Analysis of Hessian Scaling for Local and Global Costs in Variational Quantum Algorithm

Abstract: Barren plateaus are typically characterized by vanishing gradients, yet the feasibility of curvature-based optimization fundamentally relies on the statistical resolvability of the Hessian matrix. In this work, we quantify the entrywise resolvability of the Hessian for Variational Quantum Algorithms at random initialization. By leveraging exact second-order parameter-shift rules, we derive a structural representation that reduces the variance of Hessian entries to a finite covariance quadratic form of shifted cost evaluations. This framework reveals two distinct scaling regimes that govern the sample complexity required to resolve Hessian entries against shot noise. For global objectives, we prove that Hessian variances are exponentially suppressed, implying that the number of measurement shots must scale as $O(e<sup>{αn})$ with the number of qubits $n$ to maintain a constant signal-to-noise ratio. In contrast, for termwise local objectives in bounded-depth circuits, the variance decay is polynomial and explicitly controlled by the backward lightcone growth on the interaction graph, ensuring that curvature information remains statistically accessible with $O(\mathrm{poly}(n))$ shots. Extensive numerical experiments across varying system sizes and circuit depths demonstrate these theoretical bounds and the associated sampling costs. Our results provide a rigorous criterion for the computational tractability of second-order methods at initialization.