Fourier compression: A customization method for quantum control protocols

Quantum optimal control (QOC) is the field devoted to the production of external control protocols that actively guide quantum dynamics. Solutions to QOC problems were shown to constitute continuous submanifolds of control space. A solution navigation method exploiting this property to achieve secondary features in the control protocols was proposed [Larocca, Calzetta, and Wisniacki, Phys. Rev. A 101, 023410 (2020)]. Here, we present a navigation-powered protocol postprocessing mechanism allowing naturally inflexible optimization algorithms to produce user-customized solutions, for example, in terms of a Fourier decomposition. We test the performance of the method in two inherently different models: the two-level Landau-Zener system and the quantum harmonic oscillator.

Figure 1

Solution sets for the LZ model with $M=3$ control parameters. Each black point in 3D space represents a possible protocol, with ${\omega }_i$ the amplitude of the $i\text{th}$ pulse in the PWC sequence. Dim gray points represent the projections of the solutions onto the $YZ$ and $XZ$ planes. We initialize thousands of random seeds and optimize with respect to the cost function of Eq. (3). Those optimized fields with infidelity below ${10}^{-6}$ are plotted as black dots. The solutions appear to form continuous curves, one-dimensional submanifolds of parameter space.

Figure 2

Eigenvalues of the Hessian of Eq. (3) at a solution of $M=3$ ($M=48$ in the inset). We observe exactly two nonzero eigenvalues, corresponding to those eigendirections that depart from the solution set.

Figure 3

Following the null Hessian eigenvector. Black dots depict solutions found optimizing thousands of initial random seeds (see Fig. 1). Starting from one of these solutions, we follow the FD-generated Hessian eigenvector associated with the null eigenvalue generating a fidelity preserving trajectory (red curve) linking all of the scattered solutions. We set $\epsilon ={10}^{-2}$ and $h=0.01$.

Figure 4

Fourier protocol compression in the QHO. Each point in the graph depicts the value of the $i\text{th}$ 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 of Eq. (12). All of them are optimal with respect to the main objective in Eq. (8). Starting with different random solutions, we present Fourier compression trajectories with (a) $p_s=p_c=\{1,2\}$ and (b) $p_s=p_c=\left\{1\right\}$.

Figure 5

Fourier protocol compression in the LZ model. Each point in the graph depicts the value of the $i\text{th}$ component of a given protocol, corresponding to the time interval $\mathrm{\Delta }{t}_i$. The curves represent distinct protocols, which were colored relative to their secondary objective cost value of Eq. (12). All of them are optimal with respect to the main objective in Eq. (3). Starting with different random solutions, we present Fourier compression trajectories with (a) $p_s=p_c=\{1,2\}$ and (b) $p_s=p_c=\left\{1\right\}$.

Figure 6

Overlap error $E\left(\epsilon \right)$ between exact and approximate Hessian eigenvectors, as a function of the step in the FD computation, $\epsilon$, for the QHO model (dots) and the LZ model (crosses). Only those eigenvectors associated with nonzero eigenvalues are presented.

Figure 7

Calibration procedure. A solution to the QHO problem with $M=6$ parameters is initialized. A direction in control space is chosen at random and a navigation sequence following its projection onto solution space is started. After 1000 iterations with a step of $h=0.1$, the final infidelity is recorded. The solution is initialized again and a navigation following the same randomly selected direction, this time with a new step in the FD scheme, is initiated. The final infidelity is plotted as a function of $\epsilon$ (black dots), allowing for the selection of optimal $\epsilon$ without having to compute the exact eigenvectors (see Fig. 6). A second random direction (black crosses) yields similar results.

Figure 8

Exact and approximate trajectories compared: (a) Fourier cost and (b) infidelity evolution in the exact (dotted black lines) and approximate (scattered red circles) trajectories of Fig. 4. Cost trajectories are indistinguishable while the infidelity remains below ${10}^{-7}$ in both cases.

Figure 9

Run-time analysis. Fifty random fields are initialized for different values of $M$, the Hessian eigenvectors are computed, and the run time is recorded. The mean value of $\eta$, the relative duration of the FD computations with respect to the exact ones, is plotted for both the LZ model (red squares) and the QHO model (black circles). The error bars show maximum and minimum values of $\eta$.