# Generalized Product Formulas and Quantum Control

## TL;DR

This paper investigates the behavior of quantum systems evolving under alternating non-commuting operators, especially when their evolution times are scaled differently, extending existing mathematical and physical frameworks.

## Contribution

It generalizes standard product formulas for quantum evolution by considering different scaling of the two operators' evolution times in the infinite frequency limit.

## Key findings

- Derived new generalized product formulas for quantum evolutions.
- Extended mathematical techniques for non-commuting operator analysis.
- Applicable to diverse physics and mathematics scenarios involving quantum control.

## Abstract

We study the quantum evolution under the combined action of the exponentials of two not necessarily commuting operators. We consider the limit in which the two evolutions alternate at infinite frequency. This case appears in a plethora of situations, both in physics (Feynman integral) and mathematics (product formulas). We focus on the case in which the two evolution times are scaled differently in the limit and generalize standard techniques and results.