Analysis of Hessian Scaling for Local and Global Costs in Variational Quantum Algorithm
Yihan Huang, Yangshuai Wang
TL;DR
The paper develops a rigorous, variance-based framework to assess Hessian resolvability in variational quantum algorithms at random initialization. By leveraging exact parameter-shift identities, Hessian entries are expressed as finite linear combinations of shifted costs, reducing entrywise variance to a covariance-quadratic form governed by lightcone geometry. A sharp dichotomy emerges: global objective costs exhibit exponential Hessian concentration with system size, leading to prohibitive shot complexity $O(e^{\alpha n})$, while termwise $k$-local objectives retain polynomially decaying variance tied to the backward lightcone growth, enabling tractable sampling with $O(\mathrm{poly}(n))$ shots. These results are corroborated by extensive numerical experiments across system sizes, depths, and a TFIM-based VQE, highlighting the practical implications for employing second-order optimization in VQAs and guiding architectural and objective design to ensure curvature information remains statistically accessible.
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 $O(e^{αn})$ 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.