Construction Of Generalized Quantum Boolean Functions

Quantum computing is a kind of new calculation according to the theory of quantum mechanics,the basis and the principle of quantum computing and important quantum algorithm provides a possible for the computation speed beyond the turing machine mod?el.Quantum computers have great potential in dealing with complex,the realization of quantum computers need to develop quantum computing theory continually.Recently,the research on quantum information technology is not only limited to pure computation area.It has more and more been concerned through properties of quantum such as quantum communication,quantum teleportation,quantum error-correcting code and quantum cryptogram,On the other hand,k-uniform states of n qudits have very important applications in quantum information processing and quantum communication.More and more control operators are needed to be applied in the control of quantum systems.In this study,the existing construction methods of QBFs are extended and simplified.All QBFs with one qubit and all local QBFs with any qubits are constructed.And we propose the concept of GQBFs.We find all GQBFs with one qutrit and all kinds of local GQBFs with any qutrits.The number of each of four kinds of functions is uncountably infinitely many.By using diagonal matrices,we obtain uncountably infinitely many non－local QBFs with any qubits and GQBFs with any qutrits.This paper generalizes the method of constructing quantum Boolean functions by anticommuting quantum boolean functions and obtains more quantum Boolean functions and generalized quantum Boolean functions.Moreover,infinitely many families of GQBFs with any qudits are obtained from the properties of projection matrices of known saturated orthogonal arrays.This thesis consists of four chapters:Chapter 1 Introduces the research significance and research background of the thesis,some notations,the key definitions and lemmas.Chapter 2 Constructs all QBFs on one qubit and all GQBFs on one qutrit.Chapter 3 Constructs a kinds of GQBFs on n qudits by using the existence of satu-rated orthogonal arrays and obtains the number of this kind of GQBFs.Generalizing the method of constructing QBFs by anticommuting quantum Boolean functions,we obtain more QBFs and GQBFs.Moreover,By using Kronecker product,we construct all local QBFs a-nd GQBFs and obtain uncountably infinitely many non-local QBFs with any qubits and GQBFs with any qutrits by using diagonal matrices.Chapter 4 summarizes the above results,and provides some directions for research.