Multi-level evolution strategies for high-resolution black-box control

Abstract

This paper introduces a multi-level (m-lev) mechanism into Evolution Strategies (ESs) in order to address a class of global optimization problems that could benefit from fine discretization of their decision variables. Such problems arise in engineering and scientific applications, which possess a multi-resolution control nature, and thus may be formulated either by means of low-resolution variants (providing coarser approximations with presumably lower accuracy for the general problem) or by high-resolution controls. A particular scientific application concerns practical Quantum Control (QC) problems, whose targeted optimal controls may be discretized to increasingly higher resolution, which in turn carries the potential to obtain better control yields. However, state-of-the-art derivative-free optimization heuristics for high-resolution formulations nominally call for an impractically large number of objective function calls. Therefore, an effective algorithmic treatment for such problems is needed. We introduce a framework with an automated scheme to facilitate guided-search over increasingly finer levels of control resolution for the optimization problem, whose on-the-fly learned parameters require careful adaptation. We instantiate the proposed m-lev self-adaptive ES framework by two specific strategies, namely the classical elitist single-child (1+1)-ES and the non-elitist multi-child derandomized $(\mu _W,\lambda )$-sep-CMA-ES. We first show that the approach is suitable by simulation-based optimization of QC systems which were heretofore viewed as too complex to address. We also present a laboratory proof-of-concept for the proposed approach on a basic experimental QC system objective.

Appendices

The pixelation effect

The time modulation of these step-functions is attained by their inverse Fourier transform,

$$\begin{aligned} \displaystyle \mathcal {F}^{-1}\left[ \text {squ}(\nu )\right] \sim \text {sinc}(\tau ) \end{aligned}$$

where the width of $\text {sinc}(\tau )=\frac{\sin (\tau )}{\tau }$ is inversely proportional to the pixel width. Explicitly, the resulting temporal electric field in this pixelation can be described by

$$\begin{aligned} \displaystyle e(t) = \sum _{\jmath } \tilde{e}(t-\jmath \tau )\cdot \text {sinc}\left( \frac{\pi t}{\tau }\right) , \end{aligned}$$

(15)

with $\tilde{e}(t)$ as the desired electric field, and where $\tau = \frac{1}{\varDelta \nu }$ is the inverse frequency spacing per pixel. Figure 11 provides an illustration of the pixelation effect (Roth 2007).

Fig. 11

Illustration of the pixelation effect occurring in pulse shapers during typical QC experiments; shaping is carried out with 32 pixels in the domain $[-\pi ,\pi ]$. Panel a presents replica pulses with a moderate linear phase, with the envelope of the replicas in a dashed curve. Panel b shows a steeper phase, with more pronounced replica pulses (Roth 2007). Figure courtesy of Matthias Roth and Jonathan Roslund

Laboratory setup

A commercial Ti:Sapphire femtosecond laser system consisting of an oscillator and a multi-pass amplifier (KMlab, Dragon) is used to produce pulses with $\sim 25\text {fs}$ pulse width centered at $\sim 790\text {nm}$. The laser pulses are shaped with a dual-mask liquid crystal SLM (CRI, SLM-640), capable of simultaneous phase and amplitude modulation. A small fraction of the shaped laser pulse is spit and focused into a GaP photodiode (Thorlab), which generates only the TPA signal from the $790\text {nm}$ incident light. The laser power was adjusted with neutral density filters before entering the photodiode to make sure that the TPA signal does not saturate. The SLM contains $N_f=640$ pixels, and the shaper is configured at a resolution of $\sim 0.2\text {nm/pixel}$. We utilize phase-only modulation in the experiment, with all the amplitudes fixed at 1. Adjacent SLM pixels are sequentially tied together in groups of $g_{\ell }~\left( g_{\ell }=32,~16,~8,~4,~2 \right)$ to reduce the 640-dimensional space of spectral phases to the desired grid levels $n_{\ell }~\left( n_{\ell }=20,~40,~80,~160,~320 \right)$, respectively. The QC objective function is the experimental equivalence of Eq. 10, that is, to experimentally maximize the TPA signal by obtaining the optimal phase mask on the shaper. Ideally, the optimal phase should be a flat phase or a linear phase, but in laboratory practice, it contains higher order nonlinear terms to compensate for or correct those inherent in the laser system.

Figure 12 schematically illustrates an experimental QC learning loop (Nuernberger et al. 2007).

Fig. 12

The experimental quantum control learning loop. An unshaped laser pulse, approximated well by a Gaussian, is shaped by a pixel-based modulator and applied to a molecular sample. The control variables addressed in the laboratory by the evolution strategy are the individual pixels that determine the pulse shape. The measured signal reflecting the molecular response constitutes the feedback for optimization by the ES