  Building a quantum analog of classical deep neural networks represents a
fundamental challenge in quantum computing. A key issue is how to address the
inherent non-linearity of classical deep learning, a problem in the quantum
domain due to the fact that the composition of an arbitrary number of quantum
gates, consisting of a series of sequential unitary transformations, is
intrinsically linear. This problem has been variously approached in the
literature, principally via the introduction of measurements between layers of
unitary transformations. In this paper, we introduce the Quantum Path Kernel, a
formulation of quantum machine learning capable of replicating those aspects of
deep machine learning typically associated with superior generalization
performance in the classical domain, specifically, hierarchical feature
learning. Our approach generalizes the notion of Quantum Neural Tangent Kernel,
which has been used to study the dynamics of classical and quantum machine
learning models. The Quantum Path Kernel exploits the parameter trajectory,
i.e. the curve delineated by model parameters as they evolve during training,
enabling the representation of differential layer-wise convergence behaviors,
or the formation of hierarchical parametric dependencies, in terms of their
manifestation in the gradient space of the predictor function. We evaluate our
approach with respect to variants of the classification of Gaussian XOR
mixtures - an artificial but emblematic problem that intrinsically requires
multilevel learning in order to achieve optimal class separation.
