Quantum Computational Geodesics

摘要

Recent developments in the differential geometry of quantum computation offer a new approach to the analysis of quantum computation. In the Riemannian geometry of quantum computation, 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. The objective of this report is to mathematically elaborate on characteristics of geodesics describing possible minimal complexity paths in the group manifold representing the unitary evolution associated with a quantum computation. For this purpose the Jacobi equation, generic lifted Jacobi equation, lifted Jacobi equation for varying penalty parameter, and the so-called geodesic derivative are reviewed. These tools are important for investigations of the global characteristics of geodesic paths in the group manifold, and the determination of optimal quantum circuits for carrying out a quantum computation.