Numerical Solution of the Dynamic Programming Equation for the Optimal Control of Quantum Spin Systems

TL;DR

This paper develops a numerical method to solve the Hamilton-Jacobi-Bellman equation for optimal control of quantum spin systems, extending viscosity solution theory to Riemannian manifolds and demonstrating convergence with an example.

Contribution

It introduces a finite difference approximation method for the HJB equation on Lie groups, utilizing recent viscosity solution theory extensions to Riemannian manifolds.

Findings

Method converges to the true solution

Applicable to quantum spin control problems

Illustrated with a practical example

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.