The Algorithm And Application Of Computing The U-eigenvalues Of Complex Tensors

With the arrival of big data era,the amount of data to be processed in many areas of technology has exploded,which puts forward more stringent technical requirements for data storage,transmission and processing.As a high-order extension of matrix,tensor has better applicability in data storage.The research and analysis of tensor has become one of the most cutting-edge problems and research hotspots in the world.In particular,the concept of tensor eigenvalues has not only enriched tensor analysis theory,but also has a wide range of practical applications in many scientific and technological fields.Therefore,the problem of tensor eigenvalue computation has aroused wide attention in recent years.Quantum entanglement is a core problem in quantum science,which is an enabling resource in quantum communication,quantum computation,multi-body physics and other fields.Hence.how to quantify and measure the entanglement of a quantum state is of great significance to the theoretical development and applied research of quantum technology.A quantum pure state can be mathematically expressed as a complex tensor,and the entanglement eigenvalue of the quantum pure state is equal to the maximum U-eigenvalue of its corresponding complex tensor.Consequently,tensor computation can be used to characterize and quantify entanglement of quantum states.According to the above,this thesis studies a variety of methods to compute the Ueigenvalue of general complex tensors,and applies them to compute the geometric measure of entanglement of quantum pure states.The outline of the thesis is as follows.Chapter 2 introduces the related concepts and theories of U-eigenvalues of a complex tensor,rank one approximation of complex tensors,geometric measure of entanglement.Chapter 3 presents a method for calculating the maximum U-eigenvalue of complex tensors based on polynomial optimization theory.First,we show how to convert the problem of computing the maximum U-eigenvalue of a compplex tensor into a polynomial optimization problem with equality constraints in the real domain,then we solve it by the Jacobian semidefinite relaxation method.After that,we design calculation models for completely non-symmetric complex tensors and partially symmetric complex tensors,respectively.Finally,the effectiveness of the algorithm is verified by numerical experiments.In Chapter 4,we first discuss the blocking theory of complex tensors,and gives the method of symmetric embedding of non-symmetric complex tensors.Then we study the relationship between the U-eigenpair of a non-symmetric complex tensor and the USeigenpair of its symmetric embedding.After that,three algorithms for computing the U-eigenpairs of general complex tensors are proposed.More specifically,Algorithm 4.1first calculates the US-eigenpair of the symmetric complex tensor after symmetric embedding,and then obtains the U-eigenpair of the original non-symmetric tensor.Due to the large scale of symmetric complex tensors after symmetric embedding,this algorithm takes a lot of computational time and iteration steps.Algorithm 4.2 is an algorithm that directly calculates the U-eigenpairs of a non-symmetric complex tensor,thus taking less computation time and iteration steps than Algorithm 4.1.Algorithm 4.3 is an improved algorithm based on the classical Gauss-Seidel iterative method,which takes much less computation time and iteration steps than the other two algorithms.We also verify the effectiveness of the algorithms through numerical experiments and compare the computational efficiency of these algorithms.In Chapter 5,we use the algorithms in the preceding two chapters to compute the geometric measures of entanglement of quantum pure states.The effectiveness of these algorithms is verified by a large number of numerical experiments,and the computation time and iteration steps of different algorithms are compared.