Hybrid Quantum-Classical Algorithms for Complex Computational Tasks (Quantum Frontiers Journal P-ISSN 3117-6070 and E-ISSN 3117-6089, 2024)
Hybrid quantum-classical algorithms represent the dominant computational paradigm for near-term quantum advantage: classical co-processors handle data pre-processing, parameter optimisation, and post-processing, while quantum processors execute short-depth circuits that exploit superposition and entanglement for tasks classically hard to simulate. Despite growing interest, a systematic framework for designing, benchmarking, and selecting hybrid architectures across diverse computational task categories has been absent. This paper proposes the Hybrid Quantum-Classical Algorithm Design (HQCAD) framework, covering four hybrid paradigms: Variational Quantum-Classical Loops (VQCL) for optimisation and ground-state estimation; Quantum-Enhanced Machine Learning (QEML) combining quantum feature maps with classical classifiers; Quantum-Classical Divide-and-Conquer (QCDC) for graph and combinatorial problems; and Quantum-Accelerated Monte Carlo (QAMC) for sampling-based inference. HQCAD is benchmarked on six tasks -- molecular ground-state energy, binary classification, graph partitioning, financial Monte Carlo, protein fold scoring, and logistics routing -- on IBM Quantum Eagle (127-qubit) and IonQ Aria (25-qubit). VQCL achieves chemical accuracy (error < 1.6 mHa) for H2O ground-state energy. QEML achieves test AUC = 0.924 vs. classical SVM 0.884 on a quantum-advantage dataset. QAMC delivers 8.4x variance reduction for financial option pricing vs. classical Monte Carlo at equal shot budget. HQCAD provides a task-adaptive hybrid algorithm selection methodology validated across six computational domains.