Quantum Algorithms' Analyses And Physical Realizations

With the development of information technique and quantum mechanics, a new subject based on the law of quantum mechanics—quantum information theory comes into being. Because of the large improvements in computational efficiency and communication security by exploiting the superposition principle and the non-classical correlation of quantum mechanics, quantum information theory soon attracts much attention and becomes the hottest areas of modern academe. It is the quantum coherence which lies at the heart of quantum information that makes the manipulation of information-carrier qubits possible, and the occurrence of the information coding, storing and transferring in a unconventionality way. Nowadays the processing of quantum information is theoretically under intense investigation, people are eagerly to find physical system to testify potential improvements offered by quantum mechanics to the manipulation and the transmission of information. Based on the substantial development in quantum engineering and measurement of light in the past decades, quantum optical systems are ideal for the experimental test of the foundation of quantum mechanics as well as for the experimental implementation of quantum information processing.As is well known, cavity QED is another qualified system for realizing a quantum processor. A two-qubit,phase gate has been experimentally realized with resonant interaction of a two-level atom with a cavity mode. Also two schemes for the realization of quantum phase gates have been presented. Moreover, the way of the realization of the quantum controlled-not(CNOT) gate has been proposed. Schemes have been proposed for realizing a quantumdiscrete Fourier transform and a Grover search algorithm in cavity QED.This thesis investigates four quantum algorithms—Chuang's quantum clock synchronization algorithm;Grover's quantum search algorithm;Bernstein-Vazirani algorithm and Shor's quantum factoring algorithm. Moreover, we presents schemes for the realizations of these algorithms in optical and cavity QED system, respectively. Our mainly works are as follows:1. In this paper, we show that, in the framework of the quantum clock synchronization algorithm, the coherent phase 'errors' produced by the time delays between sequential operations, can be avoided by setting up the delay time intervals to satisfy certain matching conditions.2. Grover proposed the Grover's quantum search algorithm in 1997 (G97 algorithm) and the Grover's fixed-point quantum search algorithm in 2005(G05 algorithm), based on the recursion equation analysis of G97 algorithm, we analyze and compare the generalized Grover's quantum search algorithms and fixed-point quantum search algorithm.3. Based on the representation of multiple qubits by single photon, we propose optical implementation for the quantum clock synchronization (QCS) algorithm. Two kinds of linear optical realization schemes, i.e., the location-plus-polarization-qubit scheme and the all-location-qubit scheme are proposed, respectively. Linear optical realizations of three-qubit and four-qubit QCS algorithm, G97 algorithm and G05 algorithm are explicitly presented.4. We propose a scheme for implementing the quantum clock synchro-nization (QCS) algorithm in cavity QED formalism. Our method is based on three-level ladder-type atoms interacting with classical and quantized cavity fields. Atom-qubit realizations of three-qubit and four-qubit QCS algorithms, three-qubit Bernstein-Vazirani algorithm and seven-qubit Shor's algorithm are explicitly presented.5. Present a proposal for optically implementing the Shor's quantum factoring algorithm by using some linear and nonlinear optical elements. The important operations of the algorithm are the Toffoli gate and the controlled phase shift operation. They can be realized by the optical circuits consisted of beam splitters, phase shifters and additional cross Kerr medium.