Applications Of CMV Matrix And Density Evolution Operator In Quantum Ensemble Evolution

Quantum random walks,which is the counterpart of classical random walk-s,had been studied already in recent years.More results had been applicated in many fields.In this dissertation,we mainly use the Fourier analysis and Spectral representation study the one-dimensional quantum random walks based on CMV matrix.The dissertation is organized as follows:In chapter 1,we summarize the background of one-dimensional quantum ran-dom walks.We also present some general concepts,facts and fundamental methods concerning quantum walks.In chapter 2,we study the properties of Pauli gate.We also offer a application of Pauli gate in quantum computation.In chapter 3,we prove the diffusion of quantum walk under the CMV matrix has a lower bound.We also prove the upper bound of wave packet of quantum walk based on CMV matrix.In chapter 4,we study the quantum system state described by quantum en-semble density evolution operator in four cases.We also prove the existence of a minimal ensemble which includes any support and make the state appear in a fixed probability.