| Quantum computer has the ability to solve the problems which are impossible to investigate classically.Quantum circuits are the main building blocks in such a computer.Any quantum circuit with an arbitrary degree of accuracy can be realized by utilizing a union set of CNOT and single-qubit gates in which the Hadamard gate is included.Complete and precise control of single and multiple-qubit quantum gates is presumed to headship to intense perceptions in quantum physics along with modern applications,like quantum computation.The main objectives of this thesis are to prepare the two important quantum gates,i.e.,Hadamard and CNOT gates with high fidelity and fast convergence under the effects of dissipations caused by the environmental or adjacent qubits interactions,and to combine them in a quantum circuit model,to result in the maximum entangled states,i.e.,the Bell states.To accomplish either of these goals requires high precision investigations in the system time-evolution and control.On the system time-evolution side,a novel technique is proposed for preparing the Hadamard gate that is the realization of a protocol to implement the transformation of the unitary time-evolution dynamics into vector space in presence of environmental dissipations.For preparing the CNOT gate,a realization of the decomposition method is presented.In this method,the system time-evolution is designed by a novel sequence of decomposed operators through limited slices of time.On the control side,based on the Lyapunov stability theorem two new types of Lyapunov function are designed to ensure the stability of the system.Accordingly,the control laws are designed to steer the time-evolution achieving the desired gate or state.The content of the researches in this thesis mainly involves the following three parts:1)A two-level quantum gate is prepared for an open quantum system.For this purpose,one of the most important quantum gates,i.e.,the Hadamard gate is considered for preparation in the presence of environmental dissipations.First,the dynamics of the system is transferred into vector space representation to have more precise control over the manipulation of the system time-evolution.Then,stand on the Lyapunov stability theorem,the design process of control laws is carried out.The control laws are obtained to steer the unitary time-evolution operator more precisely with faster convergence.For this purpose,the matrix logarithm is used to design a Lyapunov function.The numerical simulation is done for an amplitude damping Markovian open quantum system to show the preparation of the required Hadamard gate in the new dynamical-transferred formation.Based on the proposed methods,the simulation outcomes show that the fidelity of preparation the Hadamard gate can reach noticeably 0.9985 for various environmental coupling strengths.2)A new method of preparation the quantum CNOT gate is proposed.The Cartan decomposition and the Lyapunov control methods are used to obtain the control laws for different single-qubit operations in realizing the CNOT gate.The unitary time-evolution operator is decomposed into sequences of operators and accordingly,the designed control laws are applied through several short slices of times.The numerical simulation results show that for some proper control parameters in the designed control laws,each single qubit rotation has less deviation around the intended rotation axis which make the total Fidelity of the system achieve to 1 after 1.68 a.u..3)The proposed methods in the last parts are applied to realize the maximum entangled states,i.e.,Bell states.For this purpose,the Hadamard and CNOT gates are prepared and evaluated when they are combined together in a quantum circuit model.In the simulation results,it is shown that for some well-tuned designed control laws,the combined Hadamard and CNOT gates in the quantum circuit model can be well prepared and the stable Bell states are realized after 2.55 a.u.. |
