# Tensor-Network Approach for Quantum Metrology in Many-Body Quantum Systems

## TL;DR

This paper introduces a tensor network framework using matrix product operators to efficiently identify optimal quantum metrological protocols in many-body systems with noise, enabling analysis of asymptotic precision and probe states.

## Contribution

It develops a comprehensive MPO-based approach for quantum metrology that handles noise correlations and allows direct computation of asymptotic precision in large systems.

## Key findings

- Efficient identification of optimal quantum states under noise.
- Application to atomic clock stabilization and magnetic sensing.
- Methods for calculating quantum Fisher information and fidelity susceptibility.

## Abstract

Identification of the optimal quantum metrological protocols in realistic many particle quantum models is in general a challenge that cannot be efficiently addressed by the state-of-the-art numerical and analytical methods. Here we provide a comprehensive framework exploiting matrix product operators (MPO) type tensor networks for quantum metrological problems. Thanks to the fact that the MPO formalism allows for an efficient description of short-range spatial and temporal noise correlations, the maximal achievable estimation precision in such models, as well as the optimal probe states in previously inaccessible regimes can be identified. Moreover, the application of infinite MPO (iMPO) techniques allows for a direct and efficient determination of the asymptotic precision of optimal protocols in the limit of infinite particle numbers. We illustrate the potential of our framework in terms of an atomic clock stabilization (temporal noise correlation) example as well as for magnetic field sensing in the presence of locally correlated magnetic field fluctuations (spatial noise correlations). As a byproduct, the developed methods for calculating the quantum Fisher information via MPOs may be used to calculate the fidelity susceptibility - a parameter widely used in many-body physics to study phase transitions.