# Variational quantum metrology with the Loschmidt echo

## TL;DR

This paper introduces a new quantum metrology method using the Loschmidt echo to improve measurement precision on quantum systems.

## Contribution

A variational quantum metrology scheme using the Loschmidt echo for scalable and practical quantum-enhanced precision.

## Key findings

- The proposed scheme achieved a 12.4-dB precision enhancement over uncorrelated states.
- The method is experimentally implemented on a 10-spin quantum processor.
- The protocol is applicable to various noisy intermediate-scale quantum devices.

## Abstract

By leveraging quantum effects, such as superposition and entanglement, quantum metrology promises higher precision than classical strategies. It is, however, a challenging task to achieve the higher precision on practical systems. This is mainly due to difficulties in engineering nonclassical states and performing nontrivial measurements on the system, especially when the number of particles is large. Here we propose a variational scheme with the Loschmidt echo for quantum metrology. By utilizing hardware-efficient ansatzes in the design of variational quantum circuits, the quantum Fisher information (QFI) of the probe state can be extracted from the experimentally measured Loschmidt echo in a scalable manner. This QFI is then used to guide the online optimization of the preparation of the probe state. We experimentally implement the scheme on an ensemble of 10-spin quantum processors and achieve a 12.4-dB enhancement of the measurement precision over the uncorrelated states, which is close to the theoretical limit. The scheme can also be employed on various other noisy intermediate-scale quantum devices, which provides a promising protocol to demonstrate quantum advantages.

This study introduces a variational quantum metrology scheme based on Loschmidt echo and experimentally demonstrates its quantum-enhanced characteristic, providing a promising framework for quantum metrology across diverse practical quantum systems.