Dynamical regimes of QAOA gradient response

Abstract: Characterizing the trainability of the Quantum Approximate Optimization Algorithm (QAOA) requires understanding how its gradient landscape changes across circuit parameters and problem size. Yet these gradients are usually described in terms of the native QAOA angles, making it difficult to distinguish parameter specific features from broader changes in the underlying circuit dynamics. Here we introduce a dynamical representation of the QAOA parameter space based on a norm-weighted layer strength and a cost--mixer imbalance, separating the overall scale of the evolution from the relative contribution of the two generators. Using exact-state simulations of MaxCut, we find that the gradient landscape exhibits a coarse organization in these dynamical variables that persists across changes in circuit depth and schedule structure, while the finer interference pattern remains schedule dependent. Near-optimal solutions do not simply coincide with the largest local gradients, but instead occupy a distinct intermediate dynamical regime. Uniform schedules recover the broad location of this regime, whereas nonuniform schedules mainly reorganize its fine structure. Across the system sizes studied, near-optimal solution regions remain extended in the dynamical representation while their preimages in the native QAOA angles become substantially compressed at larger sizes. These results separate the persistence of useful QAOA dynamics from their accessibility in the native parameterization, and provide a dynamical framework for interpreting QAOA trainability across circuit and problem scales.