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Basis for STM solution of quantum control problems

Yifan Xing, Xice Sun, Diyang Chu, Zhi Wang, Jun Yong Wu

Published Oct 16, 2009• 0 citations
Publisher

Abstract

Purpose This purpose of this paper is to present a new approach of solving quantum control problem. Design/methodology/approach State‐transition‐matrix (STM) is used to solute quantum control problem. Findings The paper obtains the matrix representation for wave function, normalization, and some other quantum operators, and then gives the possible methods to establish the STM model and solve it. Originality/value Based on this, many tools of the widely used control theory can be adopted to denote or analyze the quantum system.

DOI
10.1108/03684920910994097

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Purpose This purpose of this paper is to present a new approach of solving quantum control problem. Design/methodology/approach State‐transition‐matrix (STM) is used to solute quantum control problem. Findings The paper obtains the matrix representation for wave function, normalization, and some other quantum operators, and then gives the possible methods to establish the STM model and solve it. Originality/value Based on this, many tools of the widely used control theory can be adopted to denote or analyze the quantum system.
A model of quantum mechanical systems and its ensemble are established from control theory viewpoint. The state vector of wave function and density matrix of quantum mechanical systems are transformed into evolution matrix state of geometry space. The manipulation problem of an abstract physical concept is changed into a control design problem on a state of geometry space, which is easy to deal with mathematically and suitable for control. (Finally,) five basic problems about quantum sysetms control, which may be solved based on the model, are (presented.)
Optimal Control Theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This review aims at providing an introduction to key concepts of optimal control theory which is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This review describes modern aspects of optimal control theory, with a particular focus on the Pontryagin Maximum Principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The review covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.
Recent years brought a new angle of view to the problems involving matrix representations of, especially, functions. It is the utilization of the fuctuation free matrix representation of those functions, where the matrix representation of the function is taken as the image of the matrix representation of the independent variable under the concerned function as a first approximation. The quality of the approximation becomes better as the fluctuations are suppressed by changing the subspace dimension under consideration. In this study, we discuss the quantum optimal control problem of simple harmonic oscillator as a first step towards the construction of possibly an efficient new tool. Here, the expectation value based optimal control equations are formulated and the all agents of the problem are taken linear for simplicity. The resultant linear system of ordinary differantial equations have exact solutions. By using these solutions and the fluctuationlessness approximation, it is possible to evaluate the matrix representation of an analytic function of the position and momentum operators.
The purpose of this chapter is to revisit the quantum control problems and the solution methodologies discussed in the previous chapters presenting results of numerical experiments. In the following sections, we consider different model problems, discussed in Chapters 2, 3, and 4, and solution strategies based on the discretization schemes discussed in Chapter 5 and on the optimization methods illustrated in Chapter 6. We also provide some comparison of the strategies to provide information about which method is suitable in which context. Discussion on how to choose parameters characterizing the methods provide valuable insights for the practical implementation of quantum optimal control solvers.