【作者】 张陈斌；

【导师】 陈宗海；

【作者基本信息】 中国科学技术大学 ， 模式识别与智能系统， 2007， 博士

【摘要】 二十世纪末，量子信息和量子计算的蓬勃发展给人们描绘出一个崭新的时代——量子信息时代——的美好蓝图，同时也向人们提出了新的挑战：日益复杂的量子系统需要更加有效的调控手段，因为只有高效可控的量子物理过程才能成为量子计算和量子信息处理的物理资源。尽管近年来量子系统的控制问题已经引起了科学家们的广泛关注，但真正意义上的量子控制研究还处于刚刚起步阶段，许多领域还有待人们去探索。本文主要对非幺正演化条件下量子系统的能控性问题进行了分析研究，主要工作和贡献如下：(1)发展了波函数能控性，分析了能级简并的量子系统的能控性问题。在Turinici和Rabitz提出的波函数能控性问题中，引入了三个假设，其中的第二个假设要求系统不包含简并的能级跃迁，然而从实际情况来看，很多量子系统都是存在能级简并现象的，即使像氢原子这样简单的系统，第一激发态都是四重简并的。为此，本文研究了存在能级简并现象的量子系统的能控性问题。首先，以氢原子系统为例，考察了能级简并的原因，接着利用摄动分析寻找消除能级简并的方法，最后根据氢原子系统的分析经验，从对称性角度分析了一般量子系统存在能级简并的原因，提出了通过限制控制输入的类型来破坏系统的对称性、消除系统能级简并的方法，并给出了经过改进的波函数能控的判定条件。这一结果进一步发展了波函数能控性，扩大了其适用的范围。(2)首次提出了本征态能控性的概念，并借鉴量子Grover搜索算法的主要思想，创造性的利用量子测量自身的特点，给出了一种适用于本征态能控系统的量子控制策略。该策略的核心思想是对本征态能控的系统，通过分析控制目标态与本征态的关系，寻找给定目标态所属的本征态可达集，然后利用量子非结构化搜索的Grover算法，将任意给定的系统初始态经过一定次数的Grover迭代，放大该本征态所对应的概率幅(概率幅的模方对应测量时波函数塌缩到对应本征态的概率)，然后对迭代后的态进行一次测量操作，使系统以接近1的概率塌缩到所需的本征态(前面分析得到的本征态可达集所对应的本征态)，最后用容许的控制将系统从该本征态控制到期望的目标态。该策略的创新之处在于将量子测量作为一种控制手段用于量子系统的控制，而策略的意义在对原来被认为不可控的一些量子系统，找到了一种控制方法，一定程度上扩大了可控量子系统的范围。从另一个方面来看，也为Grover算法提供了一个很好的应用实例，进一步说明了量子算法的优越性。(3)针对完全能控的量子系统，提出了一种量子幅值放大和量子测量相结合的量子控制算法。量子幅值放大是Grover迭代思想的一般化，该控制方案中采用两种不同的量子幅值放大操作替换了Grover迭代，使得期望本征态的概率幅的模在经过一定次数的迭代之后精确的等于1，然后借助于测量来消除相位因子，使系统的波函数确定性的塌缩到期望的本征态，最后再借助幺正演化，将系统从本征态控制到目标态，实现期望的控制任务。该控制算法的优点在于不像基于Grover迭代的算法那样存在失败的可能性，控制算法本身没有随机性，是完全确定性的，但该算法只适用于完全能控的量子系统。(4)把本征态能控性概念进一步向前发展，提出了概率能控性的概念，将幺正演化和量子测量这两种量子控制手段统一的用概率状态转移来描述，并针对一类特殊的本征态能控的系统，提出了幺正演化和测量交替进行的量子反馈控制策略：如果单步测量所得到的本征态不是期望的本征态，则让系统以这一本征态作为新的初态进行下一步的幺正演化，然后在适当的时候再次测量，让系统塌缩到某一本征态，如此重复进行，直到达到期望的目标态为止。该算法实际上可以看作确定性幺正演化和测量导致的随机塌缩交替进行的过程，类似于开放系统的分段确定性过程。对于一类特殊的本征态能控系统(量子控制操作的不变子空间不可约的系统)，该策略可以证明是收敛的。算法中由于利用了测量的结果(经典信息)作为下一步控制的依据，因而是基于经典信息的量子反馈控制。更多还原

【Abstract】 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.更多还原