Quantum brachistochrone curves as geodesics: obtaining accurate control protocols for time-optimal quantum gates

TL;DR

This paper reformulates the quantum brachistochrone problem as a geodesic problem on the unitary group, enabling efficient numerical solutions for time-optimal quantum control protocols.

Contribution

It introduces a differential geometry-based numerical method to accurately solve the quantum brachistochrone equation for quantum control.

Findings

Successfully reformulated brachistochrone curves as geodesics.

Developed an efficient numerical solution method.

Demonstrated the method on two example problems.

Abstract

Most methods of optimal control cannot obtain accurate time-optimal protocols. The quantum brachistochrone equation is an exception, and has the potential to provide accurate time-optimal protocols for essentially any quantum control problem. So far this potential has not been realized, however, due to the inadequacy of conventional numerical methods to solve it. Here, using differential geometry, we reformulate the quantum brachistochrone curves as geodesics on the unitary group. With this identification we are able to obtain a numerical method that efficiently solves the brachistochrone problem. We apply it to two examples demonstrating its power.