Landscape for optimal control of quantum-mechanical unitary transformations

Abstract

The optimal creation of a targeted unitary transformation $W$ is considered under the influence of an external control field. The controlled dynamics produces the unitary transformation $U$ and the goal is to seek a control field that minimizes the cost $J=\ensuremath{\parallel}W\ensuremath{-}U\ensuremath{\parallel}$. The optimal control landscape is the cost $J$ as a functional of the control field. For a controllable quantum system with $N$ states and without restrictions placed on the controls, the optimal control landscape is shown to have extrema with $N+1$ possible distinct values, where the desired transformation at $U=W$ is a minimum and the maximum value is at $U=\ensuremath{-}W$. The other distinct $N\ensuremath{-}1$ extrema values of $J$ are saddle points. The results of this analysis have significance for the practical construction of unitary transformations.