Abstract
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.
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Data availability
The datasets generated and/or analyzed during the current study are not publicly available due to ongoing analyses and the inclusion of unpublished follow-up work, but are available from the corresponding author on reasonable request.
Code availability
The codes generated and/or analyzed during the current study are not publicly available due to ongoing analyses and the inclusion of unpublished follow-up work, but are available from the corresponding author on reasonable request.
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Acknowledgements
We acknowledge useful discussions with Seok-Hyung Lee, Seongjoo Cho, and Kento Tsubouchi. This research was funded by National Research Foundation of Korea (RS-2022-NR068812). This research was funded by National Research Foundation of Korea (RS-2022-NR068812) and Institute of Information \& Communications Technology Planning \& Evaluation (RS-2025-02263264). H.K. was supported by the IITP (RS-2025-02263264, RS-2025-25464252, RS-2024-00437191), the Education and Training Program of the Quantum Information Research Support Center (2021M3H3A1036573), and the NRF (RS-2025-25464492, RS-2024-00442710) funded by the Ministry of Science and ICT (MSIT), Korea. H.J. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2024-00413957, RS-2024-00438415, RS-2023-NR076733), the Institute of Information \& Communications Technology Planning \& Evaluation (IITP) grant funded by the Korea government (MSIT) (IITP-2026-RS-2020-II201606, IITP-2026-RS-2024-00437191, and RS-2025-02219034), and the Institute of Applied Physics at Seoul National University. L.J. acknowledges support from the ARO(W911NF-23-1-0077), ARO MURI (W911NF-21-1-0325), AFOSR MURI (FA9550-19-1-0399, FA9550-21-1-0209, FA9550-23-1-0338), DARPA (HR0011-24-9-0359, HR0011-24-9-0361), NSF (OMA-1936118, ERC-1941583, OMA-2137642, OSI-2326767, CCF-2312755), NTT Research, and the Packard Foundation (2020-71479). Y.L. and C.O. acknowledge support by Quantum Technology R\&D Leading Program~(Quantum Computing) (RS-2024-00431768) through the National Research Foundation of Korea~(NRF) funded by the Korean government (Ministry of Science and ICT~(MSIT)) and the Institute of Information \& Communications Technology Planning \& Evaluation (IITP) Grants funded by the Korea government (MSIT) (No. IITP-2025-RS-2025-02283189). Y.L. acknowledges Institute of Information \& Communications Technology Planning \& Evaluation (IITP) grant funded by the Korea government (MSIT) (RS-2024-00437284, No. 2022-0-00463) and National Research Foundation of Korea (2023M3K5A109480511, RS-2023-NR119931). C.O. was supported by the National Research Foundation of Korea Grants (No. RS-2025-00515456) funded by the Korean government (Ministry of Science and ICT (MSIT)) and the Institute of Information \& Communications Technology Planning \& Evaluation (IITP) Grants funded by the Korea government (MSIT) (No. IITP-2025-RS-2025-02263264 and IITP-2025-RS-2025-25464990).
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H.K. conceived the main idea of combining virtual purification and quantum error correction in quantum metrology, performed the theoretical analysis and numerical simulations, prepared all figures, and wrote the manuscript. C.O., Y.L., and H.J. contributed to the analytical derivations, assisted in developing the theoretical framework of virtual purification for metrological estimation, and proposed potential applications. S.-W.L. supervised the project, provided critical insights into quantum error correction and quantum metrology, and revised the manuscript. L.J. jointly supervised the project, contributed to theoretical discussions, and provided overall guidance and feedback on the manuscript.
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Kwon, H., Oh, C., Lim, Y. et al. Virtual purification complements quantum error correction in quantum metrology. npj Quantum Inf (2026). https://doi.org/10.1038/s41534-026-01231-0
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DOI: https://doi.org/10.1038/s41534-026-01231-0
