Lifted Jacobi equation for varying penalty parameter in the Riemannian geometry of quantum computation

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

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.

Riemannian geometry of quantum computation

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

scholarly journals Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

2021 ◽  
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng ◽  
Albert Fathi
Keyword(s):  
Riemannian Manifold ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Jacobi Equation ◽  
Compact Manifold ◽  
Complete Riemannian Manifold ◽  
Time Dependent ◽  
Jacobi Equations ◽  
Uniformly Continuous ◽  
Hamilton Jacobi Equation

AbstractIf $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in $]0,+\infty [\times M$ ] 0 , + ∞ [ × M where $U$ U is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ Σ ( U ) . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

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