  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.
