Searching for quantum optimal controls in the presence of singular critical points

TL;DR

This paper investigates how singular critical points, where the Jacobian is rank-deficient, influence the success of gradient-based quantum optimal control searches, revealing that traps only occur under specific conditions and are unlikely to hinder practical optimization.

Contribution

It demonstrates that second-order traps at singular critical points only affect gradient searches when initial controls are very close, thus unlikely to impact typical quantum control optimizations.

Findings

Second-order traps occur at constant controls.

Such traps only affect searches starting very close to them.

Practical gradient searches are generally unaffected by these traps.

Abstract

Quantum optimal control has enjoyed wide success for a variety of theoretical and experimental objectives. These favorable results have been attributed to advantageous properties of the corresponding control landscapes, which are free from local optima if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is full rank, and (3) the control field is not constrained. This paper explores how gradient searches for globally optimal control fields are affected by deviations from assumption (2). In some quantum control problems, so-called singular critical points, at which the Jacobian is rank-deficient, may exist on the landscape. Using optimal control simulations, we show that search failure is only observed when a singular critical point is also a second-order trap, which occurs if the control problem meets additional conditions involving the system Hamiltonian and/or the control objective. All known second-order traps occur at constant control fields, and we also show that they only affect searches that originate very close to them. As a result, even when such traps exist on the control landscape, they are unlikely to affect well-designed gradient optimizations under realistic searching conditions.