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

TL;DR

The quantum path kernel (QPK) is introduced, 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.

Abstract

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.

Figures (11)

• Figure 1: Computation of the Path Kernel. Bottom left: A typical parameter trajectory $\gamma$ is depicted, representing parametric evolution during the training phase. Top left: as $\mathbf{\bm{\theta}}$ evolves, it gives rise to differing NTK matrices, corresponding to distinct representations of the data. Such a sequence of matrices thus give rise to a hierarchical stack of representations in the feature learning regime. Middle: as the training approaches convergence, subsequent matrices become similar to each other, and thus their corresponding representations are correlated. Right: the Path Kernel constitutes the average over these representations.
• Figure 2: Gaussian XOR Mixture classification experiment workflow.
• Figure 3: Quantum circuit schematic of the classification model used for $d=3$ qubits and $L=2$ layers.
• Figure 4: Behavior of the quantum machine learning models $f(\mathbf{\bm{x}}; \mathbf{\bm{\theta}})$ over the training phase. (\ref{['fig:n01_dataset']}) illustrates the training dataset for the parameter selection $d=4, \epsilon=0.1$; (\ref{['fig:n01_loss']}) shows the evolving loss for each of the 20 evaluated depthwise models ($L=1, ..., 20$) during training; (\ref{['fig:n01_param']}) quantifies the deviation of the parameter vector from its initialization. (\ref{['fig:n04_dataset']}-\ref{['fig:n04_loss']}-\ref{['fig:n04_param']}) show the corresponding information when $d=4, \epsilon=0.4$; (\ref{['fig:n10_dataset']}-\ref{['fig:n10_loss']}-\ref{['fig:n10_param']}) for $d=4, \epsilon=1.0$.
• Figure 5: Respective test accuracy scores for the Quantum Path Kernel model, the Quantum NTK and the oracle. Error bars represents the standard deviation over three (otherwise identical) experiments having parametric specifications $d=4,\epsilon=0.1$, (\ref{['fig:n01_generr']}); $d=4, \epsilon=0.4$ (\ref{['fig:n04_generr']}); $d=4, \epsilon=1.0$ (\ref{['fig:n10_generr']}).