Geometric analysis of minimum time trajectories for a two-level quantum system

TL;DR

This paper provides a geometric framework for determining minimum time control trajectories in two-level quantum systems, enabling efficient synthesis of quantum operations under various control regimes.

Contribution

It introduces a simple parametrization of $SU(2)$ and the optimal front line to fully characterize time-optimal trajectories and reachable sets for controlled two-level quantum systems.

Findings

Characterizes time-optimal trajectories in $SU(2)$

Introduces the optimal front line for reachable set analysis

Provides a method for synthesizing quantum logic operations efficiently

Abstract

We consider the problem of controlling in minimum time a two-level quantum system which can be subject to a drift. The control is assumed to be bounded in magnitude, and to affect two or three independent generators of the dynamics. We describe the time optimal trajectories in $SU(2)$, the Lie group of possible evolutions for the system, by means of a particularly simple parametrization of the group. A key ingredient of our analysis is the introduction of the optimal front line. This tool allows us to fully characterize the time-evolution of the reachable sets, and to derive the worst-case operators and the corresponding times. The analysis is performed in any regime: controlled dynamics stronger, of the same magnitude or weaker than the drift term, and gives a method to synthesize quantum logic operations on a two-level system in minimum time.