Wei-Norman equations for a unitary evolution

TL;DR

This paper explores the Wei-Norman method for solving unitary evolution equations, reducing nonlinear systems to matrix Riccati equations, with significant implications for quantum control.

Contribution

It demonstrates that in the unitary case, the Wei-Norman equations can be simplified to a hierarchy of matrix Riccati equations, enhancing quantum control techniques.

Findings

Reduction of nonlinear Wei-Norman equations to Riccati equations

Application to quantum control theory

Potential computational advantages

Abstract

The Wei-Norman technique allows to express the solution of a system of linear non-autonomous differential equations in terms of product of exponentials. In particular it enables to find a time-ordered product of exponentials by solving a set of nonlinear differential equations. The method has numerous theoretical and computational advantages, in particular in optimal control theory. We show that in the unitary case, i.e.\ when the solution of the linear system is given by a unitary evolution operator, the nonlinear system can be by an appropriate choice of ordering reduced to a hierarchy of matrix Riccati equations. Our findings have a particular significance in quantum control theory since pure quantum evolution is unitary.