Gaussian kernel in quantum learning

Abstract

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.