Quantum Optimal Control via Magnus Expansion and Non-Commutative Polynomial Optimization

TL;DR

This paper introduces globally convergent methods for quantum optimal control by addressing non-commutativity with Magnus expansion and non-convexity with non-commutative polynomial optimization, enabling more effective control of complex quantum systems.

Contribution

It presents the first globally convergent algorithms for quantum optimal control, combining Magnus expansion and non-commutative polynomial optimization to overcome key challenges.

Findings

Successfully addresses non-commutativity with Magnus expansion.

Employs non-commutative polynomial optimization for non-convex problems.

Achieves globally convergent solutions for complex quantum control tasks.

Abstract

Quantum optimal control has numerous important applications ranging from pulse shaping in magnetic-resonance imagining to laser control of chemical reactions and quantum computing. Our objective is to address two major challenges that have limited the success of applications of quantum optimal control so far: non-commutativity inherent in quantum systems and non-convexity of quantum optimal control problems involving more than three quantum levels. Methodologically, we address the non-commutativity of the control Hamiltonian at different times by the use of Magnus expansion. To tackle the non-convexity, we employ non-commutative polynomial optimisation and non-commutative geometry. As a result, we present the first globally convergent methods for quantum optimal control.