Singularities of Quantum Control Landscapes

TL;DR

This paper investigates the role of singular controls in quantum control landscapes, showing they are rare and unlikely to create traps, thus supporting the effectiveness of current optimization methods.

Contribution

It provides an explicit characterization of singular controls and analyzes their landscape geometry, demonstrating their minimal impact on optimization.

Findings

Singular controls are rare and occupy a small portion of critical controls.

Numerical simulations did not find any traps associated with singular controls.

Singularities are unlikely to hinder the search for optimal quantum controls.

Abstract

A quantum control landscape is defined as the objective to be optimized as a function of the control variables. Existing empirical and theoretical studies reveal that most realistic quantum control landscapes are generally devoid of false traps. However, the impact of singular controls has yet to be investigated, which can arise due to a singularity on the mapping from the control to the final quantum state. We provide an explicit characterization of such controls that are strongly Hamiltonian-dependent and investigate their associated landscape geometry. Although in principle the singularities may correspond to local traps, we did not find any in numerical simulations. Also, as they occupy a small portion of the entire set of possible critical controls, their influence is expected to be much smaller than controls corresponding to the commonly located regular extremals. This observation supports the established ease of optimal searches to find high-quality controls in simulations and experiments.