  The purpose of this paper is to study the problem of computing unitary
eigenvalues (U-eigenvalues) of non-symmetric complex tensors. By means of
symmetric embedding of complex tensors, the relationship between U-eigenpairs
of a non-symmetric complex tensor and unitary symmetric eigenpairs
(US-eigenpairs) of its symmetric embedding tensor is established. An algorithm
(Algorithm \ref{algo:1}) is given to compute the U-eigenvalues of non-symmetric
complex tensors by means of symmetric embedding. Another algorithm, Algorithm
\ref{algo:2}, is proposed to directly compute the U-eigenvalues of
non-symmetric complex tensors, without the aid of symmetric embedding. Finally,
a tensor version of the well-known Gauss-Seidel method is developed. Efficiency
of these three algorithms are compared by means of various numerical examples.
These algorithms are applied to compute the geometric measure of entanglement
of quantum multipartite non-symmetric pure states.
