Quantum machine learning (QML) shows promise for analyzing quantum data. A notable example is the use of quantum convolutional neural networks (QCNNs), implemented as specific types of quantum circuits, to recognize phases of matter. In this approach, ground states of many-body Hamiltonians are prepared to form a quantum dataset and classified in a supervised manner using only a few labeled examples. However, this type of dataset and model differs fundamentally from typical QML paradigms based on feature maps and parameterized circuits. In this study, we demonstrate how models utilizing quantum data can be interpreted through hidden feature maps, where physical features are implicitly embedded via ground-state feature maps. By analyzing selected examples previously explored with QCNNs, we show that high performance in quantum phase recognition comes from generating a highly effective basis set with sharp features at critical points. The learning process adapts the measurement to create sharp decision boundaries. Our analysis highlights improved generalization when working with quantum data, particularly in the limited-shots regime. Furthermore, translating these insights into the domain of quantum scientific machine learning, we demonstrate that ground-state feature maps can be applied to fluid dynamics problems, expressing shock wave solutions with good generalization and proven trainability.

翻译：量子机器学习（QML）在分析量子数据方面展现出潜力。一个显著的例子是使用量子卷积神经网络（QCNN）——实现为特定类型的量子电路——来识别物质相。在此方法中，多体哈密顿量的基态被制备以形成量子数据集，并仅使用少量标记样本以监督方式进行分类。然而，这类数据集和模型与基于特征映射和参数化电路的典型QML范式存在根本差异。在本研究中，我们展示了如何通过隐式特征映射来解释利用量子数据的模型，其中物理特征通过基态特征映射隐式嵌入。通过分析先前用QCNN探索过的选定案例，我们表明量子相识别的高性能源于生成具有临界点处尖锐特征的高效基组。学习过程通过调整测量来创建尖锐的决策边界。我们的分析突显了在处理量子数据时，特别是在有限测量次数条件下，泛化能力得到提升。此外，将这些见解转化到量子科学机器学习领域，我们证明了基态特征映射可应用于流体动力学问题，能够表达激波解并展现出良好的泛化能力及可证明的可训练性。