Jacobi fields in the Riemannian geometry of quantum computation

2010 ◽  
Author(s):  
Howard E. Brandt
Keyword(s):  
Quantum Computation ◽  
Riemannian Geometry ◽  
Jacobi Fields

Riemannian geometry of quantum computation

2009 ◽  
Vol 71 (12) ◽  
pp. e474-e486 ◽  
Author(s):  
Howard E. Brandt
Keyword(s):  
Quantum Computation ◽  
Riemannian Geometry

Geodesics and Jacobi fields in singular semi-riemannian geometry

1994 ◽  
Vol 446 (1928) ◽  
pp. 441-452 ◽  
Keyword(s):  
Singular Point ◽  
Uniqueness Theorem ◽  
Riemannian Geometry ◽  
Existence And Uniqueness ◽  
Smooth Manifold ◽  
Tensor Field ◽  
Final Section ◽  
Existence And Uniqueness Theorem ◽  
Jacobi Fields ◽  
Neutral Vector

This paper proves an existence and uniqueness theorem for geodesics tangent to a neutral vector at a stable singular point of a smooth symmetric two tensor field g on a smooth manifold M . The final section is devoted to a proof of existence and uniqueness of Jacobi fields along the above mentioned geodesics.

ASPECTS OF THE RIEMANNIAN GEOMETRY OF QUANTUM COMPUTATION

2012 ◽  
Vol 26 (27n28) ◽  
pp. 1243004 ◽  
Author(s):  
HOWARD E. BRANDT
Keyword(s):  
Quantum Computation ◽  
Riemannian Geometry ◽  
Jacobi Equation ◽  
Quantum Algorithms ◽  
Geodesic Equation ◽  
Quantum Circuits ◽  
Conjugate Points ◽  
Quantum Evolution ◽  
Metric Connection

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.

Jacobi fields in nonholonomic mechanics

2021 ◽  
Author(s):  
Alexandre Anahory Simoes ◽  
Juan Carlos Marrero ◽  
David Martin de Diego
Keyword(s):  
Riemannian Geometry ◽  
Nonholonomic System ◽  
Nonholonomic Mechanics ◽  
Jacobi Fields ◽  
Similar Characterization ◽  
Complete Lift ◽  
Curvature And Torsion ◽  
Nonholonomic Connection

Abstract In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi fields in two equivalent ways: the first one, using an appropriate complete lift of the nonholonomic system and, in the second one, using the curvature and torsion of the associated nonholonomic connection.

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