The Gaussian kernel is a very popular kernel function used in many machine learning algorithms, especially in support vector machines (SVMs). It is more often used than polynomial kernels when learning from nonlinear datasets and is usually employed in formulating the classical SVM for nonlinear problems. Rebentrost et al. discussed an elegant quantum version of a least square support vector machine using quantum polynomial kernels, which is exponentially faster than the classical counterpart. This paper demonstrates a quantum version of the Gaussian kernel and analyzes its runtime complexity using the quantum random access memory (QRAM) in the context of quantum SVM. Our analysis shows that the runtime computational complexity of the quantum Gaussian kernel is approximated to [Formula: see text] and even [Formula: see text] when [Formula: see text] and the error [Formula: see text] are small enough to be ignored, where [Formula: see text] is the dimension of the training instances, [Formula: see text] is the accuracy, [Formula: see text] is the dot product of the two quantum states, and [Formula: see text] is the Taylor remainder error term. Therefore, the run time complexity of the quantum version of the Gaussian kernel seems to be significantly faster when compared with its classical version.

中文翻译：

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量子学习中的高斯核

高斯核是一种非常流行的核函数，用于许多机器学习算法，尤其是支持向量机 (SVM)。在从非线性数据集中学习时，它比多项式核更常用，并且通常用于为非线性问题制定经典的 SVM。Rebentrost 等人。讨论了使用量子多项式核的最小二乘支持向量机的优雅量子版本，它比经典对应物快成指数。本文演示了高斯核的量子版本，并在量子 SVM 的背景下使用量子随机存取存储器 (QRAM) 分析了其运行时复杂性。我们的分析表明，当[公式：见正文]时，量子高斯核的运行时计算复杂度近似为[公式：见正文]，甚至[公式：见正文]。see text] 和误差 [Formula: see text] 小到可以忽略不计，其中 [Formula: see text] 是训练实例的维度，[Formula: see text] 是准确率，[Formula: see text]是两个量子态的点积，[公式：见正文]是泰勒余数误差项。因此，与经典版本相比，高斯核的量子版本的运行时间复杂度似乎要快得多。