We propose and implement a family of iterative quantum optimization algorithms tailored to the Knapsack Problem (KP), a combinatorial optimization problem with applications in logistics, supply chain management, finance, and resource allocation. Our approach leverages the Quantum Approximate Optimization Algorithm (QAOA) to extract correlations from quantum states, providing probabilistic insights into the likelihood of items contributing to the optimal solution. These correlations inform problem-specific update steps in an iterative process to simplify the problem. 

We benchmark our algorithms against classical heuristics, such as the Greedy algorithm, across general and special scenarios using quantum simulation. General cases utilize 12 to 24 qubits and are created randomly within that boundary. Special cases utilize 12 qubits and focus on challenging configurations in that the Greedy algorithm fails to find the optimal solution, such as misleading value-to-weight ratios. Our results demonstrate that the solution quality improves with increasing circuit depth (up to 3) in many cases and showcase the competitiveness of our algorithms with classical heuristics, particularly in complex scenarios. While all of our algorithms are able to improve their solution quality for higher depths p, one of our algorithms fails to improve upon solution quality for higher depths in general cases. 

This work positions iterative quantum optimization algorithms as a framework for solving the KP, with the promise of further scalability and practicality as hardware advances.