Solution to the quantum Zermelo navigation problem

TL;DR

This paper solves the quantum Zermelo navigation problem by deriving explicit geodesic solutions on the space of unitary operators, enabling time-optimal control in the presence of uncontrollable ambient Hamiltonians.

Contribution

It provides a simple, explicit solution to the quantum Zermelo problem using Randers metrics, extending classical navigation concepts to quantum control.

Findings

Explicit geodesic solutions for quantum control under ambient Hamiltonians

Optimal control strategies align with the background Hamiltonian

Simplified form of the quantum Zermelo problem solution

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'.