Quantum Systems

Variational Quantum Algorithms rely on the Quantum Natural Gradient (QNG) to optimize parameters across the curved quantum statistical manifold defined by the Fubini-Study metric. However, standard QNG suffers from a "Flat Earth Paradox": it maps tangent vectors back to the manifold using a Euclidean retraction (an Euler step) that ignores the Levi-Civita connection. This generates a severe retraction error that forces the descent path away from the optimal direction. Furthermore, exact Riemannian methods are computationally impossible, and both succumb to the Riemannian Barren Plateau in deep circuits. To resolve these limitations, this project introduces the Dually Flat Stratified Quantum Natural Gradient (DFS-QNG). By utilizing an Abelian-stratified ansatz with mutually commuting gates and applying layer-wise training, DFS-QNG creates a localized, dually flat geometry that completely eliminates retraction errors. Empirical evaluations on VQE and QAOA benchmarks demonstrate that DFS-QNG significantly outperforms standard QNG, maintaining robust stability and convergence at scales where standard QNG diverges.