# Optimal Control of Quantum Purity for $n=2$ Systems

## TL;DR

This paper develops a numerical method for optimal control of two-level quantum systems governed by Lindblad dynamics, focusing on minimizing time and energy to reach desired states within the Bloch ball.

## Contribution

It introduces a transformation of dissipative quantum control problems into bi-linear systems on the Bloch ball and designs algorithms for optimal state transfer.

## Key findings

- Successfully constructs optimal control paths for quantum states.
- Provides a numerical framework for time- and energy-efficient quantum control.
- Enhances understanding of control strategies for dissipative quantum systems.

## Abstract

The objective of this work is to study time-minimum and energy-minimum global optimal control for dissipative open quantum systems whose dynamics is governed by the Lindblad equation. The controls appear only in the Hamiltonian. Using recent results regarding the decoupling of such dissipative dynamics into intra- and inter-unitary orbits, we transform the control system into a bi-linear control system on the Bloch ball (the unitary sphere together with its interior). We then design a numerical algorithm to construct an optimal path to achieve a desired point given initial states close to the origin (the singular point) of the Bloch ball. This is done both for the minimum-time and minimum -energy control problems.