New perspectives on quantum kernels through the lens of entangled tensor kernels

TL;DR

This work introduces entangled tensor kernels (ETKs) as a natural generalization of product kernels and proves that embedding quantum kernels are ETKs, enabling a unified analysis of quantum kernels through tensor-network concepts. It shows that ETKs can be efficiently evaluated via matrix-product-operator (MPO) methods when the MPO bond-dimension is polynomial, while also highlighting the possibility for quantum kernels to realize super-polynomial bond-dimension ETKs, which could resist classical contraction. The authors demonstrate how a Mercer decomposition can be obtained within the ETK framework for a single-layer quantum-kernel family, linking eigenstructure to inductive biases and generalization behavior, and present numerical results contrasting Haar-random and sparse pre-encoding schemes to illustrate spectrum concentration effects. Collectively, the ETK lens offers a principled route to study, design, and potentially dequantize quantum kernels, with concrete implications for kernel target alignment, generalization, and classical surrogates.

Abstract

Quantum kernel methods are one of the most explored approaches to quantum machine learning. However, the structural properties and inductive bias of quantum kernels are not fully understood. In this work, we introduce the notion of entangled tensor kernels - a generalization of product kernels from classical kernel theory - and show that all embedding quantum kernels can be understood as an entangled tensor kernel. We discuss how this perspective allows one to gain novel insights into both the unique inductive bias of quantum kernels, and potential methods for their dequantization.

Figures (6)

• Figure 1: A circuit diagram for quantum kernel evaluation -- i.e. for the evaluation of Eq. \ref{['eq:quantum_kernel']}. White boxes represent data-dependent quantum gates, while gray ones are non-parametrized gates.
• Figure 2: An overview of the conjectured landscape of ETKs, for some fixed set of local feature maps derived from data pre-processing functions $\{\phi\}$. Region (a) is the set of all such ETKs, and region (b) is the subset of those ETKs whose core tensor admits a decomposition into a polynomial bond-dimension MPO, and can therefore be evaluated efficiently classically (when given this MPO). Region (c) contains all those ETKs which can be evaluated using an embedding quantum kernel, and region (d) is the subset of those ETKs which can be evaluated using a polynomial depth quantum circuit -- i.e. those which can be evaluated efficiently quantumly. In Section \ref{['sss:inductive_bias']} we discuss the inductive bias which distinguishes quantum kernels that are in region (d), but not in region (b). Region (f) is the subset of efficient quantum kernels which also generalize efficiently (for some problem of interest). We identify these quantum kernels as naturally interesting quantum kernels for further exploration, and discuss candidates in Section \ref{['ss:summary']}. Region (e) is the subset of quantum kernels that can in principle be evaluated efficiently classically. The existence of this set suggests that ETKs with poly-bond-dimension MPOs can sometimes be used to dequantize quantum kernels, which is discussed in Section \ref{['ss:classical_surrogates']}. Whether or not region (b) is a proper subset of region (c) is an open question.
• Figure 3: The scaling of the largest eigenvalues with respect to the number of qubits. All data points are averaged over 30 different instances and shaded regions represent standard deviation. Concentrated positions of each concentrated unitary are chosen randomly for all instances.
• Figure 4: Eigenvalue spectra of quantum kernels satisfying two assumptions in Sec. \ref{['ss:example_onelayer']}. These plots are from one instance among $30$ experiments.
• Figure 5: Learning curves for kernel ridge regression tasks on the model-tailored datasets. Data points are averaged over 30 random models, and shaded regions stand for the standard deviation. Total number of training samples is $2\times 3^n$, and the test set ratio was set to $0.2$. Training was performed with the KernelRidgeRegression implementation from the scikit-learn Python package.