Tools in the Riemannian geometry of quantum computation
This paper represents a review of recent developments in the Riemannian geometry of quantum computation. A series of differential geometry concepts that were used to this end are presented and their relation with the topic is analyzed. More precisely, the Riemannian right-invariant metric, connection, curvature, geodesic equation for minimal complexity quantum circuits, Jacobi equation, and the lifted Jacobi equation for varying penalty parameter are reviewed, and tools for calculating the geodesic derivative are presented. The role of geodesic derivative in facilitating the numerical investigation of conjugate points and the global characteristics of geodesic paths in the group manifold, the determination of optimal quantum circuits for carrying out a quantum computation, and the determination of the complexity of particular quantum algorithms is also discussed. In my opinion, the article provides a very deep and useful view of the possible applications that well established concepts from differential geometry may have in the emerging field of quantum computing.
scientific article; zbMATH DE number 6084525
scientific article