标题： 卷积神经网络对量子自旋系统的低维近似解释 显示英文标题

标题： Interpreting convolutional neural networks' low dimensional approximation to quantum spin systems

摘要： 卷积神经网络（CNNs）与变分蒙特卡洛方法一起被成功应用于寻找量子多体自旋系统的基态。然而，为了实现这一点，必须绕过“维数灾难”，即一个只有线性变分参数的CNN需要成功地在一个指数级大的希尔伯特空间中近似波函数。在我们的工作中，我们提供了对CNN如何优化自旋系统学习的理论和实验分析，并研究了CNN的低维近似。我们首先量化了训练过程中底层自旋系统的物理对称性所起的作用。我们将这些见解融入到一种新的训练算法中，并展示了其更高的效率、准确性和鲁棒性。然后，我们进一步通过研究卷积核大小捕获的纠缠光谱来调查CNN近似波函数的能力。我们的洞察揭示了CNN本质上是以输入字符串的$K$模题出现统计为中心的假设。我们利用这一动机，用统一的理论解释将浅层CNN假设与其他著名的统计和物理假设（如最大熵（MaxEnt）和纠缠方块关联乘积态（EP-CPS））联系起来。通过回归分析，我们发现了CNN对不同模题期望值近似的进一步关系。我们的结果使我们能够全面、深入地理解CNN如何成功地近似量子自旋哈密顿量，并利用这种理解来提高CNN的性能。 显示英文摘要

摘要： Convolutional neural networks (CNNs) have been employed along with Variational Monte Carlo methods for finding the ground state of quantum many-body spin systems with great success. In order to do so, however, a CNN with only linearly many variational parameters has to circumvent the ``curse of dimensionality'' and successfully approximate a wavefunction on an exponentially large Hilbert space. In our work, we provide a theoretical and experimental analysis of how the CNN optimizes learning for spin systems, and investigate the CNN's low dimensional approximation. We first quantify the role played by physical symmetries of the underlying spin system during training. We incorporate our insights into a new training algorithm and demonstrate its improved efficiency, accuracy and robustness. We then further investigate the CNN's ability to approximate wavefunctions by looking at the entanglement spectrum captured by the size of the convolutional filter. Our insights reveal the CNN to be an ansatz fundamentally centered around the occurrence statistics of $K$-motifs of the input strings. We use this motivation to provide the shallow CNN ansatz with a unifying theoretical interpretation in terms of other well-known statistical and physical ansatzes such as the maximum entropy (MaxEnt) and entangled plaquette correlator product states (EP-CPS). Using regression analysis, we find further relationships between the CNN's approximations of the different motifs' expectation values. Our results allow us to gain a comprehensive, improved understanding of how CNNs successfully approximate quantum spin Hamiltonians and to use that understanding to improve CNN performance.