An alternating shifted higher order power method based algorithm for rank-R Hermitian approximation and solving Hermitian CP-decomposition problems

The Hermitian tensor is a higher order extension of the Hermitian matrix that can be used to represent quantum mixed states and solve problems such as entanglement and separability of quantum mixed states. In this paper, we propose a novel numerical algorithm, an alternating shifted higher order power method (AS-HOPM), for rank-$R$ Hermitian approximation, which can also be used to compute Hermitian Candecomp/Parafac (CP) decomposition. At the same time, for the choice of initial points, we give a Broyden–Fletcher–Goldfarb–Shanno (BFGS) method based on unconstrained optimization, and propose a BFGS-AS-HOPM algorithm for rank-$R$ Hermitian approximation. For solving the Hermitian CP-decomposition problem, numerical experiments show that using the BFGS-AS-HOPM algorithm has a higher success rate than using the AS-HOPM algorithm alone.