The matrix logarithm is one of the important matrix functions. Recently, a quantum algorithm that computes the state $|f\rangle$ corresponding to matrix-vector product $f(A)b$ is proposed in [Takahira, et al. Quantum algorithm for matrix functions by Cauchy's integral formula, QIC, Vol.20, No.1\&2, pp.14-36, 2020]. However, it can not be applied to matrix logarithm. In this paper, we propose a quantum algorithm, which uses LCU method and block-encoding technique as subroutines, to compute the state $|f\rangle = \log(A)|b\rangle / \|\log(A)|b\rangle\|$ corresponding to $\log(A)b$ via the integral representation of $\log(A)$ and the Gauss-Legendre quadrature rule.
翻译：矩阵对数是重要矩阵函数之一 。 最近, [Takahira, et al. Quantum 算法, 由 Cauchy 的整体公式, QIC, Vol.20, No.1 ⁇ 2, pp.14-36, 2020] 为矩阵对数函数提议了一个量子算法, 该算法使用 LCU 方法和块编码技术作为子例程法, 来计算国家 $f\ rangle =\ log(A)\\\\\ rangle / log(A)\\b\ rangle = $\log(A) 的矩阵函数。 但是, 它不能应用到矩阵对数法 。 在本文中, 我们提出一个量子算法, 通过 $\log(A) 和 Gaus- Legendre 矩形规则, 来计算国家 $\log(A) 和 Gaus- Legendre 等值 。
相关内容
微信扫码咨询专知VIP会员