Comparative analysis of virtual purification and quantum error correction in quantum sensing

Quantum resources such as entanglement, 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 evaluate 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 both corrects the noise and preserves the signal. We then show that VP can mitigate such indistinguishable errors 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.