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Using quantum systems as sensors or probes has been shown to greatly improvethe precision of parameter estimation by exploiting unique quantum featuressuch as entanglement. A major task in quantum sensing is to design the optimalprotocol, i.e., the most precise one. It has been solved for some specificinstances of the problem, but in general even numerical methods are not known.Here, we focus on the single-shot Bayesian setting, where the goal is to findthe optimal initial state of the probe (which can be entangled with anauxiliary system), the optimal measurement, and the optimal estimator function.We leverage the formalism of higher-order operations to develop a method basedon semidefinite programming that finds a protocol that is close to the optimalone with arbitrary precision. Crucially, our method is not restricted to anyspecific quantum evolution, cost function or prior distribution, and thus canbe applied to any estimation problem. Moreover, it can be applied to bothsingle or multiparameter estimation tasks. We demonstrate our method with threeexamples, consisting of unitary phase estimation, thermometry in a bosonicbath, and multiparameter estimation of an SU(2) transformation. Exploiting ourmethods, we extend several results from the literature. For example, in thethermometry case, we find the optimal protocol at any finite time and quantifythe usefulness of entanglement. Additionally, we show that when the costfunction is the mean squared error, projective measurements are optimal forestimation.

Designing optimal protocols in Bayesian quantum parameter estimation with higher-order operations