A key problem in the field of quantum computing is understanding whether quantum machine learning (QML) models implemented on noisy intermediate-scale quantum (NISQ) machines can achieve quantum advantages. Recently, Huang et al. [Nat Commun 12, 2631] partially answered this question by the lens of quantum kernel learning. Namely, they exhibited that quantum kernels can learn specific datasets with lower generalization error over the optimal classical kernel methods. However, most of their results are established on the ideal setting and ignore the caveats of near-term quantum machines. To this end, a crucial open question is: does the power of quantum kernels still hold under the NISQ setting? In this study, we fill this knowledge gap by exploiting the power of quantum kernels when the quantum system noise and sample error are considered. Concretely, we first prove that the advantage of quantum kernels is vanished for large size of datasets, few number of measurements, and large system noise. With the aim of preserving the superiority of quantum kernels in the NISQ era, we further devise an effective method via indefinite kernel learning. Numerical simulations accord with our theoretical results. Our work provides theoretical guidance of exploring advanced quantum kernels to attain quantum advantages on NISQ devices.

中文翻译：

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理解 NISQ 时代量子内核的力量

量子计算领域的一个关键问题是了解在嘈杂的中级量子 (NISQ) 机器上实施的量子机器学习 (QML) 模型是否可以实现量子优势。最近，黄等人。[Nat Commun 12, 2631] 从量子核学习的角度部分回答了这个问题。也就是说，他们展示了量子核可以学习特定的数据集，并且比最佳经典核方法具有更低的泛化误差。然而，他们的大部分结果都是建立在理想环境上的，而忽略了近期量子机器的警告。为此，一个关键的悬而未决的问题是：在 NISQ 设置下，量子内核的力量是否仍然有效？在这项研究中，我们通过在考虑量子系统噪声和样本误差时利用量子核的力量来填补这一知识空白。具体来说，我们首先证明了量子核的优势在数据集大、测量次数少和系统噪声大的情况下消失了。为了保持 NISQ 时代量子核的优越性，我们进一步设计了一种通过无限核学习的有效方法。数值模拟符合我们的理论结果。我们的工作为探索先进的量子内核以在 NISQ 设备上获得量子优势提供了理论指导。