Quantum kernel (QK) methods exploit quantum computers to calculate QKs for the use of kernel-based learning models. Despite a potential quantum advantage of the method, the commonly used fidelity-based QK suffers from a detrimental issue, which we call the vanishing similarity issue; the exponential decay of the expectation value and the variance of the QK deteriorates implementation feasibility and trainability of the model with the increase of the number of qubits. This implies the need to design QKs alternative to the fidelity-based one. In this work, we propose a new class of QKs called the quantum Fisher kernels (QFKs) that take into account the geometric structure of the data source. We analytically and numerically demonstrate that the QFK can avoid the issue when shallow alternating layered ansatzes are used. In addition, the Fourier analysis numerically elucidates that the QFK can have the expressivity comparable to the fidelity-based QK. Moreover, we demonstrate synthetic classification tasks where QFK outperforms the fidelity-based QK in performance due to the absence of vanishing similarity. These results indicate that QFK paves the way for practical applications of quantum machine learning toward possible quantum advantages.

中文翻译：

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用于缓解相似性消失问题的量子费希尔内核


量子核 (QK) 方法利用量子计算机来计算 QK，以便使用基于核的学习模型。尽管该方法具有潜在的量子优势，但常用的基于保真度的 QK 存在一个有害问题，我们称之为相似性消失问题；随着量子比特数量的增加，QK 的期望值和方差的指数衰减会恶化模型的实现可行性和可训练性。这意味着需要设计 QK 来替代基于保真度的 QK。在这项工作中，我们提出了一类新的 QK，称为量子费希尔核 (QFK)，它考虑了数据源的几何结构。我们通过分析和数值证明，当使用浅层交替分层结构时，QFK 可以避免该问题。此外，傅里叶分析在数值上阐明了QFK可以具有与基于保真度的QK相当的表达能力。此外，我们还演示了综合分类任务，其中由于不存在消失的相似性，QFK 在性能上优于基于保真度的 QK。这些结果表明 QFK 为量子机器学习的实际应用铺平了道路，以获得可能的量子优势。