Control-enhanced quantum metrology under Markovian noise

Quantum metrology is supposed to significantly improve the precision of parameter estimation by utilizing suitable quantum resources. However, the predicted precision can be severely distorted by realistic noises. Here, we propose a control-enhanced quantum metrology scheme to defend against these noises to improve the metrology performance. Our scheme can automatically alter the parameter-encoding dynamics with adjustable controls, thus leading to optimal resultant states that are less sensitive to the noises under consideration. As a demonstration, we numerically apply it to the problem of frequency estimation under several typical Markovian noise channels. By comparing our control-enhanced scheme with the standard scheme and the ancilla-assisted scheme, we show that our scheme performs better and can improve the estimation precision up to around one order of magnitude. Furthermore, we conduct a proof-of-principle experiment in a nuclear magnetic resonance system to verify the effectiveness of the proposed scheme. The research here is helpful for current quantum platforms to harness the power of quantum metrology in realistic noise environments.

Figure 1

Comparison of three quantum metrology schemes. (a) Standard scheme. A theoretical optimal probe state ${\rho }_0$ (usually the maximally entangled state) interacts with the encoding dynamics ${\varepsilon }_x$ over a time period $T$, which is divided into $K=T/\mathrm{\Delta }t$ equal parts ${\varepsilon }_x^k\left(\mathrm{\Delta }t\right)$, with $k=1,2,\cdots ,K$. Suitable measurements are then performed on the resultant state ${\rho }_x$ to extract the parameter information. (b) Ancilla-assisted scheme. The system and the ancillary qubit are first jointly prepared at the maximally entangled state, and then the system solely interacts with the sliced encoding dynamics ${\varepsilon }_x^k\left(\mathrm{\Delta }t\right)$; finally, a joint measurement is performed. (c) Control-enhanced scheme. The system is started from an arbitrary initial probe state ${\rho }_0$. The encoding dynamics is engineered with adjustable controls, marked as ${\varepsilon }_{x,C}^k\left(\mathrm{\Delta }t\right)$ for each time length $\mathrm{\Delta }t$. The resultant state ${\rho }_x$ is then evaluated by suitable measurements, and the controls are iteratively refreshed by a suitable optimization algorithm. This procedure automatically alters the encoding dynamics to engineer the initial probe to some optimal one that is insensitive to the noises under consideration.

Figure 2

Numerical comparison of three quantum metrology schemes on frequency estimation under parallel-dephasing noise. (a) and (c) show the QFI and the sensitivity vs the encoding time for single-qubit dephasing noise, where we set $\gamma =1/T_2=10{\text{s}}^{-1}$, and ${\omega }_0=2\pi$. (b) and (d) plot the QFI and the sensitivity vs the encoding time under two-qubit uncorrelated dephasing noise, where we set ${\gamma }^n=1/T_2^n=10{\text{s}}^{-1}$, with $n=1,2$ and ${\omega }_0=2\pi$. In all the plots, the curve of the standard scheme and that of the ancilla-assisted scheme coincide with each other.

Figure 3

Numerical comparison of four quantum metrology schemes on frequency estimation under transverse-dephasing noise. (a) and (c) show the QFI and the sensitivity vs the encoding time for $\gamma =0.1{\text{s}}^{-1}$ and ${\omega }_0=2\pi$. (b) and (d) demonstrate similar cases, but with $\gamma =10{\text{s}}^{-1}$.

Figure 4

Numerical comparison of three quantum metrology schemes on frequency estimation under amplitude-damping noise. (a) shows the QFI vs the encoding time. (b) plots the corresponding sensitivity vs the encoding time. In the simulations, we set $\gamma =2/T_1=0.2{\text{s}}^{-1}$ and ${\omega }_0=2\pi$.

Figure 5

Experimental comparison of the standard scheme and our control-enhanced scheme on frequency estimation in the NMR system. (a) The top and the bottom panels show the schematic diagrams for the standard scheme and the control-enhanced scheme, respectively. The nuclear spin ${}^1\mathrm{H}$ is decoupled, and the spin ${}^{13}\mathrm{C}$ is initialized at $|+\rangle$ for the standard scheme and a random state for the control-enhanced scheme. The encoding process is realized by freely evolving the system with an offset ${\omega }_0$ in the presence of pure parallel-dephasing noise, marked as ${\varepsilon }_{{\omega }_0}^k,k=1,2,\cdots ,5$. We set $\omega =60\times 2\pi$, and the measured coherence times is $T_2=0.149\text{s}$. In our control-enhanced scheme, the encoding dynamics is engineered by additional control for resisting the noises, marked as ${\varepsilon }_{{\omega }_0,C},k=1,2,\cdots ,5$. The final state ${\rho }_{{\omega }_0}$ and its perturbed state ${\rho }_{{\omega }_0+\delta {\omega }_0}$, with $\delta \omega =2\pi$, are measured from experiments for calculating the QFI. (b) shows the theoretically calculated QFI $F_Q^{\text{theo}}$ and the experimentally measured QFI $F_Q^{\text{exp}}$ for the standard scheme and the control-enhanced scheme. We also demonstrate the tomography results of the initial states and the final states (exact states and perturbed states) using our control-enhanced scheme.