通过Fisher信息度量的无监督机器学习检测量子和经典相变
Detecting Quantum and Classical Phase Transitions via Unsupervised Machine Learning of the Fisher Information Metric
AI总结：
提出无监督机器学习方法ClassiFIM，通过有限样本估计Fisher信息度量（FIM），在缺少序参量下可靠检测量子和经典相变，且仅需经典测量且训练数据需求更少。
摘要：
在缺少序参量的情况下，利用Fisher信息度量（FIM，也称为保真率）可以检测量子和经典相变。在此，我们提出并研究了一项无监督机器学习（ML）任务：在给定整个相图中测量所得多元概率分布的有限样本情况下，估计FIM。我们使用一种名为ClassiFIM（在配套论文中开发）的无监督ML方法来解决该任务，并利用多种自旋和费米子模型证明了其在检测量子和经典相变方面的经验有效性。为此，我们生成了几个公开可用的数据集，并附带了真实的FIM。我们发现，ClassiFIM能可靠地检测拓扑（例如XXZ链）和动力学（例如Hubbard模型中的金属-绝缘体相变）量子相变。我们与先前用于检测量子相变的无监督ML方法进行了详细的定量比较。我们证明，在适当的准确度指标方面，ClassiFIM与这些先前的方法具有竞争力，同时与先前方法的原始公式相比，所需的资源密集型训练数据显著减少。特别是，ClassiFIM仅需要经典（单基）测量。作为我们方法论开发的一部分，我们证明了几个通过等式或界限将经典和量子保真率联系起来的定理。我们还大大扩展了保真率的存在条件，例如放宽了标准的可微性条件。这些结果可能引起数学物理学界的独立兴趣。
英文摘要：
The detection of quantum and classical phase transitions in the absence of an order parameter is possible using the Fisher information metric (FIM), also known as fidelity susceptibility. Here, we propose and investigate an unsupervised machine learning (ML) task: estimating the FIM given limited samples from a multivariate probability distribution of measurements made throughout the phase diagram. We utilize an unsupervised ML method called ClassiFIM (developed in a companion paper) to solve this task and demonstrate its empirical effectiveness in detecting both quantum and classical phase transitions using a variety of spin and fermionic models, for which we generate several publicly available datasets with accompanying ground-truth FIM. We find that ClassiFIM reliably detects both topological (e.g., XXZ chain) and dynamical (e.g., metal-insulator transition in Hubbard model) quantum phase transitions. We perform a detailed quantitative comparison with prior unsupervised ML methods for detecting quantum phase transitions. We demonstrate that ClassiFIM is competitive with these prior methods in terms of appropriate accuracy metrics while requiring significantly less resource-intensive training data compared to the original formulation of the prior methods. In particular, ClassiFIM only requires classical (single-basis) measurements. As part of our methodology development, we prove several theorems connecting the classical and quantum fidelity susceptibilities through equalities or bounds. We also significantly expand the existence conditions of the fidelity susceptibility, e.g., by relaxing standard differentiability conditions. These results may be of independent interest to the mathematical physics community.
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