Spherical manifold optimization algorithm for calculating eigenvalues of complex tensors

• 1、College of Management and Economics, Hunan University, Changsha 300072

Abstract：In quantum information systems,quantum entanglement have become hot topics in the field of quantum information, M pure state of the quantum can be viewed as an M-order tensor,therefore,the quantum entanglement problem can be transformed into a tensor eigenvalue problem.This paper attempts to transform the real valued function optimization problem of complex variables with unit sphere constraints into an unconstrained optimization problem on Riemannian manifolds,Thus, we can calculate the US-eigenvalue of the complex symmetric tensor.The unconstrained optimization algorithm in Riemannian manifold space can be generalized from the unconstrained optimization algorithm in Euclidean space,In this paper, the Newton-CG method in Euclidean space is extended to Riemannian manifold space,The Riemannian Newton-CG method is used to solve the optimization problem.This method requires the first order and second order information of the objective function,The objective function of this paper is a real-valued function with complex variables based on wirtinger calculus to study the differentiability of the target function, the definition of gradient and hessian.Finally, the Numerical experiments of The Riemannian Newton-CG method are carried out,from the success rate of the algorithm, the average run time, the largest eigenvalue of the number of occurrences of terms to describe the advantages of the proposed algorithm in this paper.

Keywords： Complex symmetric tensor Quantum entanglement US-eigenvalues Wirtinger Calculus Riemannian Newton-CG