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.
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.
Estimating multiple unknown parameters simultaneously is essential for practical quantum sensing. However, it faces a fundamental challenge: the optimal strategy for estimating one parameter is often incompatible with that for another, making it impossible to simultaneously achieve the ultimate precision limits for all parameters. Here we develop a general and efficient computational framework that jointly optimizes probe states, control operations, and measurements across different strategy families, including parallel, sequential, and those with indefinite causal order. Our approach provides exact semidefinite-program formulations for several precision bounds, including the Holevo, Nagaoka-Hayashi, and quantum Cramér-Rao bounds. We demonstrate the capabilities of the framework in multiparameter magnetometry and frequency estimation, identifying optimal protocols within each class and revealing a strict hierarchy among the achievable performances of different classes in the multiparameter regime. The framework also directly incorporates resource constraints, such as energy budgets, enabling systematic investigation of experimentally realistic sensing scenarios. Furthermore, we develop a finite-memory optimization method for sequential strategies with restricted ancillary-memory dimension. By decomposing the protocol into initial probe preparation and intermediate control operations, this method provides a practical route to designing resource-constrained sequential sensing schemes. Our work establishes a versatile computational tool for determining fundamental precision limits and designing optimal quantum-sensing protocols in complex multiparameter settings.
Precise estimation of physical parameters underpins both scientific discovery and technological development. A central goal of quantum metrology and sensing is to exploit quantum resources like entanglement to devise optimal strategies for estimating physical parameters as precisely as possible. While substantial progress has been made in single-parameter quantum metrology, the multiparameter scenario remains significantly more challenging due to the issue of parameter incompatibility. In this work, we present a unified and computable approach for the simultaneous estimation of multiple parameters that attains the ultimate precision permitted by quantum mechanics. The core of our approach is to integrate the quantum tester formalism into the recently proposed tight Cramér-Rao type bound. This formulation enables us to figure out the highest achievable precision via upper and lower bounds that are computable via semidefinite programs. More importantly, within this formulation, diverse quantum resources, including entanglement, coherence, quantum control, and indefinite causal order, are treated on equal footing and systematically optimized for the purpose of achieving the ultimate precision in multiparameter estimation. As a result, our approach is applicable to various metrological strategies both in the presence and absence of noise. To demonstrate its utility, we revisit three-dimensional magnetic-field estimation, uncovering the strengths and limitations of existing analytical results and further establishing a strict hierarchy among different types of strategies.
Using quantum systems as sensors or probes has been shown to greatly improve the precision of parameter estimation by exploiting unique quantum features such as entanglement. A major task in quantum sensing is to design the optimal protocol, i.e., the most precise one. It has been solved for some specific instances 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 find the optimal initial state of the probe (which can be entangled with an auxiliary system), the optimal measurement, and the optimal estimator function. We leverage the formalism of higher-order operations to develop a method based on semidefinite programming that finds a protocol that is close to the optimal one with arbitrary precision. Crucially, our method is not restricted to any specific quantum evolution, cost function, or prior distribution, and thus can be applied to any estimation problem. Moreover, it can be applied to both single-parameter or multiparameter estimation tasks. We demonstrate our method with three examples, consisting of unitary phase estimation, thermometry in a bosonic bath, and multiparameter estimation of an SU(2) transformation. Exploiting our methods, we extend several results from the literature. For example, in the thermometry case, we find the optimal protocol at any finite time and quantify the usefulness of entanglement. Published by the American Physical Society 2024
The most general strategy for estimating parameters of a quantum channel is to probe the channel many times sequentially with the help of coherent ancillary systems. However, finding the strategy that minimizes the mean-squared error (MSE) is a notoriously hard problem, requiring optimization over the initial probe state, intermediate evolutions, final measurement, and classical postprocessing. In this work, we study how the minimum attainable MSE can be bounded from below in this setting. We propose a lower bound for the MSE of any physically implementable sequential sensing strategy that applies to estimating parameters of any physical quantum evolution, including quantum channels probed over multiple time steps under temporally correlated noise and general temporally correlated quantum processes. We show that this lower bound can be formulated as a semidefinite program (SDP). Our bound generalizes the Nagaoka-Hayashi lower bound (that applies to parameter estimation tasks with separable measurements) to the regime where the quantum channel can be accessed sequentially multiple times. For regular finite-dimensional single-parameter sequential estimation tasks admitting an optimal strategy, we show that our SDP bound is tight and provide a procedure to obtain a locally optimal strategy from the SDP optimizer. Using our optimization bound, we compute the MSE lower bound and derive new protocols for several other physically motivated problems including error-corrected quantum sensing under an unknown correlated noise model, multiparameter estimation, and estimation with qudit systems.
Abstract This tutorial introduces a systematic approach for addressing the key question of quantum metrology: For a generic task of sensing an unknown parameter, what is the ultimate precision given a constrained set of admissible strategies. The approach outputs the maximal attainable precision (in terms of the maximum of quantum Fisher information) as a semidefinite program and optimal strategies as feasible solutions thereof. Remarkably, the approach can identify the optimal precision for different sets of strategies, including parallel, sequential, quantum SWITCH‐enhanced, causally superposed, and generic indefinite‐causal‐order strategies. The tutorial consists of a pedagogic introduction to the background and mathematical tools of optimal quantum metrology, a detailed derivation of the main approach, and various concrete examples. As shown in the tutorial, applications of the approach include, but are not limited to, strict hierarchy of strategies in noisy quantum metrology, memory effect in non‐Markovian metrology, and designing optimal strategies. Compared with traditional approaches, the approach here yields the exact value of the optimal precision, offering more accurate criteria for experiments and practical applications. It also allows for the comparison between conventional strategies and the recently discovered causally‐indefinite strategies, serving as a powerful tool for exploring this new area of quantum metrology.
One of the main quests in quantum metrology is to attain the ultimate precision limit with given resources, where the resources are not only of the number of queries, but more importantly of the allowed strategies. With the same number of queries, the restrictions on the strategies constrain the achievable precision. In this Letter, we establish a systematic framework to identify the ultimate precision limit of different families of strategies, including the parallel, the sequential, and the indefinite-causal-order strategies, and provide an efficient algorithm that determines an optimal strategy within the family of strategies under consideration. With our framework, we show there exists a strict hierarchy of the precision limits for different families of strategies.