Operator Preparation, State Transfer And Convergent Control In Quantum Systems

In quantum computation, quantum states are the information carriers, while quantum logic gates, which are the unitary quantum operators, are used to deal with the information, thereby states and operators are all the research objects of quantum control. In this thesis, the operator preparation in quantum systems, the state transfer in closed quantum systems and the convergent control of states in stochastic open quantum systems are investigated. The main contents are as follows:1. The system state transfer from arbitrary pure state to target mixed state for closed quantum systems based on quantum Lyapunov control theory is investigated. The state transfer is achieved via three-step controls at most. The first step is to steer system state from initial pure state to an eigenstate, the system model is described by the Schrodinger equation. The second step is to steer system state from the eigenstate which is obtained in the first step to the target off-diagonal mixed state. In this step, an auxiliary system is used to overcome the constraint of that state evolution of closed quantum systems is unitary. If the target mixed state is a diagonal mixed state, the final state in the second step is an off-diagonal mixed state with the same purity as target diagonal mixed state. In this case, the off-diagonal mixed state obtained in the second step is as the initial state and steered to the target mixed state, which is the third step. The latter two steps are the control of mixed states, so the system model is described by the Liouville equation. The three control steps are all designed based on the Lyapunov method, and the effectiveness of the controls are verified in numerical experiments.2. The performances of state transfers in closed quantum system by using geometric control and bang-bang control are compared. The forms of bang-bang control are relevant to the relationships between the maximum amplitude M of control laws and the energy level E. For the cases of M≥E and M<E, the time characteristics of steering system state from North Pole to arbitrary point on the Bloch sphere by using geometric control and bang-bang control are investigated contrastively. The robustness when the uncertainty of system Hamiltonian is caused by perturbation is researched, the reasons of robustness are analyzed, and the results are verified in numerical experiments.3. The preparation of Hadamard gate for closed quantum systems is investigated. Hadamard gate doesn’t satisfy the canonical form of the unitary rotation gate which can be prepared by single control. In order to use single control, Hadamard gate is decomposed into two unitary rotation gate U1and U2. The operator distance is selected as the Lyapunov function, based on which the control laws Ω1and Ω2of preparing U1and U2are designed. Then, Hadamard gate is prepared by using Ω1and Ω2in turn. Numerical experiments of preparing Hadamard gate are implemented, the results are analyzed, and optimal control is compared.4. The preparations of operators for open quantum systems by Lyapunov control method are investigated. The Lyapunov function V is constructed based on the Mercators series of matrix logarithm function log(Uf+U(u)). The relationships between Ⅴ and operator distance in numerical value and convergence speed are analyzed, and two types of control laws of preparing operators are designed. Numerical experiments of preparing NOT gates for closed quantum systems and open quantum systems are implemented. A strategy of combining the two types of control laws in preparing operators is proposed. The system robustness when the Hamiltonian contains uncertainty is investigated, the scheme for the control laws to enhance the system robustness is proposed, and is demonstrate by numerical experiments.5. The state convergent control laws of states in stochastic open quantum systems are investigated in the cases of that the target state is a special eigenstates, arbitrary eigenstates and mixed states, respectively. The system state converges to any one of system equilibriums in free evolution. In order to make system state converge to the target state, the Lyapunov functions which contain state distance are proposed. The control, which is used to steer system state to converge to the target state in local state space, is designed based on the proposed Lyapunov function. In order to make the system state be the global convergent, the switching control which is composed of the control designed by the Lyapunov method and the constant control is designed. The convergence of switching control is proved based on the Lyapunov stability theorem and LaSalle’s invariance theorem for stochastic master equations. Numerical experiments of a three dimensional stochastic quantum system are implemented when the target states are eigenstates and superposition states, respectively, and the results are analyzed.