Field-Aware FMQA: Enhancing Quantum-Assisted Black-Box Optimization via Field-Weighted Factorization Machines

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Problem description and relevance

Black-box optimization (BBO) is crucial in engineering domains (e.g., aerodynamic shape design), where the objective function is evaluated through computationally expensive simulations or physical experiments. Because these evaluations are time-consuming, the total number of evaluations becomes a critical bottleneck. Thus, efficient optimization methods that minimize the number of evaluations are essential.

Surrogate-based methods, such as Bayesian optimization [1], aim to reduce costly evaluations by constructing an approximate model of the objective and iteratively selecting promising candidates. Within this context, Factorization machine with quadratic optimization annealing (FMQA) [2, 3] has been proposed as a promising approach that couples a factorization-machine surrogate with an Ising-machine based combinatorial search.

A key limitation of the conventional FM [4] surrogate is its inability to capture the field structure of input variables. Although each original design variable (e.g., a categorical or discretized continuous parameter) naturally forms a "field" consisting of its one-hot encoded bits, conventional FM treats all these bits independently. This leads to inefficient learning when binary-encoded variables are used. To address this, we propose a field-aware extension of FMQA by replacing the conventional FM surrogate with a Field-Weighted Factorization Machine (FwFM) [5], resulting in the FwFMQA framework.

By combining a more expressive surrogate with efficient Ising-based search, FwFMQA significantly improves optimization efficiency in real-world applications where evaluation cost is dominant. Many practical problems involve categorical or discrete design choices such as material composition (selection of alloy families), photonic device layout (choice of lattice structures), drug-candidate scaffolds, and aerospace component configurations-where each variable naturally forms a field of one-hot encoded bits.

In addition, continuous design variables can be incorporated by discretizing them into binary representations. While this strategy is also employed in conventional FMQA, the proposed FwFMQA more effectively captures interactions among discretized variables through its field-aware structure. By treating each original discrete or discretized continuous variable as a field, the proposed method seamlessly handles both types of variables within the same framework. This unified treatment enables more accurate surrogate modeling and provides better guidance for the Ising-based search, resulting in more efficient optimization than conventional FMQA for black-box problems.

Submission ID :

Submission Topics

Quantum methods for real-world applications

Methodology :

The proposed methodology, Field-weighted factorization machine with quadratic optimization annealing (FwFMQA), is an iterative black-box optimization framework that couples a machine learning surrogate with a quantum-inspired Ising machine. The optimization process consists of a repeated cycle of surrogate training, combinatorial optimization, and objective function evaluation.

First, to handle discrete choices and categorical parameters within a unified framework, discrete variables are represented using one-hot binary variables. Continuous variables are first uniformly discretized into sub-intervals and then encoded in the same one-hot format. The set of one-hot bits originating from the same original design variable is defined as a "field."

Next, the objective function is approximated using the Field-weighted factorization machine (FwFM) surrogate model. The FwFM is trained on a set of evaluated samples to learn linear terms and pairwise interactions between binary variables. Unlike the conventional FM, which treats all bits uniformly, the FwFM assigns field-aware weights to interactions between variables, thereby capturing structural dependencies among the original design variables.

Once the surrogate model is trained, its learned parameters are used to construct a Quadratic unconstrained binary optimization (QUBO) model. Since the FwFM models linear and pairwise interaction terms over binary variables, the learned surrogate function can be naturally formulated as a QUBO model. Furthermore, because exactly one bit must be active in each field, the one-hot constraints are incorporated into the QUBO formulation as penalty terms.

The constructed QUBO is then submitted to a quantum-inspired Ising solver. The solver searches the binary combinatorial space for a configuration that minimizes the surrogate objective. In our implementation, we use Fixstars Amplify [6], a quantum-inspired annealing platform, to solve the QUBO and obtain an optimal or near-optimal binary solution.

Finally, the obtained binary solution is decoded back into the original design variables. A high-fidelity evaluation, such as an expensive physical simulation, is then performed on the newly proposed configuration. The resulting data point is added to the training dataset, and the surrogate model is updated. By repeating this cycle, FwFMQA efficiently explores the design space while reducing the number of costly evaluations required to find high-quality solutions.

Practical demonstration :

We demonstrate the correct functioning of the proposed quantum-assisted black-box optimization method, FwFMQA, through numerical experiments on aerodynamic shape optimization of a human-powered aircraft wing [7]. This real-world benchmark is treated as an expensive black-box function.

We validate the proposed pipeline by integrating an FwFM surrogate model with an Ising machine-based solver. The FwFM surrogate, trained on binary-encoded inputs, approximates the objective function by capturing field-aware variable interactions, while the Ising machine searches for configurations that minimize the surrogate objective. This search is executed on a quantum-inspired annealing system, evaluating our approach in a realistic combinatorial optimization framework.

To quantitatively assess performance, we compare FwFMQA against the baseline FMQA under identical experimental conditions (same initial dataset and number of function evaluations).

We also evaluate the predictive performance of the surrogate models using root mean squared error (RMSE) on independent test data. The evaluation demonstrates that FwFM enables more efficient learning by incorporating field-aware interactions compared to conventional FM, resulting in a more accurate surrogate landscape that better guides the Ising machine toward high-quality solutions.

Experimental results show that FwFMQA consistently achieves lower objective values than FMQA. These findings indicate that the field-aware surrogate provides more effective guidance for Ising-based search, confirming that the proposed quantum-assisted approach functions correctly in practice and reliably addresses expensive black-box optimization problems.

Application potential :

We discuss the potential of FwFMQA to be scaled up to solve realistic, large-scale problem instances. In existing research, the foundational FMQA framework has been applied to real-world tasks such as photonic-crystal surface-emitting laser (PCSEL) design [8] and materials discovery [9–11]. However, in these previous applications, the treatable problem size was restricted. Scaling up to larger problem dimensions requires overcoming two primary constraints in the hybridization strategy: the hardware capacity of the Ising machine and the sample efficiency of the machine learning surrogate.

Regarding the hardware constraint, physical quantum annealing machines currently face strict limitations on qubit counts, which severely restricts their scalability for large problems. However, we can effectively bypass this hardware scalability problem by employing quantum-inspired Ising machines. These solvers can process a significantly larger number of bits, providing the massive capacity required to scale up to realistic combinatorial spaces.

However, even with the expanded hardware capacity provided by quantum-inspired machines, the machine learning constraint presents a severe bottleneck. In real-world engineering problems, physical experiments and detailed simulations are highly expensive, strictly limiting the available evaluation budget. As the combinatorial search space expands exponentially with problem size, the conventional FM surrogate struggles to provide accurate guidance to the solver without requiring an impractical number of additional costly evaluations. This sample-efficiency limitation prevents the framework from scaling up, regardless of hardware improvements.

FwFMQA has the potential to improve optimization efficiency. By incorporating field-aware interactions, the FwFM surrogate achieves significantly higher predictive accuracy than the conventional FM under the same severely limited evaluation budget. This improved sample efficiency ensures that the quantum-inspired Ising machine can be reliably guided toward promising regions without inflating the number of expensive evaluations. Furthermore, training the FwFM introduces negligible computational overhead and strictly preserves the required QUBO formulation. By resolving the surrogate model's sample-efficiency bottleneck, FwFMQA is perfectly positioned to leverage the massive bit-capacity of quantum-inspired hardware, offering a highly promising pathway to scale up quantum-assisted black-box optimization to larger, more complex real-world applications.