Fubini‐Study Metric Tensor Evolution and Reduced State Distinguishability in Geometry‐Aware Optimization for Reliable Near‐Term Variational Quantum Eigensolvers

ABSTRACT

The lack of both a consistently effective and broadly reliable classical optimizer for variational quantum algorithms and a clear physical understanding of how the Fubini‐Study (FS) metric evolves during optimization remains a central challenge for practical variational quantum eigensolver (VQE) applications. In this work, we systematically investigate the quantum natural gradient (QNG) as a geometry‐aware optimization strategy and benchmark its performance against commonly used classical optimizers, including gradient‐based, derivative‐free, and adaptive methods. Using representative molecular systems, we analyze energy convergence, parameter‐space trajectories, and the dynamical evolution of the FS metric tensor. QNG consistently achieves faster and more stable convergence in the studied molecules by aligning parameter updates with the intrinsic curvature of the quantum state manifold. The evolution of the metric reveals a rapid suppression of irrelevant parameter directions and a reduced state distinguishability near the ground state. Obtained under ideal and simplified noise conditions, these results clarify the geometric origin of QNG advantage and highlight the potential suitability of geometry‐aware optimization for robust VQE implementation on near‐term quantum devices.