On the similarity of bandwidth-tuned quantum kernels and classical kernels

Quantum kernels (QK) are widely used in quantum machine learning applications; yet, their potential to surpass classical machine learning methods on classical datasets remains uncertain. This limitation can be attributed to the exponential concentration phenomenon, which can impair both trainability and generalization. A common strategy to alleviate this is bandwidth tuning, which involves rescaling data points in the quantum model to improve generalization. In this work, we numerically demonstrate that optimal bandwidth tuning results in QKs that closely resemble radial basis function (RBF) kernels, leading to a lack of quantum advantage over classical methods. Moreover, we reveal that the size of optimal bandwidth tuning parameters further simplifies QKs, causing them to behave like polynomial kernels, corresponding to a low-order Taylor approximation of a RBF kernel. We thoroughly investigate this for fidelity quantum kernels and projected quantum kernels using various data encoding circuits across several classification datasets. We provide numerical evidence and derive a simple analytical model that elucidates how bandwidth tuning influences key quantities in classification tasks. Overall, our findings shed light on the mechanisms that render QK methods classically simulatable.

Quantum Kernel Methods under Scrutiny: A Benchmarking Study

Since the entry of kernel theory in the field of quantum machine learning, quantum kernel methods (QKMs) have gained increasing attention with regard to both probing promising applications and delivering intriguing research insights. Two common approaches for computing the underlying Gram matrix have emerged: fidelity quantum kernels (FQKs) and projected quantum kernels (PQKs). Benchmarking these methods is crucial to gain robust insights and to understand their practical utility. In this work, we present a comprehensive large-scale study examining QKMs based on FQKs and PQKs across a manifold of design choices. Our investigation encompasses both classification and regression tasks for five dataset families and 64 datasets, systematically comparing the use of FQKs and PQKs quantum support vector machines and kernel ridge regression. This resulted in over 20,000 models that were trained and optimized using a state-of-the-art hyperparameter search to ensure robust and comprehensive insights. We delve into the importance of hyperparameters on model performance scores and support our findings through rigorous correlation analyses. In this, we also closely inspect two data encoding strategies. Moreover, we provide an in-depth analysis addressing the design freedom of PQKs and explore the underlying principles responsible for learning. Our goal is not to identify the best-performing model for a specific task but to uncover the mechanisms that lead to effective QKMs and reveal universal patterns.

sQUlearn $\unicode{x2013}$ A Python Library for Quantum Machine Learning

sQUlearn introduces a user-friendly, NISQ-ready Python library for quantum machine learning (QML), designed for seamless integration with classical machine learning tools like scikit-learn. The library's dual-layer architecture serves both QML researchers and practitioners, enabling efficient prototyping, experimentation, and pipelining. sQUlearn provides a comprehensive toolset that includes both quantum kernel methods and quantum neural networks, along with features like customizable data encoding strategies, automated execution handling, and specialized kernel regularization techniques. By focusing on NISQ-compatibility and end-to-end automation, sQUlearn aims to bridge the gap between current quantum computing capabilities and practical machine learning applications.