Control Of High-dimensional Closed Quantum Systems And Quantum Network Systems Based On Lyapunov Theory

With the continuous emergence of new achievements in quantum satellites and quantum computers,quantum information technology has attracted more and more attention.Quantum system control plays an important role in promoting the development of quantum information technology.In particular,the control of high-dimensional quantum systems has greater practical significance,but it is also more difficult.For instance,the preparation of non-isolated eigenstates for high-dimensional closed quantum systems has not been well solved.Quantum network system is a network system whose nodes are quantum systems,the consensus of quantum network is closely related to distributed quantum computing and quantum communication.However,the quantum characteristics make the study of quantum network systems more difficult than classical network systems.Against this background,this thesis will utilize Lyapunov stability theory to study the preparation of non-isolated eigenstates of high-dimensional closed quantum systems and the consensus control of quantum network systems.The main contents can be included as follows:(1)Lyapunov control of high-dimensional closed quantum systems based on particle swarm optimization.For high-dimensional closed quantum systems,this thesis designs the control law with undetermined parameter,and proposes the scheme of particle swarm algorithm combined with path planning to determine the unknown parameter,which enables a high-probability population transfer to any target eigenstate of the controlled system.Firstly,the quadratic Lyapunov function V =<??P??)is employed to design the Lyapunov control law with undetermined parameter,which refers to the real diagonal matrix P.Secondly,based on the energy-level transition graph,a complete path to the target eigenstate from far to near is planned.Since the diagonal elements of matrix P correspond to the energy levels in the transition graph,the initial values of the diagonal elements can be designed in descending order along the planned path.Then,the initial values of the diagonal elements are used to design the initial feasible solution space for each dimension of the multi-dimensional particles,and as a constraint condition to limit the range of particle position change in the iterative process,so the convergence speed of the algorithm can be improved.Finally,the optimal matrix P can be searched according to the basic steps of the particle swarm algorithm,and the simulation experiments on a five-level quantum system and a three-qubit system are carried out to verify the effectiveness of this scheme.(2)The orbit consensus control of quantum network system.For a general quantum network system with a non-zero Hamiltonian H composed of n identical m-level quantum subsystems,any symmetric consensus state in the interaction picture exactly corresponds to an orbit in the Schrodinger picture,which is called the H-orbit of the symmetric consensus state.By using the interaction picture transformation and the tool of the LaSalle invariance principle,this thesis analyzes the orbit consensus of this quantum network and designs the corresponding swapping operators such that the system converges to the H-orbit of the target symmetric consensus state that exists in the interaction picture.In particular,we prove the convergence of the quantum network to the H-orbit when the quantum interaction graph is connected and the system Hamiltonian is permutation invariant.The orbit consensuses of a four-qubit network system and a quantum network of three identical three-level subsystems are achieved numerically,which verifies the correctness of our theoretical results and the effectiveness of the designed swapping operators.