The Quantum Path Kernel: A Generalized Neural Tangent Kernel for Deep Quantum Machine Learning

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 nonlinearity 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 literature, principally via the introduction of measurements between layers of unitary transformations. In this article, we introduce the quantum path kernel (QPK), 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, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hierarchical feature learning</i>. 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 QPK exploits the parameter trajectory, i.e., the curve delineated by model parameters as they evolve during training, enabling the representation of differential layerwise 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 <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xor</small>mixtures: an artificial but emblematic problem that intrinsically requires multilevel learning in order to achieve optimal class separation.