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  • Open Access

Quantum Error-Corrected Non-Markovian Metrology

Zachary Mann1,2,*, Ningping Cao3,1,4, Raymond Laflamme1,2,3, and Sisi Zhou3,2,1,5,†

  • *Contact author: zmann@uwaterloo.ca
  • †Contact author: sisi.zhou26@gmail.com
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PRX Quantum 6, 030321 – Published 5 August, 2025

DOI: https://doi.org/10.1103/wfyl-wtz3

Abstract

Quantum metrology aims to maximize measurement precision on quantum systems, with a wide range of applications in quantum sensing. Achieving the Heisenberg limit (HL)—the fundamental precision bound set by quantum mechanics—is often hindered by noise-induced decoherence, which typically reduces achievable precision to the standard quantum limit (SQL). While quantum error correction (QEC) can recover the HL under Markovian noise, its applicability to non-Markovian noise remains less explored. In this work, we analyze a hidden Markov model (HMM) in which a quantum probe, coupled to an inaccessible environment, undergoes joint evolution described by Lindbladian dynamics, with the inaccessible degrees of freedom serving as a memory. We derive generalized Knill-Laflamme conditions for the HMM and establish three types of sufficient conditions for achieving the HL under non-Markovian noise using QEC. Additionally, we demonstrate the attainability of the SQL when these sufficient conditions are violated, by analytical solutions for special cases and numerical methods for general scenarios. Our results not only extend prior QEC frameworks for metrology but also provide new insights into precision limits under realistic noise conditions.

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Popular Summary

Quantum metrology is the study of precision measurement in quantum mechanics. By leveraging quantum effects, such as entanglement, quantum sensors can achieve measurement precision that is inaccessible to classical methods. In practice, however, noise in the quantum system can completely erase this advantage. It has been proposed that this issue be remedied using quantum error correction, which is a protocol that protects quantum information against noise by building redundancy into the state. Protocols have been proposed for error-corrected quantum metrology under Markovian noise—noise that is not temporally correlated. However, generic noise need not be Markovian. In our work, we bridge this gap and study error-corrected quantum metrology under non-Markovian noise.

For a generic temporally correlated noise model, we derive sufficient conditions for achieving the Heisenberg limit, the optimal precision allowed by quantum mechanics. When the derived conditions are satisfied, quantum error correction eliminates the noise in the system. When the conditions fail, the Heisenberg limit may still be achievable because of memory effects from the environment. This is not possible in the Markovian case. Finally, when the Heisenberg limit is inaccessible, we study how the memory effects in the noise impact the standard measurement strategy.

Our work presents a framework for understanding error-corrected quantum metrology in more realistic noise settings. Future directions include considering noisy control operations and auxiliary systems to better reflect the experimental reality. Further, the problem of deriving general sufficient and necessary conditions for Heisenberg scaling remains open.

Article Text

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PRX Quantum 6, 030321– Published 5 August, 2025
  • Received 2 April 2025
  • Revised 10 June 2025
  • Accepted 1 July 2025
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