Identification And Control Of Typical Quantum Systems Based On Geometric Representation

Recently, much e?ort has been put into the design and realization of large scale quantum devices operating. This has been spurred by the possibilities o?ered by quantum communication and information processing, from secure transmission, simulation of quantum dynamics, and the solution of currently intractable mathematical problems. Many di?erent physical systems have been proposed as basic architectures upon which to construct quantum devices, ranging from atoms, ions, photons, quantum dots and superconductors.For large scale commercial applications, it is likely that this will involve scalable engineered and constructed devices with tailored dynamics requiring precision control. Then characterization and control of quantum systems is a key problem in quantum information and computation. The system identification and control problem of typical systems is studied in this thesis based on geometric representation. The main work and contributions are as follows.1!The hyperspherical coordinate representation and real vector representation of N-dimensional Hilbert space are constructed. The model of a quantum system is the basis of quantum system identification and control. System characteristics in the model of quantum systems is usually carried out by the operator description, which can not be directly estimated. So parameteriztion of the operator is important is quantum system identification and control. For the 2-dimensional complex Hilbert space, there are two common geometric representations representations:complex spherical coordinate representation and real vectors representation"These two representations are extended to N-dimensional Hilbert space. The recurrence formula for hyperspherical coordinates constructing is given. And the real vector representation is constructed based on the coherent vector representation of density operators. Based these two representations, the geometrical description of projective measurement and dynamics of quantum closed systems is also given.2!The identification of finite-dimensional quantum closed systems and Markovian open quantum systems is analyzed.For finite-dimensional quantum closed systems, the identification problem is studied in two experimental cases which are(1) no restriction in experimental conditions and(2) limited measurement time points. In the first case,we establish a block matrix model which depends on the reference basis chosen. And3-level quantum closed systems are chosen as a example to show the experimental design and parameter estimation in the identification. identification method. In the second case,the operator model is used and a identification method is designed based on quantum process tomography. The identifiability of the system is analyzed for this case. For finitedimensional Markovian open quantum systems, the real vector model is constructed and identification methods based on time domain analysis and frequency domain analysis are given. The identification procedure is also analyzed when only one kind of projective measurement can be used. A su?cient experimental condition for complete identification is showed. The quantum closed system is a special Markovian quantum system, so the su?cient condition is also satisfied for quantum closed systems.3!The pure state transfer and realization of unitary operators is designed.we consider state-transfer tasks in which initial and finial states are given. We show that parameterization of the initial and target states in hyperspherical coordinates yields a simple constructive control scheme for state-transfer tasks that require no complex calculations of the control parameters, i.e. all control parameters are given in terms of simple functions of the initial- and final-state coordinates. The scheme has some additional advantages over alternative geometric schemes, e.g., based on decomposition into Givens rotations in that many operations can be performed either sequentially or in parallel, reducing the time required to implement the control schemes. We introduce a parameter λ which represents the ratio of costs of time and energy, and further explore the trade-o? between time and energy optimal control using the time?energy performance index. The scheme can be generalized to implement arbitrary unitary operators, and we again find that the resulting decomposition has some advantages in that many operations commute and can be performed in parallel.