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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 κ 3 ϵ - 3 log (N m n) and O log (m n) , respectively, where κ is the condition number of the corresponding matrix of the HHL algorithm, ϵ 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 [ κ 2 log 1.5 (κ / ϵ) log (N m n) ] . [ABSTRACT FROM AUTHOR]

Quantum walks are the quantum corresponding version of classical random walks, providing polynomial and even exponential acceleration for some classical difficult problems. In recent years, there have been numerous researches on quantum walks, among which the hitting probabilities problem is another crucial problem. In 2021, Guan et al. have proposed an HHL-based algorithm to calculate the hitting probabilities of quantum walks, which can effectively transform the original problem into an inverse matrix problem, then achieve an acceleration effect on the corresponding classical algorithm under some specific conditions. However, we propose a new algorithm consisting of an improved quantum matrix inversion algorithm with complexity O κ 2 log 1. 5 (κ / ε) p o l y log (n) , where κ is the condition number of the matrix, n + 1 is the number of positions of the one-dimensional quantum walks, and ε is the expected accuracy of the output state. Compared to the HHL-based algorithm, our improved algorithm achieves an exponential improvement in the dependence on accuracy. [ABSTRACT FROM AUTHOR]