Designing metamaterials with quantum annealing and factorization machines

Automated materials design with machine learning is increasingly common in recent years. Theoretically, it is characterized as black-box optimization in the space of candidate materials. Since the difficulty of this problem grows exponentially in the number of variables, designing complex materials is often beyond the ability of classical algorithms. We show how quantum annealing can be incorporated into automated materials discovery and conduct a proof-of-principle study on designing complex thermofunctional metamaterials. Our algorithm consists of three parts: regression for a target property by factorization machine, selection of candidate metamaterial based on the regression results, and simulation of the metamaterial property. To accelerate the selection part, we rely on the D-Wave 2000Q quantum annealer. Our method is used to design complex structures of wavelength selective radiators showing much better concordance with the thermal atmospheric transparency window in comparison to existing human-designed alternatives.

Figure 1

Procedure of our automated materials discovery using a factorization machine (FM) for regression and quantum annealing (QA) for selection. This simulation can use various simulation methods or experiments, depending on the target properties.

Figure 2

Example of the target metamaterial structure for $L=6$ and $C=3$, the binary variables expressing it, and the emissive powers of target metamaterial (red curve) and blackbody (blue curve) calculated by RCWA.

Figure 3

(a) Dependence of the best FOM on the number of calculated structures (iterations) by automated materials discovery using an FM, a Gaussian process, and a random search for the $L=4$ and $C=3$ case. Inset is the optimum structure and its FOM. Blue, red, and gray squares denote ${\mathrm{SiO}}_2$, SiC, and PMMA, respectively. (b) Best FOM by FMQA using a quantum annealer and the random search for the $L=6$ and $C=3$ case. Inset is the structure with the best FOM identified by FMQA. (c) Computing time to perform 500 iterations of automated materials discovery using a quantum annealer and an exhaustive search on a classical computer with an Intel Xeon E5-2690 v3 @ 2.6GHz for the selection part, respectively. Learnings and simulations are performed by the same classical computer.

Figure 4

(a) Best FOM as a function of the iteration number for $C=3$ with various $L$. Plotted results are for a single run with the first 50 randomly generated initial structures. Parentheses denote the number of candidates. (b) Structures with a high FOM designed by FMQA depending on $L$ and $C$. Red dotted line denotes the structure with the highest FOM in our search. (c) Change in the best FOM for $L=5$ with various $C$ obtained by a single run with the first 50 randomly generated initial structures. (d) Emissive power calculated by RCWA (red curve) of the designed structure with the highest FOM for the $L=5$ and $C=6$ case, which is surrounded by the red dotted line in (b). For comparison, the blackbody emissive power (blue curve) and the multilayer optimum case with five layers (grey curve) are shown (see Supplemental note D in Ref. [56]).

Figure 5

Contour plots of (a) the normalized electric power dissipation density at select wavelengths and (b) the normalized magnetic field for the designed optimum structure with $L=5$ and $C=6$ which is shown in Fig. 4.