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.
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.
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, 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.
Quantum effect enables enhanced estimation precision in metrology, with the Heisenberg limit (HL) representing the ultimate limit allowed by quantum mechanics. Although the HL is generally unattainable in the presence of noise, quantum error correction (QEC) can recover the HL in various scenarios. A notable example is estimating a Pauli-Z signal under bit-flip noise using the repetition code, which is both optimal for metrology and robust against noise. However, previous protocols often assume noise affects only the signal accumulation step, while the QEC operations -- including state preparation and measurement -- are noiseless. To overcome this limitation, we study fault-tolerant quantum metrology where all qubit operations are subject to noise. We focus on estimating a Pauli-Z signal under bit-flip noise, together with state preparation and measurement errors in all QEC operations. We propose a fault-tolerant metrological protocol where a repetition code is prepared via repeated syndrome measurements, followed by a fault-tolerant logical measurement. We demonstrate the existence of an error threshold, below which errors are effectively suppressed and the HL is attained.
Quantum error correction has recently emerged as a tool to enhance quantum sensing under Markovian noise. It works by correcting errors in a sensor while letting a signal imprint on the logical state. This approach typically requires a specialized error-correcting code, as most existing codes correct away both the dominant errors and the signal. To date, however, few such specialized codes are known, among which most require noiseless, controllable ancillas. We show here that such ancillas are not needed when the signal Hamiltonian and the error operators commute, a common limiting type of decoherence in quantum sensors. We give a semidefinite program for finding optimal ancilla-free sensing codes in general, as well as closed-form codes for two common sensing scenarios: qubits undergoing dephasing, and a lossy bosonic mode. Finally, we analyze the sensitivity enhancement offered by the qubit code under arbitrary spatial noise correlations, beyond the ideal limit of orthogonal signal and noise operators.
Quantum error correction (QEC) codes have emerged as a powerful tool to protect quantum-enhanced metrology against noise. However, the ability to correct errors alone does not guarantee high metrological sensitivity, as the encoded states may become insensitive to the parameter of interest. Here we show that this limitation can be overcome by exploiting an intrinsic freedom of QEC codes: for a fixed set of correctable errors, the Knill-Laflamme conditions admit an equivalence class of encodings. When the correctable noise possesses unitary symmetries, these symmetries generate continuous transformations within this class, allowing systematic optimization of the encoding to increase the quantum Fisher information while preserving the correctable set of noise. Based on this observation, we develop a symmetry-based optimization approach and derive criteria identifying when such optimization can enhance metrological sensitivity. In particular, for stabilizer-sum Hamiltonians, symmetry optimization can convert a code with vanishing QFI into one achieving the standard quantum limit in general or even Heisenberg scaling in specific cases, illustrating the power of symmetry optimization for QEC-assisted quantum metrology.
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.