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. We analyze the performance of different kinds of quantum metrological protocols, including the conventional parallel, sequential, or general, which may involve an indefinite causal order. We also analyze a class of hybrid adaptive greedy strategies, which are based on classical feedforward between quantum protocols to optimize the next round. Within each class, the central question is to determine the optimal strategy -- namely, the choice of optimal input state, control operations, measurement, and estimators. Using the formalism of higher-order operations, we develop an algorithm that searches for the optimal solution, and we provide a numerical implementation based on semidefinite programming. Our benchmark cases, specifically those against existing analytical solutions, demonstrate how powerful and precise our method is. We further demonstrate the strength of our algorithm in several examples, from single to multiparameter estimation, and with various prior distributions. Particularly, we find examples where the adaptive greedy strategy matches the performance of general strategies with indefinite causal order, and at the same time examples that showcase a strict hierarchy between all different classes.