Some mathematical and algorithmic challenges in the control of quantum dynamics phenomena

The theory and practice of control over quantum mechanical phenomena is receiving increasing attention, underscored by striking experimental successes. Nevertheless, many questions of fundamental and practical relevance to the field remain unresolved. With the aim of stimulating further development, this paper formulates a number of theoretical questions, divided into three categories. First, questions related to control law design are discussed, with an emphasis on controllability and optimal control theory. This leads to the second category of open problems relevant to closed loop laboratory implementation of quantum control, including learning and feedback methods. The sensitive dependence of control on basic quantum mechanical interactions motivates the third section, which treats coherent dynamical techniques for identifying the system Hamiltonian. An open issue overarching all of these directions is the need to discover general rules for the control of quantum systems. Although the list of issues raised in this paper is extensive, it should be viewed not as a complete menu for exploration, but rather as a springboard to new challenges as the field evolves.