| A quantum control landscape is a dynamical optimization metric expressed as a function of the control variables. A landscape can be defined for various problems in quantum control, such as the transfer of population between distinct quantum states, the optimization of the expectation value of an observable operator, or the construction of a quantum unitary transformation of a specific form. The first chapter outlines the history of quantum control to the present. The second through fifth chapters develop the foundation for a theoretical analysis of quantum control landscapes, focusing particularly on the critical topology. The enumeration of the critical regions is found in general to scale favorably with the Hilbert space dimension of the system. The dimensionality of the critical regions also has desirable scaling properties, imparting intrinsic robustness to the control solutions that comprise them. The sixth chapter focuses on the local geometry of the landscape. The main result is that the landscape gradient can at any point be expanded in a low-dimensional subspace of the control function space. This provides not only a theoretical rationale for the dimension-reduction for the quantum optimal control problem, but also the basis for a qualitatively new experimental control protocol. In the seventh chapter is presented a generalization of the "toolkit" method of propagating the Schrodinger equation, used for computations in the sixth chapter, but generally applicable to many kinds of numerical computations in quantum control. A summary of findings and open questions are discussed in the eighth chapter. |
