Quantum Computational Geodesic Derivative

Dean P. Rickles, Xerxes D. Arsiwalla, Hatem Elshatlawy

Howard E. Brandt

Detailed analysis is given of the geodesic derivative, a useful tool in the Riemannian geometry of quantum computation. The geodesic derivative facilitates numerical evaluation of geodesics describing minimum complexity quantum circuits in which more than two-body interactions are prohibited.

Howard E. Brandt

A review is given of some aspects of the Riemannian geometry of quantum computation in which the quantum evolution is represented in the tangent space manifold of the special unitary unimodular group SU(2n) for n qubits. 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. Sharpened tools for calculating the geodesic derivative are presented. The geodesic derivative may facilitate 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.

T. W. B. Kibble

José F. Cariñena, Jesús Clemente-Gallardo, Giuseppe Marmo

Howard E. Brandt

Recent developments in the differential geometry of quantum computation are exposited. The quantum evolution is described in terms of the special unitary group of n-qubit unitary operators with unit determinant. The group manifold is taken to be Riemannian. In the present work, the lifted Jacobi equation and geodesic derivative are reviewed. This is applicable to investigations of conjugate points and the global characteristics of geodesic paths in the group manifold, and the determination of optimal quantum circuits for carrying out a quantum computation.