Quantumly controlled measurement, Hermitian conjugation and normalization in matrix-manipulation algorithms

In this paper, we solve three important problems that are revealed, in particular, to matrix-manipulation algorithms. The principal novelty is introducing the concept of quantumly controlled measurement that removes the post-selection problem by solving the problem of small access probability to the desired state of ancilla and possesses several remarkable properties. We also introduce separate encoding of the real and imaginary parts of a complex matrix that allows to include the Hermitian conjugation into the list of matrix manipulations. Finally, we weaken the constraints on the { modulus} of matrix elements unavoidably imposed by the normalization condition for a pure quantum state. The quantumly controlled measurement together with both other extensions are implemented into the matrix multiplication algorithm. The appropriate circuits are presented.