The Research Of Generalized Graph States Based On Hadamard Matrices

The graph state is a n-qubit quantum state corresponding to an undirected graph with n vertices and the edge set E(G).Graph states are extensively used in various fields of quantum information theory,such as one-way quantum computation,quantum error correction,entanglement theory and so on.The graph state has a structure which is closely related to some graph.Whether the definition of the graph state is given in the form of stabilizer formalism or circuit,both the two can be presented intuitively by the structure of the graph.The graph state is a special kind of stabilized state,therefore the stabilizer groups of the graph state is the abelian subgroups of the Pauli group.The starting point of this paper is to generalize the graph state,our aim is to generalize the stabilized groups of graph states from abelian groups to non-abelian groups.In this paper,we generalize the graph state based on the circuit,which is completely dependent on two preconditions,that is,the graph G and the Hadamard matrix.The basic properties of the generalized graph state are further studied.Specific research and innovation efforts are as follows:1.Introduce the theoretical basis of the graph state.Firstly,the basic theories of quantum information are introduced,then quantum entanglement is introduced in detail from three aspects:concept,measurement and application.On this basis,the related contents of graph entanglement are briefly introduced.2.Analyze the method of generalizing the graph state based on Hadamard matrix.This generalization of graph states is based on the circuit approach.The choice of Hadamard matrix is due to the importance of the matrix in mathematics and its wide application in physics.This method can be simply described as follows:given an undirected graph G with n vertices and a d×d Hadamard matrix,we can have a unique generalized graph state.In this paper,the research is based on the simple graph with four vertices and the Hadamard matrix given.3.Discuss the entanglement and structural properties of generalized graph states.The entanglement properties of generalized graph states corresponding to simple graphs of four points are calculated and analyzed.The results show that generalized graph states corresponding to the star graph and the complete graph are locally equivalent to the GHZ state of 4 quantum bits.This conclusion is applicable to the case of n points when dimension is taken as 2,3,5.Moreover,the corresponding generalized graph states of the residual graphs in the simple graph of four points cannot be locally equivalent to the GHZ state.The structural properties of the generalized graph state include the following two points.If the dimensions of two Hadamard matrices H1 and H2 are d1 and d2 respectively,then H = H1(?)H2 is also a d1d2-dimensional Hadamard matrix.The results show that the corresponding generalized graph states satisfy the same structural relationship.For any graph,the graph states corresponding to two symmetric P-equivalent Hadamard matrices are LU equivalent.