Strategy optimization for Bayesian quantum parameter estimation with finite copies: Adaptive greedy, parallel, sequential, and general strategies

In this work, we study Bayesian quantum parameter estimation given a finite number of uses of the process encoding one or more unknown physical quantities. For multiple uses, it is conventional to classify quantum metrological protocols as parallel, sequential, or indefinite causal order. Within each class, the central question is to determine the optimal strategy – namely, the choice of optimal input state, control operations, measurement, and estimator(s) – to perform the estimation task. Using the formalism of higher-order operations, we develop an algorithm that looks for the optimal solution, and we provide an efficient numerical implementation based on semidefinite programming. Our benchmark examples, specifically those against existing analytical solutions, demonstrate how powerful and precise our method is. We further explore the potential of greedy adaptive strategies, which are based on classical feedforward to design the optimal protocol for the next round. Using this framework, we compare the optimal achievable Bayesian score across classes. We demonstrate the strength of our algorithm in several examples, from single to multiparameter estimation and with various prior distributions. Particularly, we find examples in which there is a strict hierarchy between different classes. Nonetheless, the performance of the different quantum memory-assisted classes are not significantly different, while they may significantly outperform the adaptive greedy strategy.

💡 Research Summary

This paper addresses the problem of Bayesian quantum parameter estimation when only a finite number of uses of the unknown quantum process (the “channel”) are available. While the Bayesian framework is well‑known for its ability to combine prior knowledge with measurement outcomes, analytical solutions for the optimal estimation protocol are scarce except for very special cases (e.g., quadratic cost functions or single‑shot scenarios). The authors therefore develop a unified, numerically tractable method that can find the optimal protocol for any number of channel copies, any prior distribution, and any cost function.

The core of the method is the “tester” formalism. By representing the joint preparation‑measurement procedure as a set of positive semidefinite operators ({T_i}) (the testers), the Bayesian score \