Hybrid Quantum–Classical Framework for Quantum Complexity

Abstract

This paper examines quantum algorithms and their computational complexity through a unified framework combining mathematical modeling, system architecture, and empirical evaluation. Key complexity measures circuit depth, gate count, and query complexity are analyzed under NISQ constraints. A hybrid quantum classical optimization framework is introduced to improve efficiency and stability. Results show the full model achieves 94.2% accuracy with baseline runtime, while removing optimization lowers accuracy to 85.6% and increases runtime by 30%. Reducing qubits decreases cost but drops accuracy to 78.3%, and disabling error mitigation causes unstable performance at 70.1%. Comparative analysis indicates strong advantages of quantum algorithms for structured problems, especially in scalability and asymptotic complexity. However, performance remains sensitive to noise, limited qubits, and circuit depth, emphasizing the need for hardware–algorithm co-design.