# Estimating the time evolution of NMR systems via quantum speed limit-like expression

## TL;DR

This paper introduces a novel method using quantum speed limit theory to accurately estimate the evolution time of NMR systems, outperforming previous approaches by up to four orders of magnitude.

## Contribution

It applies quantum speed limit theory in a new way to estimate quantum process durations, specifically for NMR systems, with improved accuracy.

## Key findings

- Estimated evolution time is more accurate than previous methods.
- Method successfully tested on nuclear spins 1/2 and 3/2 in NMR.
- Achieved up to four orders of magnitude improvement in estimation accuracy.

## Abstract

Finding the solutions of the equations that describe the dynamics of a given physical system is crucial in order to obtain important information about its evolution. However, by using estimation theory, it is possible to obtain, under certain limitations, some information on its dynamics. The quantum-speed-limit (QSL) theory was originally used to estimate the shortest time in which a Hamiltonian drives an initial state to a final one for a given fidelity. Using the QSL theory in a slightly different way, we are able to estimate the running time of a given quantum process. For that purpose, we impose the saturation of the Anandan-Aharonov bound in a rotating frame of reference where the state of the system travels slower than in the original frame (laboratory frame). Through this procedure it is possible to estimate the actual evolution time in the laboratory frame of reference with good accuracy when compared to previous methods. Our method is tested successfully to predict the time spent in the evolution of nuclear spins 1/2 and 3/2 in NMR systems. We find that the estimated time according to our method is better than previous approaches by up to four orders of magnitude. One disadvantage of our method is that we need to solve a number of transcendental equations, which increases with the system dimension and parameter discretization used to solve such equations numerically.