Abstract
Quantum Fisher Information (QFI) can be used to quantify how sensitive a quantum state reacts to changes in its variational parameters, making it a natural diagnostic for algorithms such as the Quantum Approximate Optimization Algorithm (QAOA). We perform a systematic QFI analysis of QAOA for Max-Cut on cyclic and complete graphs with qubits. Two mixer families are studied, RX-only and hybrid RX-RY, with depths and , respectively, and with up to three entanglement stages implemented through cyclic- or complete-entangling patterns. Complete graphs consistently yield larger QFI eigenvalues than cyclic graphs; none of the settings reaches the Heisenberg limit (), but several exceed the linear bound (). Introducing entanglement primarily redistributes QFI from diagonal to off-diagonal entries: non-entangled circuits maximize per-parameter (diagonal) sensitivity, whereas entangling layers increase the covariance fraction and thus cross-parameter correlations, with diminishing returns beyond the first stage. Leveraging these observations, we propose, as a proof of concept, a QFI-Informed Mutation (QIm) heuristic that sets mutation probabilities and step sizes from the normalized diagonal QFI. On 7- and 10-qubit instances, QIm attains higher mean energies and lower variance than equal-probability and random-restart baselines over 100 runs, underscoring QFI as a lightweight, problem-aware preconditioner for QAOA and other variational quantum algorithms.