Quantum kernel method is one of the key approaches to quantum machine learning, which has the advantages that it does not require optimization and has theoretical simplicity. By virtue of these properties, several experimental demonstrations and discussions of the potential advantages have been developed so far. However, as is the case in classical machine learning, not all quantum machine learning models could be regarded as kernel methods. In this work, we explore a quantum machine learning model with a deep parameterized quantum circuit and aim to go beyond the conventional quantum kernel method. In this case, the representation power and performance are expected to be enhanced, while the training process might be a bottleneck because of the barren plateaus issue. However, we find that parameters of a deep enough quantum circuit do not move much from its initial values during training, allowing first-order expansion with respect to the parameters. This behavior is similar to the neural tangent kernel in the classical literatures, and such a deep variational quantum machine learning can be described by another emergent kernel, quantum tangent kernel. Numerical simulations show that the proposed quantum tangent kernel outperforms the conventional quantum kernel method for an ansatz-generated dataset. This work provides a new direction beyond the conventional quantum kernel method and explores potential power of quantum machine learning with deep parameterized quantum circuits.
Quantum kernel method is one of the key approaches to quantum machine learning, which has the advantages that it does not require optimization and has theoretical simplicity. By virtue of these properties, several experimental demonstrations and discussions of the potential advantages have been developed so far. However, as is the case in classical machine learning, not all quantum machine learning models could be regarded as kernel methods. In this work, we explore a quantum machine learning model with a deep parameterized quantum circuit and aim to go beyond the conventional quantum kernel method. In this case, the representation power and performance are expected to be enhanced, while the training process might be a bottleneck because of the barren plateaus issue. However, we find that parameters of a deep enough quantum circuit do not move much from its initial values during training, allowing first-order expansion with respect to the parameters. This behavior is similar to the neural tangent kernel in the classical literatures, and such a deep variational quantum machine learning can be described by another emergent kernel, quantum tangent kernel. Numerical simulations show that the proposed quantum tangent kernel outperforms the conventional quantum kernel method for an ansatz-generated dataset. This work provides a new direction beyond the conventional quantum kernel method and explores potential power of quantum machine learning with deep parameterized quantum circuits.
The quantum kernel method is one of the key approaches to quantum machine learning, which has the advantage of not requiring optimization and its theoretical simplicity. By virtue of these properties, several experimental demonstrations and discussions of the potential advantages have been developed so far. However, as is the case in classical machine learning, not all quantum machine learning models could be regarded as kernel methods. In this work, we explore a quantum machine learning model with a deep parametrized quantum circuit and aim to go beyond the conventional quantum kernel method. In this case, the expressive power and performance are expected to be enhanced, while the training process might be a bottleneck because of the barren plateaus. Moreover, the high computational cost of gradient-based optimization and the large search space of the gradient-free optimization directly make the training intractable. However, we find that parameters of a deep enough quantum circuit do not move much from their initial values during training, allowing for a first-order expansion with respect to the parameters. This behavior is similar to that of the neural tangent kernel in classical machine learning, and such a quantum machine learning with deep variational quantum circuits can be described by another emergent kernel, the . We show that the proposed quantum tangent kernel has the potential to outperform the conventional quantum kernel method by performing a classification task on an ansatz-generated dataset. This work provides a different direction beyond the conventional quantum kernel method and explores the potential power of quantum machine learning with deep parametrized quantum circuits. Published by the American Physical Society 2024
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.
Quantum machine learning methods, including quantum kernels and variational quantum algorithms (VQAs), are poised to demonstrate quantum advantages in machine learning, chemistry, condensed matter physics, and materials science. However, these methods face critical challenges hindering practical quantum advantates: quantum kernels require meticulous design to outperform classical alternatives, while VQA training is obstructed by barren plateaus (BPs) in the optimization landscape. To address these issues, this thesis proposes two novel frameworks to guide quantum kernel design and tackle quantum neural network (QNN) training challenges in VQAs. Specifically, to resolve quantum kernel design challenges, we propose a data-driven method for automatic quantum kernel design, leveraging classical machine learning to learn from input data and identify optimal kernel structures. Experiments show this method finds suitable quantum kernels that avoid vanishing similarity, a common flaw in traditional designs. To improve VQA trainability, we propose an encoder-decoder parameter prediction framework, where a graph neural network (GNN) acts as the encoder to process information based on the QNN’s directed acyclic graph (DAG). Experiments confirm the parameters from this framework significantly enhance VQAs’ robustness, convergence speed, and trainability. In conclusion, we propose two schemes, QuKerNet and QuGHN, for automatic quantum kernel design and enhanced VQA trainability. QuKerNet considers optimal circuit layouts and variational parameters, and is more resource-efficient for contemporary quantum hardware by accounting for qubit topology and count. QuGHN is versatile, applicable to circuits with varying qubits, ansatzes, and tasks; its parameters enable faster convergence and more stable VQA solutions. These advancements enhance quantum machine learning practicality on near-term machines and its utility in large-scale quantum circuit simulations.
Quantum kernel methods (QKMs) have emerged as a prominent framework for supervised quantum machine learning. Unlike variational quantum algorithms, which rely on gradient-based optimisation and may suffer from issues such as barren plateaus, non-variational QKMs employ fixed quantum feature maps, with model selection performed classically via convex optimisation and cross-validation. This separation of quantum feature embedding from classical training ensures stable optimisation while leveraging quantum circuits to encode data in high-dimensional Hilbert spaces. In this review, we provide a thorough analysis of non-variational supervised QKMs, covering their foundations in classical kernel theory, constructions of fidelity and projected quantum kernels, and methods for their estimation in practice. We examine frameworks for assessing quantum advantage, including generalisation bounds and necessary conditions for separation from classical models, and analyse key challenges such as exponential concentration, dequantisation via tensor-network methods, and the spectral properties of kernel integral operators. We further discuss structured problem classes that may enable advantage, and synthesise insights from comparative and hardware studies. Overall, this review aims to clarify the regimes in which QKMs may offer genuine advantages, and to delineate the conceptual, methodological, and technical obstacles that must be overcome for practical quantum-enhanced learning.
Abstract Predicting the performance of a Quantum Neural Network (QNN) on a given dataset before its training can save critical resources. However, obtaining such a priori information is, in general, very challenging. The Quantum Neural Tangent Kernel (QNTK) has recently emerged as a tool to mathematically describe the behavior of QNNs. Here, we provide practical guidelines for using the QNTK to diagnose the performance of a given QNN model. In particular, we link QNTK spectral features to the training dynamics, and we show how a first-order kernel approximation can predict the expected inference performance. By analyzing the interplay between the number of model parameters and of training samples, we identify a mismatch at the interpolation threshold: while the QNTK exhibits a double descent in generalization error, we observe that QNNs do not show this behavior in practice. This suggests that QNTK diagnostics are mostly reliable for overparametrized models well above the interpolation regime. Extensive numerical simulations across different architectures and datasets validate our approach. Our results demonstrate that QNTK diagnostics yields useful information about QNN behavior for sufficiently deep circuits, and enable detecting and addressing potential shortcomings in model design.
Abstract Recently, quantum neural networks or quantum–classical neural networks (qcNN) have been actively studied, as a possible alternative to the conventional classical neural network (cNN), but their practical and theoretically-guaranteed performance is still to be investigated. In contrast, cNNs and especially deep cNNs, have acquired several solid theoretical basis; one of those basis is the neural tangent kernel (NTK) theory, which can successfully explain the mechanism of various desirable properties of cNNs, particularly the global convergence in the training process. In this paper, we study a class of qcNN composed of a quantum data-encoder followed by a cNN. The quantum part is randomly initialized according to unitary 2-designs, which is an effective feature extraction process for quantum states, and the classical part is also randomly initialized according to Gaussian distributions; then, in the NTK regime where the number of nodes of the cNN becomes infinitely large, the output of the entire qcNN becomes a nonlinear function of the so-called projected quantum kernel. That is, the NTK theory is used to construct an effective quantum kernel, which is in general nontrivial to design. Moreover, NTK defined for the qcNN is identical to the covariance matrix of a Gaussian process, which allows us to analytically study the learning process. These properties are investigated in thorough numerical experiments; particularly, we demonstrate that the qcNN shows a clear advantage over fully classical NNs and qNNs for the problem of learning the quantum data-generating process.