A matrix can be converted into a doubly stochastic matrix by using two diagonal matrices. And a doubly stochastic matrix can be written as a sum of permutation matrices. In this paper, we describe a method to write a given generic matrix in terms of quantum gates based on the block encoding. In particular, we first show how to convert a matrix into doubly stochastic matrices and by using Birkhoff's algorithm, we express that matrix in terms of a linear combination of permutations which can be mapped to quantum circuits. We then discuss a few optimization techniques that can be applied in a possibly future quantum compiler software based on the method described here.

翻译：利用两个对角矩阵可将任意矩阵转化为双随机矩阵，而双随机矩阵又可表示为置换矩阵的和。本文提出一种基于块编码理论将给定通用矩阵用量子门表示的方法。具体而言，我们首先展示如何将矩阵转换为双随机矩阵，并利用Birkhoff算法将该矩阵表示为可映射至量子电路的置换矩阵线性组合。随后，我们讨论若干优化技术，这些技术可应用于基于本文方法构建的未来量子编译器软件中。