Numerical Solution of the Dynamic Programming Equation for the Optimal Control of Quantum Spin Systems
arXiv:1002.3067 · doi:10.1016/j.sysconle.2011.05.010
Abstract
The purpose of this paper is to describe the numerical solution of the Hamilton-Jacobi-Bellman (HJB) for an optimal control problem for quantum spin systems. This HJB equation is a first order nonlinear partial differential equation defined on a Lie group. We employ recent extensions of the theory of viscosity solutions from Euclidean space to Riemannian manifolds to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used to develop a finite difference approximation method, which is shown to converge using viscosity solution techniques. An example is provided to illustrate the method.
11 pages, 5 figures
References in corpus (6)
- Quantum Computation as Geometry
- Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field
- Optimal control, geometry, and quantum computing
- A reduced complexity numerical method for optimal gate synthesis
- Gate complexity using Dynamic Programming
- Viscosity solutions to second order partial differential equations on Riemannian manifolds