Some Mathematical Problems of Control of Quantum Systems

Oleg Vasilevich Morzhin, Alexander Nikolaevich Pechen

Mathematical problems of optimal control of quantum systems attract high interest in connection with both fundamental questions and existing and prospective applications. An important problem is the development of methods for constructing controls for quantum systems. One of the widely used methods is the Krotov method. The article is the first part of the review devoted to mathematical aspects of this method for optimal control of quantum systems. Beyond quantum control this method was suggested in the articles [V.F. Krotov, I.N. Feldman, Eng. Cybern. 1983. Vol. 21, 123], [A.I. Konnov, V.F. Krotov, Automat. Remote Control, 1999], and then adopted to optimal quantum control in papers by V.A. Kazakov, V.F. Krotov, R. Kosloff, C.P. Koch, D.J. Tannor, and others. The review considers this method, including cases with constraints on control spectrum and on states of the quantum system, various ways for choosing the improvement function, in particular, linear or linear-quadratic form, etc. We also discuss GRAPE (GRadient Ascent Pulse Engineering) and CRAB (Chopped Random-Basis) methods, as well as combined methods such as the method suggested by I. Maday and G. Turinici for generalizing the 1st order Krotov method and the Zhu-Rabitz method. Nowadays, various versions of the Krotov method are used as a key tool in optimal quantum control, and the review describes applications of this method to mathematical control problems for molecular dynamics, Bose-Einstein condensate, nuclear-magnetic resonance, quantum gates generation, etc. The first part of the review is devoted to closed quantum systems.

Re-Bing Wu, Jing Zhang, Chunwen Li +2 more

Navin Khaneja

Shuang Cong

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.)

Alexander Nikolaevich Pechen

Re-Bing Wu, Tzyh‐Jong Tarn, Chunwen Li

Manipulation of infinite-dimensional quantum systems is important to controlling complex quantum dynamics with many practical physical and chemical backgrounds. In this paper, a general investigation is casted to the controllability problem of quantum systems evolving on infinite-dimensional manifolds. Recognizing that such problems are related with infinite-dimensional controllability algebras, we introduce an algebraic mathematical framework to describe quantum control systems possessing such controllability algebras. Then we present the concept of smooth controllability on infinite-dimensional manifolds, and draw the main result on approximate strong smooth controllability. This is a nontrivial extension of the existing controllability results based on the analysis over finite-dimensional vector spaces to analysis over infinite-dimensional manifolds. It also opens up many interesting problems for future studies.