title: Solution to the quantum Zermelo navigation problem.
abstract: The solution to the problem of finding a time-optimal control Hamiltonian to generate a given unitary gate, in an environment in which there exists an uncontrollable ambient Hamiltonian (e.g., a background field), is obtained. In the classical context, finding the time-optimal way to steer a ship in the presence of a background wind or current is known as the Zermelo navigation problem, whose solution can be obtained by working out geodesic curves on a space equipped with a Randers metric. The solution to the quantum Zermelo problem, which is shown here to take a remarkably simple form, is likewise obtained by finding explicit solutions to the geodesic equations of motion associated with a Randers metric on the space of unitary operators. The result reveals that the optimal control in a sense "goes along with the wind."
authors: Department of Mathematics, Brunel University, Uxbridge UB8 3PH, United Kingdom.|St. Petersburg National Research University of Information Technologies, Mechanics and Optics, Kronverksky Avenue 49, St. Petersburg 197101, Russia.
authors: Dorje C
authors: DC
authors: Brody
authors: Department of Mathematics, Brunel University, Uxbridge UB8 3PH, United Kingdom.
authors: David M
authors: DM
authors: Meier