Resource Analysis of Quantum Optimization Algorithms via Variational Methods

Variational Quantum Algorithms (VQAs) are promising approaches for solving combinatorial optimization problems on near-term quantum devices. However, the role played by quantum resources such as entanglement and coherence during the optimization process remains poorly understood. In this work, we investigate the evolution of these resources in two VQAs--the Quantum Approximate Optimization Algorithm (QAOA) and the Feedback-based Algorithm for Quantum Optimization (FALQON)--applied to the Minimum Vertex Cover problem. We employ a variational resource-quantification framework based on the Bures metric and Uhlmann's theorem, in which an auxiliary quantum circuit is optimized to identify the closest resource-free states. This approach allows us to quantify Bures entanglement and Bures coherence of the algorithmic states without requiring full quantum state tomography. For the considered problem instance, QAOA exhibits a pronounced accumulation of entanglement in its initial layers, followed by a progressive reduction as the algorithm converges toward the optimal solution. Its coherence likewise decreases overall with increasing circuit depth, although a finite residual value persists due to the degeneracy of the optimal-solution subspace. In contrast, FALQON exhibits a substantially lower entanglement peak, reached at larger circuit depth, followed by a decay as convergence proceeds. Its coherence decreases smoothly and monotonically together with the optimization error. These results reveal distinct quantum-resource trajectories associated with the two optimization strategies and illustrate how entanglement and coherence evolve as variational quantum states approach solutions of a classical combinatorial optimization problem.