The mathematical structure of quantum control landscapes.

摘要

Quantum control typically deals with the interaction of an external field with a quantum system, with the goal of bringing about desired behavior at the atomic or molecular level. Closed-loop algorithms in the laboratory have been surprisingly successful at discovering optimal controls for a variety of applications. However, questions persist concerning the physical and mathematical relationships among the system, control, and the controlled dynamics. Many of these questions are best addressed through analysis of the control landscapes, i.e. objective functions on control space.;Prior work on the structure of quantum control landscapes showed that (away from singular controls) the studied landscapes have global extrema and saddle points, but no suboptimal extrema that can act as "traps" for the gradient flow. That work also high-lighted the vast multiplicity of distinct controls at every level of the landscape, including the optimal value. This dissertation presents new results that further elucidate the mathematical structure of quantum control landscapes and the physical and algorithmic implications these features hold.;First, a numerical method is presented for exploring the multiplicity of controls yielding a target final-time unitary evolution operator, the general solution of the Schrodinger equation. This tool is used to explore the role of the final time T in quantum control. Next, several families of landscapes are considered for the generation of target unitary transformations, which are important in quantum computing. The critical point structures of the landscapes are analyzed, yielding important information about the associated gradient flows. Following this, a dynamic homotopy theory is presented, revealing the global topological structure of level sets and other "dynamical" sets in the space of controls. This global structure has implications for understanding and designing quantum control algorithms, as well as the physical mechanism of control. Finally, the role of the many saddles in quantum control landscapes is evaluated by estimating and bounding the volume fraction of the halo of near-critical points (those with small gradient) surrounding each saddle. This volume fraction is interpreted as the probability of lying in these flat regions where the gradient flow and other optimal control methods can become inefficient.