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.

翻译：自核理论进入量子机器学习领域以来，量子核方法（QKMs）在探索潜在应用和提供引人入胜的研究见解方面日益受到关注。目前计算基础Gram矩阵主要有两种方法：保真度量子核（FQKs）和投影量子核（PQKs）。对这些方法进行基准测试对于获得稳健的见解并理解其实际效用至关重要。本研究基于FQKs和PQKs，针对多种设计方案，对量子核方法进行了全面的大规模考察。我们的研究涵盖了五个数据集族、64个数据集的分类与回归任务，系统比较了基于FQKs和PQKs的量子支持向量机与核岭回归的应用。这产生了超过20,000个模型，我们采用最先进的超参数搜索对其进行训练和优化，以确保获得稳健且全面的见解。我们深入探讨了超参数对模型性能评分的重要性，并通过严谨的相关性分析支持我们的发现。在此过程中，我们还详细考察了两种数据编码策略。此外，我们提供了深入分析，探讨了PQKs的设计自由度，并探究了其学习背后的基本原理。我们的目标并非为特定任务找出性能最佳的模型，而是揭示导致有效量子核方法的机制，并发现普遍性规律。