Optimal strategies for local quantum metrology -- including the preparation of optimal probe states, implementation of optimal control and measurement strategies, are well established. However, for distributed quantum metrology, where the goal is to estimate global properties of multiple spatially distributed parameters, the optimal scheme -- particularly the role of optimal control -- remains poorly understood. In this work, we address this challenge by developing optimal schemes for distributed quantum metrology that characterize the ultimate precision limits in distributed systems. We derive the optimal probe state, optimal control protocols, and measurement strategies in estimating a linear combination of $N$ independent unknown parameters coupled to $d$ networked sensors. Crucially, we prove that the optimal control operations can be implemented locally on each sensor, eliminating the need for non-local control operations across distant nodes. This result significantly reduces the complexity of implementing optimal strategies in distributed quantum metrology. To demonstrate the power of our framework, we apply it to several key scenarios.
Optimal strategies for local quantum metrology -- including the preparation of optimal probe states, implementation of optimal control and measurement strategies, are well established. However, for distributed quantum metrology, where the goal is to estimate global properties of multiple spatially distributed parameters, the optimal scheme -- particularly the role of optimal control -- remains poorly understood. In this work, we address this challenge by developing optimal schemes for distributed quantum metrology that characterize the ultimate precision limits in distributed systems. We derive the optimal probe state, optimal control protocols, and measurement strategies in estimating a linear combination of N independent unknown parameters coupled to d networked sensors. Crucially, we prove that the optimal control operations can be implemented locally on each sensor, eliminating the need for non-local control operations across distant nodes. This result significantly reduces the complexity of implementing optimal strategies in distributed quantum metrology. To demonstrate the power of our framework, we apply it to several key scenarios.
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.
Nature imposes fundamental limits on how precisely we can probe the physical world. Quantum metrology is the study of achieving higher precision and accuracy in estimating the physical parameters by harnessing quantum resources. It is considered one of the most promising quantum technologies, offering significant advantages over classical ones in both near‑term and long‑term applications. However, existing theoretical research in quantum metrology often falls short of providing a unified and rigorous understanding of the ultimate precision under realistic experimental limitations on available quantum resources. In this thesis, we conduct a systematic investigation of the optimal precision limits in quantum metrology and identify the metrological strategies that achieve optimality under specified resource constraints. We primarily focus on the single-parameter estimation of quantum processes, a cornerstone task in quantum metrology. We first establish a general framework of strictly optimal quantum metrology using semidefinite programming within the formalism of higher-order quantum operations. This framework reveals a strict hierarchy for the performance of different families of strategies---parallel, sequential, and indefinite-causal-order ones---each subject to specific causal constraints. Our results rigorously demonstrate a quantum metrological advantage of indefinite causal order in finite-dimensional systems, which has recently aroused rapidly growing interest. In realistic scenarios, a fully optimized quantum metrology protocol may be experimentally demanding due to the complexity of state preparation and control. Tailored to practical constraints on the allowed ancilla and control operations, we propose a highly efficient tensor network algorithm to optimize the probe state and a long sequence of control for quantum metrology. When the number of control steps tends to infinity, we develop an asymptotic theory of quantum metrology for achieving the ultimate precision limit---Heisenberg limit---without ancilla. This contrasts with the traditional approach using quantum error correction for metrology that typically requires noiseless and controllable ancilla. Our results pave the way for exploiting the full potential of quantum resources in metrology. In the near term, our methods enable high-precision estimation with limited interaction with the unknown process using experimentally feasible control techniques. In the long term, they lay the foundation for scalable and fault-tolerant quantum metrology.
Quantum metrology has emerged as a powerful tool for timekeeping, field sensing, and precision measurements in fundamental physics. With the advent of distributed quantum metrology, its capabilities have extended to probing spatially distributed parameters across networked quantum systems. However, scalable implementations of distributed quantum metrology with multiparameter estimation remain limited, particularly due to the challenges of generating and distributing entanglement across a quantum network and dealing with incompatibilities in multiparameter quantum metrology. Here we demonstrate distributed multiparameter quantum metrology on a modular superconducting quantum network with low-loss microwave interconnects, a platform that uniquely combines fast gate operations, adaptive control, and deterministic non-local entanglement generation. Using a control-enhanced sequential protocol, we estimate all three components of a remote vector field, achieving up to 13.72 dB improvement in precision over the individual strategy. We further perform direct estimation of vector field gradients along two directions across spatially separated nodes, realizing a 3.44 dB gain over local entanglement strategies. These results establish superconducting quantum networks as a competitive and reconfigurable platform for scalable multiparameter distributed quantum metrology.
Measurement noise is a major source of noise in quantum metrology.Here, we explore preprocessing protocols that apply quantum controls to the quantum sensor state prior to the final noisy measurement (but after the unknown parameter has been imparted), aiming to maximize the estimation precision.We define the quantum preprocessing-optimized Fisher information, which determines the ultimate precision limit for quantum sensors under measurement noise, and conduct a thorough investigation into optimal preprocessing protocols.First, we formulate the preprocessing optimization problem as a biconvex optimization using the error observable formalism, based on which we prove that unitary controls are optimal for pure states and derive analytical solutions of the optimal controls in several practically relevant cases.Then we prove that for classically mixed states (whose eigenvalues encode the unknown parameter) under commuting-operator measurements, coarse-graining controls are optimal, while unitary controls are suboptimal in certain cases.Finally, we demonstrate that in multiprobe systems where noisy measurements act independently on each probe, the noiseless precision limit can be asymptotically recovered using global controls for a wide range of quantum states and measurements.Applications to noisy Ramsey interferometry and thermometry are presented, as well as explicit circuit constructions of optimal controls.
A naturally multiparameter sensing scenario arises in spatially distributed networks of quantum sensors, where each node probes a distinct parameter of interest. In such settings, the central challenges are to achieve quantum-enhanced sensitivity under well-defined resource constraints and to explicitly identify measurement strategies that attain the ultimate precision limits. Here, we analytically demonstrate quantum-enhanced sensitivity for a broad class of distributed quantum probes designed to simultaneously estimate a complete set of unknown reference-relative phase differences encoded locally. Although such locally encoded parameters can, in principle, be optimally estimated using separable strategies, our distributed protocol offers distinct advantages. In particular, for fixed network size and identical bare probe Hamiltonians, it is energetically more efficient and exhibits stroboscopic disentanglement at specific evolution times, enabling the accumulated pairwise information to be coherently mapped onto a specific subsystem. Consequently, all set of pairwise reference-relative phase differences can be accessed through a single collective measurement without requiring independent local estimation. We further construct the corresponding optimal measurement strategies and show that, within this class of probes, the ultimate multiparameter precision limits—the Holevo and quantum Cramér-Rao bounds—are saturable. We illustrate the framework with two concrete applications: simultaneous gravimetry across spatially separated locations and estimation of distributed coupling strengths. Feasibility analyses indicate that the proposed distributed quantum-enhanced sensing protocols operate within experimentally accessible regimes.
Recent years have witnessed a growing interest in understating the limitations imposed by quantum noise in precision measurements and devising techniques to reduce it. The attention is currently turning to the simultaneously estimation of several parameters of interest, driven by its promising potential across a wide range of sensing applications as well as fueled by experimental progress in various optical and atomic platforms. Here, we provide a comprehensive overview of key research directions in multiparameter quantum sensing and metrology, highlighting both opportunities and challenges. We introduce the basic framework, discuss ultimate sensitivity bounds, optimal measurement strategies, and the role of quantum incompatibility, showing important differences with respect to single-parameter estimation. Additionally, we discuss emerging experimental implementations in distributed quantum sensing, including cutting-edge optimization techniques. This review aims to bridge the gap between theory and experiments, paving the way for the next-generation of quantum sensors and their integration with other quantum technologies.