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：泛化是机器学习模型通过学习训练数据对新数据进行准确预测的能力。然而，理解量子机器学习模型的泛化能力一直是一个重大挑战。本文提出了数据量子Fisher信息度量（DQFIM），它描述了依赖于变分拟设、训练数据及其对称性的变分量子算法的容量。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。通过使用动力学李代数，我们解释了如何利用少量训练态实现泛化。反直觉的是，打破训练数据的对称性有助于改善泛化效果。最后，我们发现当训练数据和测试数据来自不同数据分布时（即分布外泛化），其效果可能优于使用相同分布的情况。本研究为探索量子机器学习模型的能力提供了有用的框架。