Optimal Multiparameter Metrology: The Quantum Compass Solution

TL;DR

This paper develops a systematic method to identify optimal quantum sensors for multiparameter estimation, introducing the 'quantum compass' solution that achieves the Heisenberg limit and is practical for implementation.

Contribution

It extends Fisher information-based quantum metrology with a new optimality criterion, proposing the quantum compass as a multiparameter optimal sensor with analytical and numerical solutions.

Findings

Exact quantum compass solutions for two and three parameter sensing.

Quantum circuits achieving the Heisenberg limit on a trapped-ion platform.

A new metrological cost function enabling variational quantum sensor design.

Abstract

We study optimal quantum sensing of multiple physical parameters using repeated measurements. In this scenario, the Fisher information framework sets the fundamental limits on sensing performance, yet the optimal states and corresponding measurements that attain these limits remain to be discovered. To address this, we extend the Fisher information approach with a second optimality requirement for a sensor to provide unambiguous estimation of unknown parameters. We propose a systematic method integrating Fisher information and Bayesian approaches to quantum metrology to identify the combination of input states and measurements that satisfies both optimality criteria. Specifically, we frame the optimal sensing problem as an optimization of an asymptotic Bayesian cost function that can be efficiently solved numerically and, in many cases, analytically. We refer to the resulting optimal sensor as a `quantum compass' solution, which serves as a direct multiparameter counterpart to the Greenberger-Horne-Zeilinger state-based interferometer, renowned for achieving the Heisenberg limit in single-parameter metrology. We provide exact quantum compass solutions for paradigmatic multiparameter problem of sensing two and three parameters using an SU(2) sensor. Our metrological cost function opens avenues for quantum variational techniques to design low-depth quantum circuits approaching the optimal sensing performance in the many-repetition scenario. We demonstrate this by constructing simple quantum circuits that achieve the Heisenberg limit for vector field and 3D rotations estimation using a limited set of gates available on a trapped-ion platform. Our work introduces and optimizes sensors for a practical notion of optimality, keeping in mind the ultimate goal of quantum sensors to precisely estimate unknown parameters.