Entanglement-Enhanced Quantum Metrology in Colored Noise by Quantum Zeno Effect

In open quantum systems, the precision of metrology inevitably suffers from the noise. In Markovian open quantum dynamics, the precision can not be improved by using entangled probes although the measurement time is effectively shortened. However, it was predicted over one decade ago that in a non-Markovian one, the error can be significantly reduced by the quantum Zeno effect (QZE) [Chin, Huelga, and Plenio, Phys. Rev. Lett. 109, 233601 (2012)]. In this work, we apply a recently developed quantum simulation approach to experimentally verify that entangled probes can improve the precision of metrology by the QZE. Up to $n=7$ qubits, we demonstrate that the precision has been improved by a factor of $n^{1/4}$, which is consistent with the theoretical prediction. Our quantum simulation approach may provide an intriguing platform for experimental verification of various quantum metrology schemes.

Figure 1

(a) Single-qubit Ramsey experiment for quantum metrology under non-Markovian noise. The free precession is governed by non-Markovian dynamics, which can be simulated using the ensemble-averaging technique. This averaging can be visualized in the Bloch sphere: A large number of identical initial probes undergo phase diffusion owing to their different modulated Hamiltonians, leading to an attenuation in amplitude for the overall spin signal. This attenuation gives rise to the error in quantum-metrology experiments. (b) Quantum circuits for multiqubit Ramsey experiment under non-Markovian noise with unentangled and entangled probes, respectively. The evolution $U$ is the simulation of non-Markovian dynamics. For the case of entangled probes, the readout is typically some collective measurement [5, 6].

Figure 2

(a) Molecular structure and relevant parameters of the seven-qubit NMR processor. Chemical shifts (diagonal, Hz), scalar coupling strengths (off-diagonal, Hz), and relaxation times ($T_1$ and $T_2$) are all listed in the table. (b) Pulse sequence for the 7-qubit Ramsey experiment using the entangled probe under non-Markovian noise. To simulate the non-Markovian dynamics, the rotating angles ${\theta }_m^j$ are carefully designed by ensemble-averaging modulation. A step-by-step description of the sequence can be found in the SM [43].

Figure 3

(a) Dynamics of population $P(t)$ for the unentangled (top panel) and entangled (bottom panel) probes. Blue curves are obtained by numerically fitting the experimental data (yellow triangles) using Eq. (4), and red lines are envelopes with the form $(1-e^{-\mathrm{\Gamma }(t)})/2$, which depicts the time-dependent decoherence factor $\mathrm{\Gamma }(t)$. The star symbol represents the optimal measurement point determined by the experiment. (b) Standard deviation $\delta {\omega }_0$ of the measured frequency with respect to the variation of $t$. Dashed (solid) curves are fitting results corresponding to the unentangled (entangled) probes for different numbers of qubits. Triangles and dots are the optimal measurement points. The scaling behavior of these optimal times when changing $n$ is shown in the inset.

Figure 4

Experimental results of the ratio $r=\delta {\omega }_0{|}_u/\delta {\omega }_0{|}_e$ (dots) and the minimum standard deviation $\delta {\omega }_0{|}_e$ (triangles) for different numbers of qubits $n$. Under the non-Markovian noise, the sensitivity is enhanced by $r=n^{1/4}$ (solid red curve) with the entangled probes, reaching the QZE limit; $\delta {\omega }_0{|}_e$ scales as $n^{-3/4}$ (solid blue curve). In the absence of noise, the sensitivity enhancement is governed by the Heisenberg limit $r=n^{1/2}$ (dashed red curve), and $\delta {\omega }_0{|}_e$ scales as $n^{-1}$ (dashed blue curve).