Generalization is the ability of machine learning models to make accurate predictions on new data by learning from training data. However, understanding generalization of quantum machine learning models has been a major challenge. Here, we introduce the data quantum Fisher information metric (DQFIM). It describes the capacity of variational quantum algorithms depending on variational ansatz, training data and their symmetries. We apply the DQFIM to quantify circuit parameters and training data needed to successfully train and generalize. Using the dynamical Lie algebra, we explain how to generalize using a low number of training states. Counter-intuitively, breaking symmetries of the training data can help to improve generalization. Finally, we find that out-of-distribution generalization, where training and testing data are drawn from different data distributions, can be better than using the same distribution. Our work provides a useful framework to explore the power of quantum machine learning models.

翻译：泛化性是指机器学习模型通过从训练数据中学习，对新数据做出准确预测的能力。然而，理解量子机器学习模型的泛化性一直是一个重大挑战。本文引入了数据量子Fisher信息度量（DQFIM），该度量描述了变分量子算法的能力，该能力取决于变分拟设、训练数据及其对称性。我们应用DQFIM来量化成功训练和泛化所需的电路参数和训练数据。利用动力学李代数，我们解释了如何使用少量训练状态实现泛化。反直觉的是，打破训练数据的对称性可能有助于提升泛化性能。最后，我们发现，在训练数据与测试数据来自不同数据分布的分布外泛化场景中，其性能可能优于使用相同分布的情况。我们的工作为探索量子机器学习模型的能力提供了一个有用的理论框架。