An improved quantum algorithm for support matrix machines

Abstract

With the rapid growth of video and image technology, the binary classification of matrices has attracted much attention. In 2017, Duan et al. proposed a quantum algorithm for support matrix machines (QSMM) that efficiently addresses image classification problems. The QSMM consists of two core subroutines: an HHL algorithm and a quantum singularity threshold (QSVT) algorithm with complexities of $O\left[ \kappa ^{3} \epsilon ^{-3} \log (N m n)\right]$ and $O\left[ \log (m n)\right]$, respectively, where $\kappa$ is the condition number of the corresponding matrix of the HHL algorithm, $\epsilon$ is the expected accuracy of the output state, N is the number of samples in the training set and mn is the size of the feature space. The QSMM achieves an exponential increase in speed over classical counterparts. However, we find that Duan’s QSMM can be improved by applying an improved quantum matrix inversion (QMI) algorithm instead of the HHL algorithm. Compared with that of Duan’s QSMM, the dependence on precision of our improved QSMM (IQSMM) is exponentially improved, and the complexity of our first subroutine is as low as $O[\kappa ^2\log ^{1.5}(\kappa / \epsilon ) \log (Nmn)]$.