# Kick and fix: the roots of quantum control

## TL;DR

This paper surveys the mathematical techniques for approximating the exponential of sums of non-commuting operators, crucial for quantum control and related fields, highlighting their historical development and interconnections.

## Contribution

It provides a comprehensive historical overview and unifies pulsed and continuous approaches in quantum operator exponentiation.

## Key findings

- Highlights the link between operator exponentiation and the strong coupling regime.
- Shows that pulsed and continuous formulations are fundamentally connected.
- Discusses the importance of these techniques in quantum control and technologies.

## Abstract

When two operators $A$ and $B$ do not commute, the calculation of the exponential operator $e^{A+B}$ is a difficult and crucial problem. The applications are vast and diversified: to name but a few examples, quantum evolutions, product formulas, quantum control, Zeno effect. The latter are of great interest in quantum applications and quantum technologies. We present here a historical survey of results and techniques, and discuss differences and similarities. We also highlight the link with the strong coupling regime, via the adiabatic theorem, and contend that the "pulsed" and "continuous" formulations differ only in the order by which two limits are taken, and are but two faces of the same coin.