Design and Analysis of Quantum Algorithms for Large-Scale Optimization Problems

Abstract

Large-scale combinatorial and continuous optimisation problems remain intractable for classical algorithms as problem dimensionality scales. Quantum computing offers advantages through parallelism, amplitude amplification, and tunnelling, but practical realisation on near-term noisy intermediate-scale quantum (NISQ) devices demands careful algorithm design balancing advantage against noise resilience and circuit depth constraints. This paper presents the Quantum Optimisation Algorithm Suite (QOAS), a systematic framework across four algorithmic paradigms: Adaptive QAOA with dynamic depth scheduling; Variational Quantum Eigensolver (VQE) for QUBO problems; hybrid Quantum Annealing (QA); and Grover-enhanced Branch and Bound (GBB) for constrained integer programming. QOAS is benchmarked on five problem classes (MaxCut, TSP, Portfolio, Drug-Target, Feature Selection) across n = 12-84 qubits on IBM Quantum Eagle (127-qubit) and Falcon (27-qubit). Adaptive QAOA achieves approximation ratio 0.924 (SD = 0.018) for MaxCut at n = 48 -- outperforming standard QAOA (0.848) and Goemans-Williamson (0.878) at equivalent wall-clock time. GBB achieves 28.4x speedup over classical branch-and-bound for the n = 24 drug-target problem. QOAS provides systematic methodology for algorithm selection, circuit depth optimisation, and noise mitigation for NISQ workloads.