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We formalize Quantum Hybrid Modules (QHM) to evaluate the insertion of quantum subroutines into classical AI pipelines with explicit assumptions about data loading, noise, and measurement overhead. The framework comprises three components: (1) a Hadamard Quantum Self-Attention (HQSA) module computing sparse attention via Hadamard-test inner products in $O(n\sqrt{n/k} \cdot \log d/\epsilon)$ time assuming quantum random access memory (qRAM); (2) a Quantum Approximate Optimization Algorithm (QAOA) module for combinatorial optimization sub-problems; and (3) a Grover-accelerated neurosymbolic verification loop for quadratically faster constraint checking. We evaluate QHM through 22 experiments spanning statevector and GPU simulation up to 30 qubits, cloud simulators, and five quantum processing unit (QPU) experiments on Rigetti Cepheus and IonQ Forte, including 8--100-qubit superconducting and 8--30-qubit trapped-ion QAOA. Simulations confirm the intended mechanisms: QAOA exceeds the Goemans--Williamson bound on selected 3-regular instances, Hadamard-test attention reproduces classical softmax, and Grover verification reaches a $20{,}862\times$ query reduction at 30 qubits. Classical, dequantized, and qRAM-free baselines expose current limits: small quantum kernels lag tuned classical baselines, hybrid variational quantum circuit layers add training cost without consistent accuracy gains, and shot-based noisy intermediate-scale quantum (NISQ) attention is limited by systematic bias. Hardware results are depth- and platform-dependent: shallow circuits retain measurable signal, Rigetti QAOA depolarizes to the random baseline and is not recovered by post-hoc mitigation, and IonQ Forte preserves QAOA signal but remains below noiseless performance. We do not claim near-term quantum advantage; we characterize the gap between asymptotic theory and present-day hardware.