Robust Quantum Metrological Schemes

-qubit errors after the signal sensing. In comparison at least five qubits are required for correcting arbitrary 1-qubit errors in standard quantum error correction.

Related papers

Optimal quantum metrology protocols with erasure qubits

Michal Arieli, Alex Retzker, Tuvia Gefen

We investigate the precision limits and optimal protocols for sensing single qubit signals in the presence of erasure noise. We study a hierarchy of precision limits achievable with metrological strategies of differing complexity, and identify the optimal protocol for each. The detectability of erasure noise is shown to lead to enhanced precision limits and simplified sensing protocols. For energy gap estimation, we demonstrate that a simple product-state strategy with continuous erasure detection yields significant improvements, outperforming optimal passive entangled protocols over a broad range of finite system sizes. We show that for other single-qubit signals, quantum error correction provides a substantial advantage by correcting the dominant erasure processes, and can restore Heisenberg-limited precision in certain erasure configurations. As a byproduct of our analysis, we find erasure-conversion schemes for qubits subject to thermal noise that attain the corresponding ultimate precision limits.We investigate the precision limits and optimal protocols for sensing single qubit signals in the presence of erasure noise. We study a hierarchy of precision limits achievable with metrological strategies of differing complexity, and identify the optimal protocol for each. The detectability of erasure noise is shown to lead to enhanced precision limits and simplified sensing protocols. For energy gap estimation, we demonstrate that a simple product-state strategy with continuous erasure detection yields significant improvements, outperforming optimal passive entangled protocols over a broad range of finite system sizes. We show that for other single-qubit signals, quantum error correction provides a substantial advantage by correcting the dominant erasure processes, and can restore Heisenberg-limited precision in certain erasure configurations. As a byproduct of our analysis, we find erasure-conversion schemes for qubits subject to thermal noise that attain the corresponding ultimate precision limits.

quant-pharXiv:2603.11807Mar 13, 20260 citations

Robust quantum metrological schemes based on protection of quantum Fisher information

Xiao-Ming Lu, Sixia Yu, C. H. Oh

Fragile quantum features such as entanglement are employed to improve the precision of parameter estimation and as a consequence the quantum gain becomes vulnerable to noise. As an established tool to subdue noise, quantum error correction is unfortunately overprotective because the quantum enhancement can still be achieved even if the states are irrecoverably affected, provided that the quantum Fisher information, which sets the ultimate limit to the precision of metrological schemes, is preserved and attained. Here we develop a theory of robust metrological schemes that preserve the quantum Fisher information instead of the quantum states themselves against noise. After deriving a minimal set of testable conditions on this kind of robustness, we construct a family of 2t+1 qubits metrological schemes being immune to t-qubit errors after the signal sensing. In comparison, at least five qubits are required for correcting arbitrary 1-qubit errors in standard quantum error correction. Quantum features such as entanglement can be used to improve metrological precision, but noise affects their performances. Here the authors construct schemes for exploiting quantum error correction to preserve the quantum Fisher information, thereby shielding quantum gain efficiently.Fragile quantum features such as entanglement are employed to improve the precision of parameter estimation and as a consequence the quantum gain becomes vulnerable to noise. As an established tool to subdue noise, quantum error correction is unfortunately overprotective because the quantum enhancement can still be achieved even if the states are irrecoverably affected, provided that the quantum Fisher information, which sets the ultimate limit to the precision of metrological schemes, is preserved and attained. Here we develop a theory of robust metrological schemes that preserve the quantum Fisher information instead of the quantum states themselves against noise. After deriving a minimal set of testable conditions on this kind of robustness, we construct a family of 2t+1 qubits metrological schemes being immune to t-qubit errors after the signal sensing. In comparison, at least five qubits are required for correcting arbitrary 1-qubit errors in standard quantum error correction. Quantum features such as entanglement can be used to improve metrological precision, but noise affects their performances. Here the authors construct schemes for exploiting quantum error correction to preserve the quantum Fisher information, thereby shielding quantum gain efficiently.

arXiv:1405.4052Nature CommunicationsJun 8, 201583 citations

Preserving Heisenberg-Limited Metrological Information during Storage via Correlated-Noise Correction

Hang Xu, Xue-Ke Song, Jingzheng Huang +2 more

Quantum error correction has become an indispensable tool for restoring Heisenberg-limited precision in noisy quantum metrology. Existing protocols, however, almost exclusively focus on correcting noise during the signal-encoding stage and implicitly assume that the probe is measured immediately after sensing. In many quantum information processing tasks, the encoded probe must instead be stored before subsequent quantum operations, during which environmental noise can significantly degrade the accumulated metrological information. Here, we propose a correlated-noise correction (CNC) protocol for protecting quantum probes during the storage stage. By correlating probe errors with auxiliary qubits through fixed two-body entangling gates, memory errors are converted into measurable syndromes that are extracted only once after storage. We show that the protocol naturally extends from single-qubit to multi-qubit probes and protects the stored quantum Fisher information against dephasing, bit-flip, and amplitude-damping noise. Furthermore, we demonstrate that preserving the quantum Fisher information does not necessarily require restoring the entire quantum state when the probe is measured immediately after storage, whereas full state recovery becomes essential for subsequent rounds of quantum signal processing. Our results establish correlated-noise correction as a practical framework for protecting metrological information during quantum memory and provide a useful building block for sensing-enabled quantum information processing.Quantum error correction has become an indispensable tool for restoring Heisenberg-limited precision in noisy quantum metrology. Existing protocols, however, almost exclusively focus on correcting noise during the signal-encoding stage and implicitly assume that the probe is measured immediately after sensing. In many quantum information processing tasks, the encoded probe must instead be stored before subsequent quantum operations, during which environmental noise can significantly degrade the accumulated metrological information. Here, we propose a correlated-noise correction (CNC) protocol for protecting quantum probes during the storage stage. By correlating probe errors with auxiliary qubits through fixed two-body entangling gates, memory errors are converted into measurable syndromes that are extracted only once after storage. We show that the protocol naturally extends from single-qubit to multi-qubit probes and protects the stored quantum Fisher information against dephasing, bit-flip, and amplitude-damping noise. Furthermore, we demonstrate that preserving the quantum Fisher information does not necessarily require restoring the entire quantum state when the probe is measured immediately after storage, whereas full state recovery becomes essential for subsequent rounds of quantum signal processing. Our results establish correlated-noise correction as a practical framework for protecting metrological information during quantum memory and provide a useful building block for sensing-enabled quantum information processing.

quant-pharXiv:2608.08130Aug 11, 20260 citations

Bypassing the protection-sensitivity incompatibility in quantum-error-corrected metrology via asymmetric codes

Junjie Chen, Rui Luo, Zhenyu Du +3 more

Quantum metrology surpasses the classical precision limit by encoding signals in a probe state such that signal contributions are indistinguishable and their phases accumulate coherently as a collective response. In realistic settings, scalable quantum metrology requires quantum error correction to protect the probe against noise. However, quantum error correction relies on syndrome information that distinguishes errors for identification and correction. Because signals and errors act on the same physical degrees of freedom, there is a structural incompatibility between signal-sensitivity and noise-protection in quantum-error-corrected metrology. We quantify this incompatibility by establishing trade-offs between code distance and the quantum Fisher information of code states for non-degenerate codes, quantum low-density parity-check codes, and generalized Shor codes. We bypass this limitation with asymmetric quantum error correction, in which protection is relaxed along the signal direction while being maintained in complementary directions. We construct such codes for local sensing Hamiltonians, restoring Heisenberg-limited precision while retaining a growing distance and hence protection against local perturbations in complementary directions. Strongly asymmetric quantum low-density parity-check and concatenated asymmetric constructions make the framework sparse, scalable, and continuously tunable. The associated probe states can be prepared by constant-depth adaptive circuits with optimal resource scaling.Quantum metrology surpasses the classical precision limit by encoding signals in a probe state such that signal contributions are indistinguishable and their phases accumulate coherently as a collective response. In realistic settings, scalable quantum metrology requires quantum error correction to protect the probe against noise. However, quantum error correction relies on syndrome information that distinguishes errors for identification and correction. Because signals and errors act on the same physical degrees of freedom, there is a structural incompatibility between signal-sensitivity and noise-protection in quantum-error-corrected metrology. We quantify this incompatibility by establishing trade-offs between code distance and the quantum Fisher information of code states for non-degenerate codes, quantum low-density parity-check codes, and generalized Shor codes. We bypass this limitation with asymmetric quantum error correction, in which protection is relaxed along the signal direction while being maintained in complementary directions. We construct such codes for local sensing Hamiltonians, restoring Heisenberg-limited precision while retaining a growing distance and hence protection against local perturbations in complementary directions. Strongly asymmetric quantum low-density parity-check and concatenated asymmetric constructions make the framework sparse, scalable, and continuously tunable. The associated probe states can be prepared by constant-depth adaptive circuits with optimal resource scaling.

quant-pharXiv:2512.20426Dec 24, 20250 citations

Achieving the Heisenberg limit using fault-tolerant quantum error correction

Himanshu Sahu, Qian Xu, Sisi Zhou

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 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.

quant-pharXiv:2601.05457Jan 12, 20260 citations

Precision bounds in noisy quantum metrology

Jan Kołodyński

In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as $1/N^2$ with the number of particles involved---substantially surpassing the $1/N$ -scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic $N$ limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large $N$ the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as $1/N^2$ with the number of particles involved---substantially surpassing the $1/N$-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic $N$ limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large $N$ the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.

arXiv:1409.0535Sep 1, 201422 citations

2t+1

qubits metrological schemes being immune to

t

-qubit errors after the signal sensing. In comparison at least five qubits are required for correcting arbitrary 1-qubit errors in standard quantum error correction.

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.

In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as

1/N^2

with the number of particles involved---substantially surpassing the

1/N

-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic

N

limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large

N

the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.