Quantum Control Landscapes: A Closer Look

TL;DR

This paper investigates the structure of quantum control landscapes, revealing the presence of traps in certain optimization scenarios and analyzing how different constraints affect the landscape's critical points.

Contribution

It provides a detailed analysis of the critical points and traps in quantum control landscapes for pure-state transfer and unitary optimization problems, highlighting the effects of domain restrictions.

Findings

No traps for pure-state transfer over the unitary group

Traps emerge when optimizing over the special unitary group

Modifying the performance index can eliminate some traps

Abstract

The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in $L^2(0,T)$, however, there are critical points for which the fidelity can assume any value in (0,1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over U(N), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in $L^2(0,T)$ is larger and includes traps, some of which remain traps even when the target time is allowed to vary.