Lessons from the quantum control landscape: Robust optimal control of quantum systems and optimal control of nonlinear Schrödinger equations

Abstract

The control of quantum systems occurs across a broad range of length and energy scales in modern science, and efforts have demonstrated that locating suitable controls to perform a range of objectives has been widely successful. The justification for this success arises from a favorable topology of a quantum control landscape, defined as a mapping of the controls to a cost function measuring the success of the operation. This is summarized in the landscape principle that no suboptimal extrema exist on the landscape for well-suited control problems, explaining a trend of successful optimizations in both theory and experiment.

This dissertation explores what additional lessons may be gleaned from the quantum control landscape through numerical and theoretical studies. The first topic examines the experimentally relevant problem of assessing and reducing disturbances due to noise. The local curvature of the landscape is found to play an important role on noise effects in the control of targeted quantum unitary operations, and provides a conceptual framework for assessing robustness to noise. Software for assessing noise effects in quantum computing architectures was also developed and applied to survey the performance of current quantum control techniques for quantum computing. A lack of competition between robustness and perfect unitary control operation was discovered to fundamentally limit noise effects, and highlights a renewed focus upon system engineering for reducing noise. This convergent behavior generally arises for any secondary objective in the situation of high primary objective fidelity.

The other dissertation topic examines the utility of quantum control for a class of nonlinear Hamiltonians not previously considered under the landscape principle. Nonlinear Schrödinger equations are commonly used to model the dynamics of Bose-Einstein condensates (BECs), one of the largest known quantum objects. Optimizations of BEC dynamics were performed in which the nonlinearity itself was harnessed as a control, leading to successful optimization of coherent mode-to-mode transformations. Such success strengthens further extension of the landscape principle to wider classes of control.