Exploiting landscape geometry to enhance quantum optimal control

The successful application of quantum optimal control (QOC) over the past decades has unlocked the possibility of directing the dynamics of quantum systems. Nevertheless, solutions obtained from QOC algorithms are usually highly irregular, making them unsuitable for direct experimental implementation. In this paper, we propose a method to reshape those unattractive optimal controls. The approach is based on the fact that solutions to QOC problems are not isolated policies but constitute multidimensional submanifolds of control space. This was originally shown for finite-dimensional systems. Here, we analytically prove that this property is still valid in a continuous variable system. The degenerate subspace can be effectively traversed by moving in the null subspace of the Hessian of the cost function, allowing for the pursuit of secondary objectives. To demonstrate the usefulness of this procedure, we apply the method to smooth and compress optimal protocols in order to meet laboratory demands.

Figure 1

Each point in three-dimensional (3D) space represents a possible protocol, where ${\omega }_i$ is the amplitude of the $i\mathrm{th}$ pulse in the sequence. Solutions are scattered colored dots. They form continuous curves, 1D submanifolds of 3D parameter space. An expansion task with $\omega \left(0\right)=1, \omega \left(T\right)=0.25$, and $T=1.8$ was chosen.

Figure 2

Navigating in solution space to achieve secondary features. Each point in the graph depicts the value of the $i\mathrm{th}$ component of a given protocol, corresponding to the time interval $\mathrm{\Delta }{t}_i$. That is, the curves represent distinct protocols, which were colored relative to their secondary-objective cost value. All of them are optimal with respect to the main objective in Eq. (4). (a) A smoothing descent with $C_1$, and (b) a compression procedure with $C_2$. Both processes start with the same (dark blue) protocol and achieve remarkably different optimal fields (dark red) depending on which secondary cost was targeted.