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Register a new userPractical Limits of Error Correction for Quantum Metrology
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Noise is the greatest obstacle in quantum metrology that limits it achievable precision and sensitivity. There are many techniques to mitigate the effect of noise, but this can never be done completely. One commonly proposed technique is to repeatedly apply quantum error correction. Unfortunately, the required repetition frequency needed to recover the Heisenberg limit is unachievable with the existing quantum technologies. In this article we explore the discrete application of quantum error correction with current technological limitations in mind. We establish that quantum error correction can be beneficial and highlight the factors which need to be improved so one can reliably reach the Heisenberg limit level precision.
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For a generic set of Markovian noise models, the estimation precision of a parameter associated with the Hamiltonian is limited by the $1/sqrt{t}$ scaling where $t$ is the total probing time, in which case the maximal possible quantum improvement in the asymptotic limit of large $t$ is restricted to a constant factor. However, situations arise where the constant factor improvement could be significant, yet no effective quantum strategies are known. Here we propose an optimal approximate quantum error correction (AQEC) strategy asymptotically saturating the precision lower bound in the most general adaptive parameter estimation scheme where arbitrary and frequent quantum controls are allowed. We also provide an efficient numerical algorithm finding the optimal code. Finally, we consider highly-biased noise and show that using the optimal AQEC strategy, strong noises are fully corrected, while the estimation precision depends only on the strength of weak noises in the limiting case.
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We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramer-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.
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