Optimal State Manipulation for a Two-Qubit System Driven by Coherent and Incoherent Controls

TL;DR

This paper explores optimal control strategies for a two-qubit quantum system using both coherent and incoherent controls, aiming to maximize the overlap with a target state, with applications in quantum information processing.

Contribution

It introduces a framework combining coherent and incoherent controls governed by the Lindblad equation, analyzing optimality conditions and developing gradient-based methods for state manipulation.

Findings

Zero controls can satisfy the Pontryagin maximum principle.

Identified conditions for stationary points and global minima.

Developed gradient projection algorithms for control optimization.

Abstract

Optimal control of two-qubit quantum systems attracts high interest due to applications ranging from two-qubit gate generation to optimization of receiver for transferring coherence matrices along spin chains. State preparation and manipulation is among important tasks to study for such systems. Typically coherent control, e.g. a shaped laser pulse, is used to manipulate two-qubit systems. However, the environment can also be used $\unicode{x2013}$ as an incoherent control resource. In this article, we consider optimal state manipulation for a two-qubit system whose dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation, where coherent control enters into the Hamiltonian and incoherent control into both the Hamiltonian (via Lamb shift) and the superoperator of dissipation. We exploit two physically different classes of interaction with coherent control and optimize the Hilbert-Schmidt overlap between final and target density matrices, including optimization of its steering to a given value. We find the conditions when zero coherent and incoherent controls satisfy the Pontryagin maximum principle, and in addition, when they form a stationary point of the objective functional. Moreover, we find a case when this stationary point provides the globally minimal value of the overlap. Using upper and lower bounds for the overlap, we develop one- and two-step gradient projection methods operating with functional controls.