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Current Developments in Lens Design and Optical Engineering

Designing selective thermal absorbers with factorization machines with quantum annealing

Jhih-Sheng Wu, Jia-Jun Hu

Published Sep 18, 2025• 0 citations
Publisher

Abstract

Quantum annealing (QA) is a heuristic optimization method for efficiently solving complex combinatorial problems with discrete variables. Thus, quantum annealing is proper for optical designs with multiple materials. The possible designs with multiple materials and various geometries form a huge solution space that is challenging for optimization. Due to their wave-like behavior, optical systems are typically coupled over long distances and can interfere with one another. Mathematically, this leads to high-order couplings in quantum annealing that are not supported by current quantum hardware. Factorization machines (FMs) can reduce high-order terms and provide an effective quadratic form suitable for quantum hardware. In this work, we adopt the previously proposed factorization machines with quantum annealing (FMQA) to design selective thermal absorbers. The FQMA, consisting of iterations between QA and FMs, has many hyperparameters, such as the initial inputs, latent factors, and selection of trained data. We study how these hyperparameters improve the convergence of the FMQA. We find that a balance between the initial random samplings and the sampling by QA is crucial for optimization. The principle underlying the entire optimization process of the FMQA requires a deliberate balance between depth and breadth, alongside an adiabatic approach. We design selective thermal absorbers to only absorb light with wavelengths ranging from 8 to 13 microns. We design the absorbers with silicon dioxide, PMMA, and silver. The FMQA with the optimized hyperparameters shows stably better figures of merit.

DOI
10.1117/12.3064298

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Quantum annealing (QA) is a heuristic optimization method for efficiently solving complex combinatorial problems with discrete variables. Thus, quantum annealing is proper for optical designs with multiple materials. The possible designs with multiple materials and various geometries form a huge solution space that is challenging for optimization. Due to their wave-like behavior, optical systems are typically coupled over long distances and can interfere with one another. Mathematically, this leads to high-order couplings in quantum annealing that are not supported by current quantum hardware. Factorization machines (FMs) can reduce high-order terms and provide an effective quadratic form suitable for quantum hardware. In this work, we adopt the previously proposed factorization machines with quantum annealing (FMQA) to design selective thermal absorbers. The FQMA, consisting of iterations between QA and FMs, has many hyperparameters, such as the initial inputs, latent factors, and selection of trained data. We study how these hyperparameters improve the convergence of the FMQA. We find that a balance between the initial random samplings and the sampling by QA is crucial for optimization. The principle underlying the entire optimization process of the FMQA requires a deliberate balance between depth and breadth, alongside an adiabatic approach. We design selective thermal absorbers to only absorb light with wavelengths ranging from 8 to 13 microns. We design the absorbers with silicon dioxide, PMMA, and silver. The FMQA with the optimized hyperparameters shows stably better figures of merit.
Black-box optimization (BBO) is used in materials design, drug discovery, and hyperparameter tuning in machine learning. The world is experiencing several of these problems. In this review, a factorization machine with quantum annealing or with quadratic-optimization annealing (FMQA) algorithm to realize fast computations of BBO using Ising machines (IMs) is discussed. The FMQA algorithm uses a factorization machine (FM) as a surrogate model for BBO. The FM model can be directly transformed into a quadratic unconstrained binary optimization model that can be solved using IMs. This makes it possible to optimize the acquisition function in BBO, which is a difficult task using conventional methods without IMs. Consequently, it has the advantage of handling large BBO problems. To be able to perform BBO with the FMQA algorithm immediately, we introduce the FMQA algorithm along with Python packages to run it. In addition, we review examples of applications of the FMQA algorithm in various fields, including physics, chemistry, materials science, and social sciences. These successful examples include binary and integer optimization problems, as well as more general optimization problems involving graphs, networks, and strings, using a binary variational autoencoder. We believe that BBO using the FMQA algorithm will become a key technology in IMs, including quantum annealers.
Complexity of materials designed by machine learning is currently limited by the inefficiency of classical computers. We show how quantum annealing can be incorporated into automated materials discovery and conduct a proof-of-principle study on designing complex thermofunctional metamaterials consisting of SiO2, SiC, and Poly(methyl methacrylate). Empirical computing time of our quantum-classical hybrid algorithm involving a factorization machine, a rigorous coupled wave analysis, and a D-Wave 2000Q quantum annealer was insensitive to the problem size, while a classical counterpart experienced rapid increase. Our method was 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. Our result shows that quantum annealing provides scientists gigantic computational power that may change how materials are designed.
Solving continuous variable optimization problems by factorization machine quantum annealing (FMQA) demonstrates the potential of Ising machines to be extended as a solver for integer and real optimization problems. However, the details of the Hamiltonian function surface obtained by the factorization machine (FM) have been overlooked. This study shows that in the widely used case where real numbers are represented by a combination of binary variables, the function surface of the Hamiltonian obtained by FM can be very noisy. This noise interferes with the inherent capabilities of quantum annealing and is likely to be a substantial cause of problems previously considered unsolvable because of the limitations of FMQA performance. The origin of the noise is identified and a simple, general method is proposed to prevent its occurrence. The generalization performance of the proposed method and its ability to solve practical problems is demonstrated.
This paper explores the applications of quantum annealing (QA) and classical simulated annealing (SA) to a suite of combinatorial optimization problems in machine learning, namely feature selection, instance selection, and clustering. We formulate each task as a Quadratic Unconstrained Binary Optimization (QUBO) problem and implement both quantum and classical solvers to compare their effectiveness. For feature selection, we propose several QUBO configurations that balance feature importance and redundancy, showing that quantum annealing (QA) produces solutions that are computationally more efficient. In instance selection, we propose a few novel heuristics for instance-level importance measures that extend existing methods. For clustering, we embed a classical-to-quantum pipeline, using classical clustering followed by QUBO-based medoid refinement, and demonstrate consistent improvements in cluster compactness and retrieval metrics. Our results suggest that QA can be a competitive and efficient tool for discrete machine learning optimization, even within the constraints of current quantum hardware.
Quantum computing and machine learning are state-of-the-art technologies that have been investigated intensively in both academia and industry. The hybrid technology of these two ingredients is expected to be a powerful tool to solve complex problems in many branches of science and engineering such as combinatorial optimization problems and accelerate the creation of next-generation technologies. In this work, we develop an algorithm to solve a black-box optimization problem by improving Factorization Machine Annealing (FMA) such that the training of a machine learning model called factorization machine is performed not by a full dataset but by a subdataset that is sampled from a full dataset: Subsampling Factorization Machine Annealing (SFMA). According to such a probabilistic training process, the performance of FMA on exploring a solution space gets enhanced. As a result, SFMA exhibits balanced performance of exploration and exploitation, which we call exploitation-exploration functionality. We conduct numerical benchmarking tests to compare the performance of SFMA with that of FMA. Consequently, SFMA certainly exhibits the exploration-exploitation functionality and outperforms FMA in speed and accuracy. In addition, the performance of SFMA can be further improved by sequentially using two subsampling datasets with different sizes such that the size of the latter dataset is substantially smaller than the former. Such a substantial reduction not only enhances the exploration performance of SFMA but also enables us to run it with correspondingly low computational cost even for a large-scale problem. These results indicate the effectiveness of SFMA in a certain class of black-box optimization problems of significant size: the potential scalability of SFMA in solving large-scale problems with correspondingly low computational cost. The exploitation-exploration functionality of SFMA is expected to pave the way for solving various complex combinatorial optimization problems in the real world and designing next-generation technologies that become the basic building blocks for industrial advancements.
Factorization Machine (FM) is the most commonly used model to build a recommendation system since it can incorporate side information to improve performance. However, producing item suggestions for a given user with a trained FM is time-consuming. To address this problem, we propose a quadratic unconstrained binary optimization (QUBO) scheme to combine with FM and apply quantum annealing (QA) computation. Compared to classical methods, this hybrid algorithm provides a fast sub-optimal sampling of good user suggestions. We then demonstrate the aforementioned computational behavior on current noisy intermediate-scale quantum (NISQ) hardware by experimenting with a real example on a D-Wave annealer.