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 the search required for reaching particular goals 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 plethora of optimal controls on 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 on the top manifold where an ancillary objective can also be optimized while maintaining the full optimality of the original objective that defined the landscape.

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• Quantum control