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arXiv:2506.07618 · Communications Physics

Error-mitigated quantum metrology via enhanced virtual purification

Xiaodie Lin, Haidong Yuan

Published Nov 26, 2025• 3 citations
arXivHTMLPDF

Abstract

Abstract Quantum metrology stands as a leading application of quantum science and technology, yet noise often constrains its precision and sensitivity. In near-term quantum metrology, existing protocols largely depend on virtual state purification, but significant noise accumulation and additional noise from the implementations of these protocols can impede their effectiveness. We propose enhanced virtual channel purification to address these problems, yielding enhanced virtual state purification as a by-product. Within sequential quantum metrology schemes, our error analysis reveals substantial bias reduction and quantum advantages in sampling cost when the number of encoding channels is $${\mathcal{O}}({p}^{-1})$$ O ( p − 1 ) , where p represents the error rate of encoding channels. In this range, our methods demonstrate significant improvements in parameter estimation precision and robustness against practical noise, as evidenced by numerical simulations for both single- and multi-parameter tasks. Particularly, these methods can naturally extend beyond quantum metrology, indicating their broad applicability in quantum information and quantum computation.

DOI
10.1038/s42005-025-02383-9

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•Dec 24, 2025•0 citations
Abstract Quantum metrology stands as a leading application of quantum science and technology, yet noise often constrains its precision and sensitivity. In near-term quantum metrology, existing protocols largely depend on virtual state purification, but significant noise accumulation and additional noise from the implementations of these protocols can impede their effectiveness. We propose enhanced virtual channel purification to address these problems, yielding enhanced virtual state purification as a by-product. Within sequential quantum metrology schemes, our error analysis reveals substantial bias reduction and quantum advantages in sampling cost when the number of encoding channels is O(p−1){\mathcal{O}}({p}^{-1})O(p−1) O ( p − 1 ) , where p represents the error rate of encoding channels. In this range, our methods demonstrate significant improvements in parameter estimation precision and robustness against practical noise, as evidenced by numerical simulations for both single- and multi-parameter tasks. Particularly, these methods can naturally extend beyond quantum metrology, indicating their broad applicability in quantum information and quantum computation.
The precision and sensitivity achievable in quantum metrology are often compromised by the presence of noise. While quantum error correction has emerged as a promising strategy, it is ineffective in addressing noise that is indistinguishable from the signal. To address this challenge, virtual state purification was introduced as a complementary approach to quantum error correction. However, significant noise accumulation can impede its performance. To overcome this limitation, we propose a swap test-based method specifically designed to address indistinguishable noise, even under high noise levels. A systematic error-scaling analysis demonstrates that this method enables quantum-enhanced precision in certain scenarios. Furthermore, numerical simulations demonstrate that our method surpasses virtual state purification in both single- and multi-parameter estimation tasks. The significant improvements in precision across diverse settings underscore the robustness and practicality of our method for real-world applications.
Quantum metrology with entangled resources aims to achieve sensitivity beyond the standard quantum limit by harnessing quantum effects even in the presence of environmental noise. So far, sensitivity has been mainly discussed from the viewpoint of reducing statistical errors under the assumption of perfect knowledge of a noise model. However, we cannot always obtain complete information about a noise model due to coherence time fluctuations, which are frequently observed in experiments. Such unknown fluctuating noise leads to systematic errors and nullifies the quantum advantages. Here, we propose an error-mitigated quantum metrology that can filter out unknown fluctuating noise with the aid of purification-based quantum error mitigation. We demonstrate that our protocol mitigates systematic errors and recovers superclassical scaling in a practical situation with time-inhomogeneous bias-inducing noise. Our result is the first demonstration to reveal the usefulness of purification-based error mitigation for unknown fluctuating noise, thus paving the way not only for practical quantum metrology but also for quantum computation affected by such noise.
Noise is the main source that hinders us from fully exploiting quantum advantages in various quantum informational tasks. However, characterizing and calibrating the effect of noise is not always feasible in practice. Especially for quantum parameter estimation, an estimator constructed without precise knowledge of noise entails an inevitable bias. Recently, virtual purification-based error mitigation (VPEM) has been proposed to apply for quantum metrology to reduce such a bias occurring from unknown noise. While it was demonstrated to work for particular cases, whether VPEM always reduces a bias for general estimation schemes is unclear. For more general applications of VPEM to quantum metrology, we study factors determining whether VPEM can reduce the bias. We find that the closeness between the dominant eigenvector of a noisy state and the ideal quantum probe (without noise) with respect to an observable determines the reducible amount of bias by VPEM. Next, we show that one should carefully choose the reference point of the target parameter, which gives a smaller bias than others because the bias depends on the reference point. Otherwise, even if the dominant eigenvector and the ideal quantum probe are close, the bias of the mitigated case could be larger than the nonmitigated one. Finally, we analyze the error mitigation for a phase estimation scheme under various noises. Based on our analysis, we predict whether VPEM can effectively reduce a bias and numerically verify our results.
Quantum metrology, which addresses parameter estimation in quantum systems, has broad applications across science and technology. Conventional metrology protocols for multi-qubit states in the multi-parameter regime typically require highly complex quantum measurements, leading to substantial quantum-resource costs. In this work, we introduce a family of metrology protocols that use only few-qubit measurements, thereby significantly reducing the required resources. For arbitrary pure states, one of our protocols approaches the quantum Cramér-Rao bound up to an overhead in sample complexity that scales linearly with the number of qubits, irrespective of the number of parameters to be estimated. For typical Haar-random states, this overhead can be reduced to a constant. Our results build on recent advances in quantum state certification protocols with few-qubit measurements: we establish a universal connection between certification and metrology in which the precision of the certification protocol determines the metrological overhead. We also illustrate our approach through an example of Hamiltonian estimation from ground states.
Quantum resources enable one to achieve quantum-enhanced estimation sensitivity beyond its classical counterpart. Many studies mainly focus on reducing statistical error, under the assumption that one can always set an unbiased estimator. However, setting an unbiased estimator is not always feasible, especially when one cannot fully characterize noise. Such incomplete noise characterization induces a bias and eventually makes it impossible to attain the enhanced-estimation. In this work, we explore two systematic approaches; quantum error correction (QEC) and the virtual purification (VP) to reduce the bias, and compare their performance. First, we show that when the noise is indistinguishable from the signal, QEC cannot reduce the bias since it is impossible to construct a QEC code that corrects the noise while preserving the signal. We then show that VP can mitigate indistinguishable error that eventually enable a more accurate estimation compared to QEC. Our findings reveal that VP offers a robust alternative to QEC in scenarios where indistinguishable errors pose significant challenges. We then demonstrate that VP with a stabilizer state probe can efficiently suppress the bias under local depolarizing noise, thereby yielding a significant improvement in estimation performance compared to the QEC-based approach.
Abstract Quantum computing has made remarkable strides in recent years, as demonstrated by quantum supremacy experiments and the realization of high-fidelity, fault-tolerant gates. However, a major obstacle persists: practical real-world applications remain scarce, largely due to the inefficiency of loading classical data into quantum processors. Here, we propose an alternative strategy that shifts the focus from classical data encoding to directly processing quantum data. We target quantum metrology, a practical quantum technology whose precision is often constrained by realistic noise. We develop an experimentally feasible scheme in which a quantum computer optimizes information acquired from quantum metrology, thereby enhancing performance in noisy quantum metrology tasks and overcoming the classical-data-loading bottleneck. We demonstrate this approach through experimental implementation with nitrogen-vacancy centers in diamond and numerical simulations using models of distributed superconducting quantum processors. Our results show that this method improves the accuracy of sensing estimates and significantly boosts sensitivity, as quantified by the quantum Fisher information, thus offering a new pathway to harness near-term quantum computers for realistic quantum metrology.