André J. Ferreira-Martins, Renato M. S. Farias, Giancarlo Camilo, Thiago O. Maciel, Allan Tosta, Ruge Lin, Abdulla Alhajri, Tobias Haug, Leandro Aolita (Jun 24 2025).

Abstract: Variational quantum algorithms (VQAs) are promising candidates for near-term applications of quantum computers, but their training represents a major challenge in practice. We introduce exact-geodesic VQAs, a curvature-aware framework that enables analytic Riemannian optimization of variational quantum circuits through a convenient choice of circuit ansatz. Our method exploits the exact metric to find a parameter optimization path based on exact geodesic transport with conjugate gradients (EGT-CG). This supersedes the quantum natural gradient method, in fact recovering it as its first-order approximation. Further, the exact-geodesic updates for our circuit ansatz have the same measurement cost as standard gradient descent. This contrasts with previous metric-aware methods, which require resource-intensive estimations of the metric tensor using quantum hardware. In numerical simulations for electronic structure problems of up to 14 spin-orbitals, our framework allows us to achieve up to a 20x reduction in the number of iterations over Adam or quantum natural gradient methods. Moreover, for degenerate cases, which are notoriously difficult to optimize with conventional methods, we achieve rapid convergence to the global minima. Our work demonstrates that the cost of VQA optimization can be drastically reduced by harnessing the Riemannian geometry of the manifold expressed by the circuit ansatz, with potential implications at the interface between quantum machine learning, differential geometry, and optimal control theory.

Arxiv: https://arxiv.org/abs/2506.17395