Discovering configurations that are both high-utility and structurally diverse under expensive black-box evaluation and strict query budgets remains a central challenge in data-driven discovery. Many classical optimizers concentrate on dominant modes, while quality-diversity methods require large evaluation budgets to populate high-dimensional archives. Quantum Approximate Optimization Algorithm (QAOA) provides distributional sampling but requires an explicit problem Hamiltonian, which is unavailable in black-box settings. Practical quantum circuits favor quadratic Hamiltonians since higher-order interaction terms are costly to realize. Learned quadratic surrogates such as Factorization Machines (FM) have been used as proxies, but are limited to pairwise structure. We extend this surrogate-to-Hamiltonian approach by modelling higher-order variable dependencies via self-attention and projects them into a valid Positive Semi-Definite quadratic form compatible with QAOA. This enables diversity-oriented quantum sampling from learned energy landscapes while capturing interaction structure beyond pairwise terms. We evaluate on risk discovery for enterprise document processing systems against diverse classical optimizers. Quantum-guided samplers achieve competitive utility while consistently improving structural diversity and exclusive discovery. FM surrogates provide stronger early coverage, whereas ours yields higher-fidelity surrogate landscapes and better extreme-case discovery. Our method recovers roughly twice as many structurally tail-risk outliers as most classical baselines and identify an exclusive non-overlapping fraction of high-utility configurations not found by competing methods, highlighting that an effective mechanism for learning higher-order interaction structure and projecting it into quadratic surrogate Hamiltonians for quantum-assisted black-box discovery.
Discovering configurations that are both high-utility and structurally diverse under expensive black-box evaluation and strict query budgets remains a central challenge in data-driven discovery. Many classical optimizers concentrate on dominant modes, while quality-diversity methods require large evaluation budgets to populate high-dimensional archives. Quantum Approximate Optimization Algorithm (QAOA) provides distributional sampling but requires an explicit problem Hamiltonian, which is unavailable in black-box settings. Practical quantum circuits favor quadratic Hamiltonians since higher-order interaction terms are costly to realize. Learned quadratic surrogates such as Factorization Machines (FM) have been used as proxies, but are limited to pairwise structure. We extend this surrogate-to-Hamiltonian approach by modelling higher-order variable dependencies via self-attention and projects them into a valid Positive Semi-Definite quadratic form compatible with QAOA. This enables diversity-oriented quantum sampling from learned energy landscapes while capturing interaction structure beyond pairwise terms. We evaluate on risk discovery for enterprise document processing systems against diverse classical optimizers. Quantum-guided samplers achieve competitive utility while consistently improving structural diversity and exclusive discovery. FM surrogates provide stronger early coverage, whereas ours yields higher-fidelity surrogate landscapes and better extreme-case discovery. Our method recovers roughly twice as many structurally tail-risk outliers as most classical baselines and identify an exclusive non-overlapping fraction of high-utility configurations not found by competing methods, highlighting that an effective mechanism for learning higher-order interaction structure and projecting it into quadratic surrogate Hamiltonians for quantum-assisted black-box discovery.
The recently proposed QAOA-GPT framework demonstrated that generative pre-trained transformers can learn mappings between problem graphs and optimized quantum circuits for the Quantum Approximate Optimization Algorithm (QAOA). In this work, we extend QAOA-GPT to Higher-Order Unconstrained Binary Optimization (HUBO) problems, focusing on spin-glass Hamiltonians that include cubic interaction terms. Using FEATHER graph embeddings to encode topological information, we train the model on graph-circuit pairs generated via ADAPT-QAOA and evaluate its performance on 8- and 16-qubit instances embedded on heavy-hex lattices. The generative model produces adaptive QAOA-like circuits and corresponding variational parameters in a single forward pass, bypassing the iterative classical optimization loop. The generated circuits achieve average approximation ratios exceeding 0.95, closely matching classically optimized ADAPT-QAOA results, while maintaining consistent parameter distributions across circuit depths. These results demonstrate that QAOA-GPT generalizes effectively to higher-order cost Hamiltonians and complex energy landscapes, establishing generative modeling as a scalable pathway toward autonomous variational circuit design and quantum algorithm discovery in the NISQ era.
Many practically relevant applications of quantum machine learning involve classical data, for which performance depends critically on how inputs are embedded into quantum states. Yet the use of a fixed embedding circuit ansatz remains standard practice. We propose an energy-based generative learning framework that synthesizes gate sequences to optimize embedding structures and refine data-tailored parameters, using a fidelity-based surrogate objective to guide the search toward improved class distinguishability. Empirically, the method improves classification performance across diverse settings, while also revealing datasets where architecture search within the present embedding family yields only limited additional gains. We explain this saturation by deriving bounds on the achievable empirical risk in terms of the Wasserstein distance in the input space, showing that classical data geometry provides an \emph{a priori} diagnostic for regimes in which substantial gains from embedding optimization are unlikely. The results establish a practically useful and theoretically motivated framework for searching effective quantum data embeddings through generative optimization, with the attainable gains diagnosed through the geometry of the underlying classical data.
We introduce Quantum-Informed Surrogate Sampling (QISS), a post-processing framework that generates candidate solutions to combinatorial optimization problems from low-weight correlations of shallow quantum circuits. The quantum device estimates local observables, which are directly accessible by repeated measurements and for which a wide range of error-mitigation tools are available, while candidate solutions are generated classically without explicit dependence on the combinatorial optimization problem itself. We evaluate QISS on Maximum Cut and Maximum Independent Set problems on N variables and show that only O(N) low-order correlators from shallow circuits suffice to produce competitive solutions that surpass vanilla QAOA. For MaxCut on 3-regular graphs, QISS from p=3 QAOA correlators outperforms vanilla QAOA at p=17 on average, with further improvements possible by warm-starting QAOA. We validate the procedure on the 54-qubit IQM Emerald quantum device and demonstrate its noise resilience. Our results support a regime for near-term optimization in which shallow circuits serve not as direct samplers but as generators of informative statistics for scalable classical sampling.
In black-box combinatorial optimization, objective evaluations are often expensive, so high quality solutions must be found under a limited budget. Factorization machine with quantum annealing (FMQA) builds a quadratic surrogate model from evaluated samples and optimizes it on an Ising machine. However, FMQA requires binary decision variables, and for nonbinary structures such as integer permutations, the choice of binary encoding strongly affects search efficiency. If the encoding fails to reflect the original neighborhood structure, small Hamming moves may not correspond to meaningful modifications in the original solution space, and constrained problems can yield many infeasible candidates that waste evaluations. Recent work combines FMQA with a binary autoencoder (bAE) that learns a compact binary latent code from feasible solutions, yet the mechanism behind its performance gains is unclear. Using a small traveling salesman problem as an interpretable testbed, we show that the bAE reconstructs feasible tours accurately and, compared with manually designed encodings at similar compression, better aligns tour distances with latent Hamming distances, yields smoother neighborhoods under small bit flips, and produces fewer local optima. These geometric properties explain why bAE+FMQA improves the approximation ratio faster while maintaining feasibility throughout optimization, and they provide guidance for designing latent representations for black-box optimization.
Abstract Optimization is a promising candidate for studying the utility of variational quantum algorithms (VQAs). However, evaluating cost functions using quantum hardware introduces runtime overheads that limit exploration. Surrogate-based methods can reduce calls to a quantum computer, yet existing approaches require hyperparameter pre-training and have been tested only on small problems. Here, we show that surrogate-based methods can enable successful optimization at scale, without pre-training, by using radial basis function interpolation (RBF) to construct an adaptive, hyperparameter-free surrogate. Using the surrogate as an acquisition function drives hardware queries to the vicinity of the true optima. For 16-qubit random 3-regular Max-Cut instances with the Quantum Approximate Optimization Algorithm (QAOA), our method outperforms state-of-the-art approaches, without considering their upfront training costs. Furthermore, we successfully optimize QAOA circuits for 127-qubit random Ising models on an IBM processor using 10 4 −10 5 measurements. Strong empirical performance demonstrates the promise of automated surrogate-based learning for large-scale VQA applications.
In supervised quantum machine learning with classical data, classification performance critically depends on the geometry induced by quantum data embedding. Existing data-adaptive approaches rely on pairwise fidelity evaluations between samples, which scale as O(B2) for B samples and create a computational bottleneck in iterative embedding optimization. To address this issue, we propose a prototype-guided framework for efficient data-adaptive quantum embedding. Using class-prototype separation and sample-to-prototype compactness, we derive a measurable lower certificate for class-ensemble distinguishability and construct a prototype-based objective requiring only O(B) fidelity evaluations. We integrate this objective into an end-to-end pipeline that combines task-specific quantum circuit architecture optimization with continuous refinement of the data-embedding parameters. Across five benchmark datasets with up to 20 qubits, our method retains competitive quantum kernel support vector machine (SVM) performance while reducing fidelity-evaluation cost, with particularly strong gains over budget-matched pairwise optimization on T-MNIST. Geometric analyses examine how the learned embeddings relate to kernel alignment and classification margins, revealing dataset-dependent benefits and limitations. We further derive a prototype-based classifier in the reproducing kernel Hilbert space that provides an explicit witness for the SVM objective and a low-cost proxy for model selection, and extend the framework to multiple prototypes for multimodal class structures.