四元数、张量即超矩阵和方程是代数学中的三个重要研究对象，并在实际问题中有广泛应用。四元数代数上的张量方程的可解性及其解的结构表示是代数学中全新的重要研究内容，在数学本身及高阶系统控制和量子计算等领域中有重要应用。.本项目是四元数代数上的张量方程组与高阶系统控制和量子计算等问题紧密结合的国际前沿性课题。旨在研究四元数代数上的张量方程组及其相关应用。首先研究四元数代数上的张量分解和张量广义逆的结构、性质、快速有效算法，然后用四元数张量分解和张量广义逆研究四元数张量方程组的相容性、通解及厄米特等特殊解的结构表示和快速有效算法及其在高阶系统控制和量子计算中的应用。本项目的实施将有助于非交换代数的进一步发展及应用。

Quaternion, tensor or hypermatrix and equation are three important research objects in algebra, and they are widely used in practical problems. The solvability of tensor equations over the quaternion algebra and the structural representation of their solutions are new and important contents in algebra, which have important applications in the fields of mathematics itself, high order system control and quantum computing..This project is an international frontier subject that system of quaternion tensor equations is closely connected with higher-order system control and quantum computing problems. This project aims to investigate some systems of tensor equations over the quaternion algebra and their related applications. Firstly, this project is to investigate the structures, properties and fast and efficient algorithms of some tensor decompositions and tensor generalized inverses over the quaternion algebra, which play important roles in investigating the solutions to quaternion tensor equations. Then using the results on quaternion tensor decompositions and quaternion tensor generalized inverses, this project is to establish the solvability conditions, the structural representation of the general solutions and some special solutions, such as Hermitian solutions, of some systems of quaternion tensor equations. The developed main theory and methods of this project will also be applied to high-order system control and quantum computing..The implementation of this project will contribute to the further development and application of noncommutative algebra.