The paper is devoted to acceleration of the path-following interior point polynomial time method for optimization over the cone of positive semidefinite matrices, with applications to quadratically constrained problems and extensions onto the general self-concordant case. In particular, we demonstrate that in a problem involving m of general type m x m linear matrix inequalities with n 3 m scalar control variables the conjugate-gradient-based acceleration allows to reduce the arithmetic cost of an e-solution by a factor of order of max {n1/3 m-1/6, n1/5}, for the Karmarkar-type acceleration this factor is of order of min {n, m1/2}. The conjugate-gradient-based acceleration turns out to be efficient also in the case of several specific "structured" problems coming from applications in control and graph theory.