Research On Controllability And Control Strategy Of Quantum Systems Under No-Unitary Evolution

In the last two decades of the 20^th century, a blueprint of a new era—Quantum Information Era—was exhibited before us. With the rapid development of quantum information and quantum computation, a new challenge also appears: the more complex quantum systems demand more powerful control method and strategy since only the highly controllable quantum processes can be used for quantum information processing. Although in recent years more and more attention has been paid to control problem of quantum systems, the quantum control research is just at the beginning and there are still many directions need to be explored. In this thesis, the controllability of quantum system is studied from the perspective of non-unitary evolution. The main work and contributions are as follows.(1) The wavefunction controllability is improved and the controllability of degenerate quantum system is analyzed. In their paper of quantum wavefunction controllability, Turinici and Rabitz placed three constraints on the system being studied one of which is "there is no degenerate transitions in the system". However, the degeneration of energy levels is a common phenomenon of practical systems. For example, the first excited state of hydrogen atom is fourfold degenerated. So the controllability of degenerate quantum system is studied. First, the origin of degeneration of energy levels is analyzed by using perturbation method on hydrogen atom system. Then the general problem of energy degeneration of quantum system is analyzed from the perspective of symmetry and a method is presented to modify the controllability of degenerate quantum system. In this method, the control input has been restricted in order to break the symmetry of the system, since without a certain kind of symmetry the degeneration of energy levels will disappear. This method has improved the wavefunction controllability and extended its application field.(2) The eigenstate controllability is presented and a control algorithm based on Grover iteration and quantum measurement is designed for eigenstate controllable system. The process of this algorithm is as follows. First, one. analyzes the eigenstates-from reachable sets and seeks the one which the target state belongs to. Then using the Grover iteration to amplify the probability amplitude of the desired eigenstate (the modul square of which is the probability of the corresponding eigenstate that the system will collapse to when it is measured). By measuring, the system will then collapse to the desired eigenstate with a probability of almost unity. Finally, one can use the admissible control to drive the system from the eigenstate to the target state. The innovation of this algorithm is that the quantum measurement has been used as a control, not just a method to acquire evolution information of the system. This method also enlarges the range of controllable systems in some sense. On the other hand, it's also a good application of Grover search algorithm and the advantage of quantum algorithm has been qualified again.(3) A quantum control algorithm based on quantum amplitude amplification and quantum measurement is presented. The quantum amplitude amplification is the generalization of quantum Grover search algorithm. In this strategy, the Grover iteration has been replaced by two different quantum amplitude amplification operations. This makes it possible to collapse the system with probability 1, in other words, with certainty. The measurement operation in this scheme is only used to cancel the phase factor. Finally, one can use the admissible control to drive the system from the eigenstate to the target state and the control task is accomplished. The advantage of this algorithm is that it has no failure possibility but it is only useful for completely controllable quantum systems.(4) The probability controllability is presented and a control strategy is also presented for a certain kind of eigenstate controllable system. In this scheme, the description of unitary evolution and quantum measurement is unified and the quantum control operation is represented by probability state transition. The algorithm works as follows. At first a measurement is applied on the system. If the state after collapse is not the desired eigenstate, then system evolution will go on. After a period of time, the system is measured again. The process will go on like this until we get the desired eigenstate by measurement. Finally the system is driven to the target state with admissible control as before. In this scheme, the unitary evolution is followed by a measurement and vise versa. The whole process is in fact a piecewise deterministic process which is famous in open system theory. For a certain kind of eigenstate controllable system, the convergence of this algorithm can be proved. This whole strategy is actually a quantum feedback control with classical information since the measurement results are used as feedback information in the process.