Frequentist Parameter Estimation with Supervised Learning

TL;DR

This paper demonstrates that supervised learning for parameter estimation in quantum sensors converges to the Bayesian MAP estimator and aligns with MLE under large training data, with quantum noise setting fundamental limits.

Contribution

It clarifies the frequentist interpretation of machine-learned estimators and connects them to classical statistical bounds like Cramér-Rao, advancing quantum sensor calibration methods.

Findings

Estimator converges to Bayesian MAP under regularity conditions.

Cramér-Rao bound applies to the mean-square error of the estimator.

Quantum noise imposes a fundamental limit on training grid resolution.

Abstract

Recently there has been a great deal of interest surrounding the calibration of quantum sensors using machine learning techniques. In this work, we explore the use of regression to infer a machine-learned point estimate of an unknown parameter. Although the analysis is neccessarily frequentist - relying on repeated esitmates to build up statistics - we clarify that this machine-learned estimator converges to the Bayesian maximum a-posterori estimator (subject to some regularity conditions). When the number of training measurements are large, this is identical to the well-known maximum-likelihood estimator (MLE), and using this fact, we argue that the Cram{\'e}r-Rao sensitivity bound applies to the mean-square error cost function and can therefore be used to select optimal model and training parameters. We show that the machine-learned estimator inherits the desirable asymptotic properties of the MLE, up to a limit imposed by the resolution of the training grid. Furthermore, we investigate the role of quantum noise the training process, and show that this noise imposes a fundamental limit on number of grid points. This manuscript paves the way for machine-learning to assist the calibration of quantum sensors, thereby allowing maximum-likelihood inference to play a more prominent role in the design and operation of the next generation of ultra-precise sensors.