Critical points of the optimal quantum control landscape: a propagator approach

TL;DR

This paper investigates the conditions under which the control-to-propagator mapping in quantum control is non-singular, providing criteria and specific 'way-points' that guarantee non-singularity in bilinear quantum control problems.

Contribution

It introduces sufficient conditions and identifies specific 'way-points' that ensure non-singularity of control trajectories, advancing understanding of quantum control landscapes.

Findings

Identifies two lists of 'way-points' ensuring non-singularity.

Shows one list is independent of coupling operator matrix under certain conditions.

Provides criteria based on elements of the evolution semigroup.

Abstract

Numerical and experimental realizations of quantum control are closely connected to the properties of the mapping from the control to the unitary propagator. For bilinear quantum control problems, no general results are available to fully determine when this mapping is singular or not. In this paper we give suffcient conditions, in terms of elements of the evolution semigroup, for a trajectory to be non-singular. We identify two lists of "way-points" that, when reached, ensure the non-singularity of the control trajectory. It is found that under appropriate hypotheses one of those lists does not depend on the values of the coupling operator matrix.