Quantum Error Correction for Metrology

We propose and analyze a new approach based on quantum error correction (QEC) to improve quantum metrology in the presence of noise. We identify the conditions under which QEC allows one to improve the signal-to-noise ratio in quantum-limited measurements, and we demonstrate that it enables, in certain situations, Heisenberg-limited sensitivity. We discuss specific applications to nanoscale sensing using nitrogen-vacancy centers in diamond in which QEC can significantly improve the measurement sensitivity and bandwidth under realistic experimental conditions.

Figure 1

Circuit model of the QEC for the model described by Eq. (4). The code state is prepared by application of a Hadamard ($H$) and CNOT gate. After each segment of free evolution of duration $\alpha =T/r$, the QEC operation $\mathcal{R}$ is applied. After the final decoding and measurement ($D$), the effective error rate has been reduced by a factor $T/\alpha$.

Figure 2

Normalized estimation error $\delta \omega \sqrt{\tau }$ for the model of transversal noise (see text), for sufficiently large $\tau$. In the standard approach (dashed-dotted line) the interrogation time has an optimal value $T\approx {\gamma }^{-1}$, limiting the achievable sensitivity. Ideally, QEC (solid line) can restore the noise-free scaling (dashed line) ($\propto 1/\sqrt{T}$) even for relatively small QEC repetition rates $1/\alpha =\gamma$. The dotted lines show the achievable sensitivity in the presence of a small parallel noise component ${\gamma }_{\parallel }={10}^{-3}\gamma$, and a probability $p_{\text{error}}={10}^{-3}$ that the QEC operation fails.