Quantum positive matrix-positive matrix multiplication algorithm

Abstract In quantum computing,
matrix multiplication occurs as a subroutine in more complex algorithms. 
We improve the quantum linear system algorithm 
[Harrow et al . 2009 Phys. Rev. Lett. 103 150502] 
to perform sparse matrix-matrix multiplication and propose a subroutine for parallel vector addition. 
Based on them, a quantum algorithm is exhibited for N × N positive matrix-positive matrix multiplication.
For some small ratio of the maximum element to the minimum element of the positive matrices (i.e., O (poly log N )), 
this algorithm achieves exponential acceleration.