From Theory to Practice: Analyzing Variational Quantum Power Method for Quantum Optimization of QUBO Problems

Abstract

The variational quantum power method (VQPM), which adapts the classical power iteration algorithm for quantum settings, has shown promise for eigenvector estimation and optimization on quantum hardware. In this work, we provide a comprehensive theoretical and numerical analysis of VQPM by investigating its convergence, robustness, and qubit locking mechanisms. We present detailed strategies for applying VQPM to QUBO problems by leveraging these locking mechanisms, establishing systematic guidelines for their practical applications. Furthermore, we provide a comparative study against the Quantum Approximate Optimization Algorithm (QAOA). Our analysis evaluates classical optimization behaviors and evaluates performance using localized Hamming distance (bit difference of the combinatorial solution). Scaling simulations up to $n=18$ qubits demonstrate that the success probability in VQPM exhibits notable resilience. Finally, we evaluate VQPM under realistic quantum noise using the IBM Qiskit Aer framework. Our results indicate that VQPM serves as an effective quantum optimization algorithm for combinatorial problems, and this work can serve as an initial guideline for such applications.

19 pages, 11 figures. v3: IBM qiskit-aer noisy simulation is added. v4: The comparison sec. to QAOA rewritten and extended by using L-BFGS-B in the simulations of QAOA and adding scaling comparisons. All simulation code is publicly available at https://github.com/adaskin/vqpm