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We present a hybrid quantum-classical data processing pipeline for a clinically meaningful task: biomarker discovery for precision oncology. By formulating feature selection as a higher-order polynomial constrained binary optimization (PCBO) problem, our approach captures first-, second-, and third-order biomarker interactions often ignored by traditional methods. To address constraints on quantum computational resources, we apply parameter transfer techniques to the Recursive Quantum Approximate Optimization Algorithm (RQAOA), effectively eliminating costly variational loops and reducing quantum evaluations by orders of magnitude. Through co-design of data encoding, Hamiltonian sparsification, and topology-aware compilation, the method aligns algorithmic needs with real device constraints. We benchmark the performance of classical solvers against the hybrid approach, demonstrating the potential for quantum computers to boost classical solver performance by effectively reducing problem size. We also compare our proposed PCBO approach to feature selection against classical baselines on the basis of yielding smaller, interpretable, and higher-quality feature sets. Focusing on cancer classification and treatment-response prediction, we perform resource estimations to determine when and how quantum hardware platforms may function as effective accelerators for biomedical discovery.