# Geometric quantum speed limits and short-time accessibility to unitary operations

## TL;DR

This paper develops a geometric quantum speed limit framework for unitary transformations and introduces a new method to bound the minimum control time, providing better insights into short-time quantum control dynamics.

## Contribution

It presents a novel geometric QSL for unitaries and a framework to more accurately bound the minimum control time beyond traditional QSL limitations.

## Key findings

- A geometric QSL for unitary operations is formulated.
- A new framework for bounding the minimum control time is proposed.
- The approach offers more practical insights for quantum control at short times.

## Abstract

The notion of quantum speed limit (QSL) refers to the fundamental fact that two quantum states become completely distinguishable upon dynamical evolution only after a finite amount time, called the QSL time. A different, but related concept is that of minimum control time (MCT), which is the minimum evolution time needed for a state to be driven (by suitable, generally time-dependent, control fields) to a given target state. While the QSL can give information about the MCT, it usually imposes little restrictions to it, and is thus unpractical for control purposes. In this work we revisit this issue by first presenting a theory of geometrical QSL for unitary transformations, rather than for states, and discuss its implications and limitations. Then, we propose a framework for bounding the MCT for realizing unitary transformations that goes beyond the QSL results and gives much more meaningful information to understand the controlled dynamics of the system at short times.