Landscape for optimal control of quantum-mechanical unitary transformations

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=∥W−U∥. 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=−W. The other distinct N−1 extrema values of J are saddle points. The results of this analysis have significance for the practical construction of unitary transformations.