  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.
