The Top Manifold Connectedness of Quantum Control Landscapes

TL;DR

This study investigates the connectedness of optimal solutions in quantum control landscapes, revealing that the top manifold of solutions is generally path-connected, which has implications for multi-objective optimization.

Contribution

It provides numerical evidence that the top manifold of quantum control landscapes is path-connected across various objectives, enhancing understanding of landscape geometry.

Findings

Top manifold solutions are path-connected for different quantum objectives.

Continuous paths between optimal controls can be found in the landscape.

Implications for multi-objective optimization in quantum control.

Abstract

The control of quantum systems has been proven to possess trap-free optimization landscapes under the satisfaction of proper assumptions. However, many details of the landscape geometry and their influence on search efficiency still need to be fully understood. This paper numerically explores the path-connectedness of globally optimal control solutions forming the top manifold of the landscape. We randomly sample a plurality of optimal controls in the top manifold to assess the existence of a continuous path at the top of the landscape that connects two arbitrary optimal solutions. It is shown that for different quantum control objectives including state-to-state transition probabilities, observable expectation values and unitary transformations, such a continuous path can be readily found, implying that these top manifolds are fundamentally path-connected. The significance of the latter conjecture lies in seeking locations in the top manifold where an ancillary objective can also be optimized while maintaining the full optimality of the original objective that defined the landscape.