Tensor-network approach for quantum metrology in many-body quantum systems

Krzysztof Chabuda,Jacek Dziarmaga,Tobias J Osborne,Rafał Demkowicz-Dobrzański

doi:10.1038/s41467-019-13735-9

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. The maximal achievable estimation precision as well as the optimal probe states in previously inaccessible regimes can be identified including models with short-range noise correlations. Moreover, the application of infinite MPO (iMPO) techniques allows for a direct and efficient determination of the asymptotic precision 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 magnetic field sensing (spatial noise correlations). As a byproduct, the developed methods may be used to calculate the fidelity susceptibility—a parameter widely used to study phase transitions.

Highlights

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-ofthe-art numerical and analytical methods
That led to unparalleled insights into the physics of quantum manybody systems, the most relevant for the present work is the matrix product operator (MPO) ansatz for density operators[28] and its infinite particle limit known as infinite MPO29
In this paper, building upon experience obtained from uncorrelated noise metrological studies[30], where the optimal input states where shown to be efficiently described as matrix product states (MPS), we develop a comprehensive tensor-network-based framework allowing to (i) calculate relevant metrological quantities (such as the Quantum Fisher Information (QFI) or a Bayesiantype cost), (ii) optimize input probe states and as a result (iii) identify the optimal metrological protocol

Summary

Introduction

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-ofthe-art numerical and analytical methods. Apart from idealized noiseless models[1] as well as models operating within the fully symmetric subspace[9] (where the Hilbert space dimension grows linearly with the number of particles) only small-scale problems are feasible via direct numerical study, and even a slight increase in the number of elementary objects makes such an approach intractable. In cases when one deals with metrological models involving correlated noise, or whenever states outside the fully symmetric subspace are involved, there are no efficient methods that can be applied. Temporal noise correlations are present in the atomic clock stabilization problem[17], making identification of the optimal quantum clock stabilization strategies a highly non-trivial task[18,19,20]. That led to unparalleled insights into the physics of quantum manybody systems, the most relevant for the present work is the matrix product operator (MPO) ansatz for density operators[28] and its infinite particle limit known as infinite MPO (iMPO)[29]

Methods

Results

Conclusion