Extensive research has been dedicated to the asymptotic theory of quantum metrology, where the goal is to determine the ultimate precision limit of quantum channel estimation when many accesses to the channel are allowed. The ultimate limit has been well established, but in general, a noiseless and controllable ancilla is required for attaining it. Little is known about the metrological performance without noiseless ancillae, which is more relevant in practical circumstances. In this Letter, we present a novel theoretical framework to address this problem, bridging quantum metrology and the asymptotic theory of quantum channels. Leveraging this framework, we prove sufficient conditions for achieving the Heisenberg limit with repeated applications of the channel to be estimated both with and without applying interleaved unitary control operations. For the latter case, we design an algorithm to identify the control operation explicitly.
Extensive research has been dedicated to the asymptotic theory of quantum metrology, where the goal is to determine the ultimate precision limit of quantum channel estimation when many accesses to the channel are allowed. The ultimate limit has been well established, but in general, a noiseless and controllable ancilla is required for attaining it. Little is known about the metrological performance without noiseless ancillae, which is more relevant in practical circumstances. In this Letter, we present a novel theoretical framework to address this problem, bridging quantum metrology and the asymptotic theory of quantum channels. Leveraging this framework, we prove sufficient conditions for achieving the Heisenberg limit with repeated applications of the channel to be estimated both with and without applying interleaved unitary control operations. For the latter case, we design an algorithm to identify the control operation explicitly.
Quantum error correction (QEC) is theoretically capable of achieving the ultimate estimation limits in noisy quantum metrology. However, existing quantum error-correcting codes designed for noisy quantum metrology generally exploit entanglement between one probe and one noiseless ancilla of the same dimension, and the requirement of noiseless ancillas is one of the major obstacles to implementing the QEC metrological protocol in practice. Here we successfully lift this requirement by explicitly constructing two types of multiprobe quantum error-correcting codes, where the first one utilizes a negligible amount of ancillas and the second one is ancilla free. Specifically, we consider Hamiltonian estimation under Markovian noise and show that (i) when the Heisenberg limit (HL) is achievable our codes can achieve the HL and its optimal asymptotic coefficient and (ii) when only the standard quantum limit (SQL) is achievable (even with arbitrary adaptive quantum strategies) the optimal asymptotic coefficient of the SQL is also achievable by our codes under slight modifications.
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 error correction has been successfully applied to enhance the precision of parameter estimation in the presence of noise. Nonetheless, existing methods require a number of noiseless, controllable ancillae and lack efficient encoding and decoding procedures. In this Letter, we demonstrate that subsystem error correction provides a new direction that can substantially simplify the metrological protocol. We derive general conditions under which subsystem stabilizer codes achieve the Heisenberg limit and show that, for broad classes of noise, this can be realized by syndrome-free protocols using at most a single ancilla qubit. Furthermore, we extend this framework to dynamical error correction and show that Floquet codes can protect time-dependent metrological signals in reaching the Heisenberg limit.
Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the Heisenberg limit using quantum probes subject to Markovian noise, assuming that noiseless ancilla systems are available, and that fast, accurate quantum processing can be performed. When the sufficient condition is satisfied, the quantum error-correcting code achieving the best possible precision can be found by solving a semidefinite program. We also show that noiseless ancilla are not needed when the signal Hamiltonian and the error operators commute. Finally we provide two explicit, archetypal examples of quantum sensors: qubits undergoing dephasing and a lossy bosonic mode.
Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the Heisenberg limit using quantum probes subject to Markovian noise, assuming that noiseless ancilla systems are available, and that fast, accurate quantum processing can be performed. When the sufficient condition is satisfied, a quantum error-correcting code can be constructed that suppresses the noise without obscuring the signal; the optimal code, achieving the best possible precision, can be found by solving a semidefinite program.
Measuring stochastic signals ("noise metrology") constitutes a central task in quantum sensing and the characterization of open quantum systems. Here we establish ultimate precision bounds for multiparameter estimation of stochastic signals encoded through Markovian Lindblad dynamics, allowing for arbitrary quantum control and noiseless ancillae. Although Markovianity enforces standard-quantum-limit scaling with sensing time T, our bounds reveal Heisenberg-type scaling in the number of dissipative channels, R: when the stochastic signal exhibits high-rank correlations across the R channels and the probe is entangled, the average variance (per parameter) scales no better than Ω(1/(TR2)). For collective k-body dissipation, R=Θ(Nk), signifying super-Heisenberg scaling with the system size N. We further show that, when the unknown parameters enter through the dissipative eigenrates, a Rapid Prepare-and-Measure (RPM) protocol that tracks many distinct quantum jumps in parallel attains these limits. In this regime, the estimation problem reduces to a multi-Poisson counting model, providing a conceptually clean route to optimal quantum noise metrology. We illustrate the breadth of the framework with applications to networked noise metrology, collective many-body dissipation, learning Pauli noise, and subdiffraction quantum imaging.