Quantum kernel methods are promising candidates for achieving a practical quantum advantage for certain machine learning tasks. Similar to classical machine learning, an exact form of a quantum kernel is expected to have a great impact on the model performance. In this work we combine all trace-induced quantum kernels, including the commonly-used global fidelity and local projected quantum kernels, into a common framework. We show how generalized trace-induced quantum kernels can be constructed as combinations of the fundamental building blocks we coin "Lego" kernels, which impose an inductive bias on the resulting quantum models. We relate the expressive power and generalization ability to the number of non-zero weight Lego kernels and propose a systematic approach to increase the complexity of a quantum kernel model, leading to a new form of the local projected kernels that require fewer quantum resources in terms of the number of quantum gates and measurement shots. We show numerically that models based on local projected kernels can achieve comparable performance to the global fidelity quantum kernel. Our work unifies existing quantum kernels and provides a systematic framework to compare their properties.

翻译：量子核方法被认为是实现某些机器学习任务实际量子优势的有前景候选方法。类似于经典机器学习，量子核的精确形式对模型性能具有重要影响。在本工作中，我们将所有迹诱导量子核（包括常用的全局保真度核和局部投影量子核）统一至一个共同框架。我们展示了如何将广义迹诱导量子核构建为基本构建块（我们称之为“乐高”核）的组合，这些构建块为生成的量子模型施加了归纳偏差。我们将表达能力和泛化能力与非零权重乐高核的数量相关联，并提出一种系统方法来增加量子核模型的复杂度，从而衍生出一种新型局部投影核，该核在量子门数量和测量次数方面需要更少的量子资源。我们的数值实验表明，基于局部投影核的模型可实现与全局保真度量子核相当的性能。本工作统一了现有量子核，并提供了比较其性质的系统框架。