Quantum Subroutine for Efficient Matrix Multiplication

We propose an efficient quantum subroutine for matrix multiplication that computes a state vector encoding the entries of the product of two matrices in superposition. The subroutine exploits efficient state preparation techniques and shows a potential speed-up with respect to classical methods. The most important benefit of our subroutine is that it encodes the entries of the matrix product directly in the state vector, which can be used for further computations within the same quantum circuit. All scenarios involving the computation of non-homomorphic functions of the product of two matrices can benefit from our technique. As a possible application, we discuss the computation of the variance of the entries of a matrix product, which can be a useful tool for some machine learning algorithms.