QuantumQUBO Agent: Automating Quadratic Unconstrained Binary Optimization (QUBO) Formulation Generation from Natural Language

This paper presents an Optuna-based formulation-level auto-tuning framework for support vector machines (SVMs) implemented on multiple quantum-inspired annealers. In an annealing-based SVM, continuous dual variables are discretized and converted into a quadratic unconstrained binary optimization (QUBO) model. This transformation introduces three coupled classes of parameters: representation parameters-the encoding base B and bit depth K-which determine numerical range, resolution, and QUBO size; the RBF kernel parameter γ, which determines classifier geometry; and the equality-constraint penalty ξ, which controls feasibility and coefficient balance. We formulate their joint selection as a mixed discrete-continuous black-box optimization problem. The framework has two optimization levels: an inner annealer minimizes the generated QUBO, while an outer Optuna loop reconstructs the formulation in every trial and maximizes validation accuracy. The same solver-agnostic procedure is applied to Fixstars Amplify Annealing Engine, Toshiba SQBM+, and Fujitsu Digital Annealer using TPE and Gaussian-process samplers and is compared with conventional grid search. Experiments on linear and nonlinear classification tasks with 0-20% label noise show mean gains over grid search of approximately 0.8 and 2.1 percentage points, respectively. The results demonstrate that formulation quality and backend capability must be evaluated jointly and that task-level feedback can compensate for discretization, penalty imbalance, and backend-dependent approximate optimization.

We introduce a physics-inspired continuous relaxation framework that yields substantially improved solutions for NP-hard combinatorial optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), binary sparse coding, and planted-solution Ising models. By parameterizing discrete binary variables as continuous wave-like states on the complex unit circle, we inherently smooth highly non-convex energy landscapes. We show that representing binary variables as complex phases reveals an implicit regularization mechanism that promotes convergence toward discrete states. Extracting this mechanism yields significant improvements even within standard real-valued optimization frameworks, using this regularizer explicitly. Empirically, this regularization yields vastly higher ground-state convergence rates than standard real-valued alternatives. Our models achieved zero error in large-scale 160x160 QUBO tasks under severe noise (sigma=0.25), and outperformed traditional algorithms (OMP and LASSO) in underdefined sparse coding with perfect recovery at sigma=0.15. The solver's robustness was further validated by recovering exact ground-state configurations in 8 out of 11 rigorously engineered planted-solution benchmarks.

As multi-drone fleets scale, zone assignment rapidly evolves into an intractable NP-hard combinatorial problem that overwhelms classical exhaustive search. While quantum optimization promises to shatter these classical bottlenecks, mapping complex spatial tasks from human intent to restricted quantum hardware remains a severe challenge. To bridge this gap, we present an end-to-end framework integrating a fine-tuned Large Language Model (LLM) front-end with a highly scalable, domain-specific quantum-classical backend. The front-end utilizes Supervised Fine-Tuning (SFT) and Direct Preference Optimization (DPO) to translate free-form natural language instructions into structurally robust Quadratic Unconstrained Binary Optimization (QUBO) constraints without false negatives. To overcome the strict qubit limits of near-term quantum devices, our framework features a novel constraint-preserving graph partitioner and a compressed separator-based dynamic programming (DP) merge. By structurally encoding constraints via W-state initialization and XY-mixers in Conditional Value-at-Risk Quantum Approximate Optimization (CVaR-QAOA), the pipeline stays highly compact. Empirical results demonstrate that this architecture circumvents classical scaling walls, recovering the global optimum on 100% of idealized oracle cases and 96.3% under real QAOA sampling, enabling natural-language-guided task allocation at previously intractable scales.