisely describe the sign distributions of the ground-state wave functions. (3) For frustrated spin models and strongly correlated fermion models, leading-order sign rules can partially describe the sign distributions of the ground-state wave functions, quantum fluctuations only lead to a decrease in the accuracy of the leading-order sign rules but do not make them completely ineffective. For
all qubit lattice models, we are able to use shn-FNN to summarize the general forms of the leading-order sign rules, which will provide precise or approximate sign information for the inference of ground-state wave functions in subsequent research.

We employed neural networks to study the properties of the amplitudes of the expansion coefficients in ground-state wave function for qubit lattice models with U(1) symmetry. When a qubit system possesses U(1) symmetry, The Hilbert space can be divided into different sectors according to the conserved charges, the ground state of the qubit lattice models we are interested in often lies in the sector with the largest dimension. We have found that, neural networks can effectively infer the amplitudes of the expansion coefficients in ground-state wave functions, by studying the amplitude of the expansion coefficients of the lowest-energy quantum states in small sectors. We discuss several spin-1/2 and fermion models, including spinless free fermion rings, antiferromagnetic Heisenberg models on chain and square lattice, frustrated Heisenberg rings and Fermi-Hubbard rings. For these models, neural networks can efficiently infer the amplitudes of the expansion coefficients in ground-state wave functions. Additionally, we explore factors that affect the inference accuracy, such as ensemble learning, data normalization, construction of the data set, and neural network structure. Finally, by combining the inferred amplitudes with the leading-order sign rules, we can construct trial wave functions that closely approximate the ground-state wave functions.