Fluctuation-enhanced quantum metrology

The fluctuation of a field, which can induce the dephasing on the system, is one of the most common noises in quantum dynamics and widely regarded as detrimental in quantum metrology. Contrary to this belief, we show that while the fluctuation may be harmful for the estimation of the magnitude of a field, it can actually be helpful for the estimation of various other parameters, including the parameters that represent the direction of a field and the frequency of a rotating field. In particular we show that the precision achievable at the presence of the fluctuation can even surpass the highest precision achievable under the ideal unitary dynamics, which has been widely believed as the ultimate limit. We also provide an analysis for general dynamics where the noise and the Hamiltonian can be correlated and present a general formula for the precision achievable with such correlated parametrization. Our study provides a path in quantum metrology that can lead to the higher precision limits beyond what believed to be possible. It can have wide implications in practical applications, such as quantum gyroscope, quantum reference frame alignment etc, where fluctuations are in general unavoidable.

The article is organized as following. In Sec. II we give a brief introduction of the essential tools in quantum estimation and introduce the model of a spin interacting with a magnetic field. In Sec.III we study the estimation of the direction of the field and derive the highest precision achievable under the unitary dynamics. In Sec.IV we study the precision that can be achieved at the presence of the fluctuation and show that the precision outperforms the highest value obtained under the unitary dynamics. We then show that the fluctuation can also improve the precision of estimating the frequency of a rotating field in Sec.V. The optimal measurement and numerical simulations are provided in Sec.VI and Sec.VII respectively. In Sec.VIII we study the general Markovian dynamics where the noise and the Hamiltonian are correlated and provide an analytical analysis on the precision achievable in the asymptotic limit. Sec.IX concludes.

We consider the general sequential scheme for quantum metrology, as illustrated in Fig.1(b), which allows adaptive operations during the evolution. The widely used parallel scheme, as shown in Fig.1(a), is a special case of the general sequential scheme with the control operations taken as SWAP operations between the system and different ancillas.

We consider using spins to measure a fluctuating magnetic field where the dynamics can be described as

$$\frac{d|\psi\rangle}{dt}=-i[B+\xi(t)]\sigma_{\vec{n}}|\psi\rangle,$$

(4)

here $\sigma_{\vec{n}}=\vec{n}\cdot\vec{\sigma}$ with $\vec{n}$ denotes the direction of the field with respect to pre-fixed axes, $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ is the vector of Pauli matrices, $\xi(t)$ represents the fluctuation which can either arise from the fluctuation of the magnetic field itself or from the environment. We assume that the fluctuation is Markovian with a white spectrum, $E[\xi(t)]=0$, $E[\xi(t)\xi(\tau)]=\gamma\delta(t-\tau)$, which has the most detrimental effects. For simplicity, we consider a magnetic field in the $XZ$-plane, in which case $\sigma_{\vec{n}(\theta)}=\cos\theta\sigma_{1}+\sin\theta\sigma_{3}$ with $\theta$ representing the direction of the magnetic field in the $XZ$-plane. The estimations of $\theta$ corresponds to the estimation of the direction of the field. It can also corresponds to the orientation of an object in a plane. For example, by attaching a field source (with a known strength) to the object, we can then infer the orientation of the object from the estimation of $\theta$. We are going to show that the estimation of $\theta$ has very different behavior at the presence of the fluctuation, compared with the estimation of $B$.

For the estimation of $\theta$, under the free evolution we have

$\displaystyle\begin{aligned} h_{\theta}&=\int_{0}^{t}U^{\dagger}(\tau)\frac{\partial H}{\partial\theta}U(\tau)d\tau\\ &=\sin(Bt)\vec{n}_{\theta}\cdot\vec{\sigma},\\ \end{aligned}$

(6)

where $\vec{n}_{\theta}=(-\cos Bt\sin\theta,-\sin Bt,\cos Bt\cos\theta)$. The maximal QFI is then $J_{Q}^{\theta}=4\sin^{2}(Bt)$Pang2014PRA ; Jing2015 . Similarly the optimal state can be taken as $|\psi(0)\rangle=\frac{|\lambda_{\max}\rangle+|\lambda_{\min}\rangle}{\sqrt{2}}$ with $|\lambda_{\max/\min}\rangle$ as the eigenvector of $h_{\theta}$. This state depends on $\theta$ thus can only be prepared adaptively according to $h_{\hat{\theta}}$ with $\hat{\theta}$ as the estimated value obtained from accumulated data. But if an ancillary spin is available, the optimal probe state can be taken as $|\psi(0)\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}$, no adaptation is then needed. We note that as $U(\tau)$ and $\frac{\partial H}{\partial\theta}$ do not commute with each other, $h_{\theta}$ does not increase linearly with $t$ which makes $J_{Q}^{\theta}$ smaller. However, if controls can be employed, proper controls can make $U(\tau)$ commute with $\frac{\partial H}{\partial\theta}$. One choice of such control is to reverse the dynamics and make $U(\tau)=I$yuan2015optimal ; Wiebe2014 . Under such control $h_{\theta}=t\frac{\partial H}{\partial\theta}=Bt(-\sin\theta\sigma_{1}+\cos\theta\sigma_{3})$, the maximal variance of $h_{\theta}$ can reach $B^{2}t^{2}$. Such control typically depends on $\theta$ and can only be implemented adaptively, for example, it can be realized by adding a control Hamiltonian $H_{c}=-B\sigma_{\vec{n}(\hat{\theta})}$ with $\hat{\theta}$ as the estimated value, which converges to the optimal control in the asymptotic limit(see Supplemental Material). The highest QFI for the estimation of $\theta$ achievable under the unitary dynamics with the optimal control strategy is then $J_{Q}^{\theta}=4B^{2}t^{2}$, which can be achieved when $\hat{\theta}\rightarrow\theta$ in the asymptotic limit.

At first glance, this does not seem possible. As to design controls that make $U(\tau)=I$, we need to track each trajectory precisely in order to reverse it. This requires a precise knowledge of the fluctuation along each trajectory, which is typically beyond the reach. Surprisingly we show that this improvement actually can be achieved. Before we present an explicit control protocol, we first examine the precision that can be achieved under the free evolution at the presence of the fluctuation, i.e., we first compute the precision achievable under the sequential scheme without adding any control operations.

The dynamics of the spin at the presence of the fluctuation can be equivalently described by the master equation

$$\frac{d\rho}{dt}=-i[B\sigma_{\vec{n}(\theta)},\rho]+\gamma[\sigma_{\vec{n}(\theta)}\rho\sigma_{\vec{n}(\theta)}-\rho].$$

(7)

From the master equation, we can immediately see the difference between the estimation of $B$ and the estimation of $\theta$, and understand why the fluctuation can not improve the estimation of $B$. As the parameter $B$ is only encoded in the Hamiltonian, the fluctuation can only affects the precision of $B$ negatively. This is also implicitly assumed in many schemes of quantum metrology, which leads to the belief that the unitary dynamics sets the ultimate limit on the achievable precisions. This assumption, however, does not hold in general. In particular we can see that the parameter $\theta$ is encoded in both the Hamiltonian and the noise. In such correlated parametrization both the Hamiltonian and the noise can contribute to the precision, which makes it possible to achieve higher precision beyond what can be reached under the unitary dynamics. As shown in Fig.2, such correlated parametrization differs from the Hamiltonian parameter estimation and the noise spectroscopySinitsyn_2016 ; Norris2016 studied in literature, which either assume the parameter is only in the Hamiltonian or assume the parameter is only in the noise. The correlated parametrization is also different from the environment-assisted quantum metrology studied previouslyGoldstein2011 ; Cappellaro2012 which utilizes additional spins in the environment as part of the system with limited control.

We now compute the precision that can be achieved under the free evolution. By using an ancilla and preparing the initial state as $|\psi(0)\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}$(for example in NV-center the nuclear spin can be taken as the ancilla as its evolution is much slower than the electron spin, the evolution on the nuclear spin thus can be approximated as Identity), it is straightforward to obtain the QFI as(see Sec. II of the Supplemental material for detail)

$$J_{Q}^{\theta}(t)=2[1-e^{-2\gamma t}\cos(2Bt)].$$

(8)

By contrast, with the maximally entangled state the QFI under the free unitary dynamics, which corresponds to $\gamma=0$, is

$$J_{QU}^{\theta}(t)=2[1-\cos(2Bt)]=4\sin^{2}Bt.$$

(9)

Comparing the two QFI, we can see that the fluctuation displays two competing effects. On the one hand, it provides an extra channel for parametrization which can help improving the precision. This is manifested in the short time regime with $J_{Q}^{\theta}(t)-J_{QU}^{\theta}(t)\approx 4\gamma t>0$ for $t\ll 1$, exactly the improvement predicted from the heuristic argument. On the other hand, the fluctuation mixes the state which can decrease the precision. This is manifested in the long time regime where under the free evolution the noisy dynamics can perform worse than the unitary dynamics. We now design explicit control strategies that can get rid of the negative effect of the fluctuation but maintain its positive contribution, and show that the improvement from the fluctuation can be kept in the long time regime as well.

We use the quantum error correction(QEC) as the control strategy. At first sight, QEC may seem incompatible with the idea of using the fluctuation to improve the precision. How can the noisy dynamics outperform the unitary ones if the noise is being corrected? The key again lies at the fact that the noise changes with the parameter in the correlated parametrization. In the conventional schemes, the noise is assumed to be independent of the parameter, the QEC corrects the noise regardless of the value of the parameter. After applying the QEC, the dynamics become effectively unitary the noise thus can not contribute to the precisionPlenio2000 ; Dur2014 ; Arrad2014 ; Kessler2014 ; Ozeri2013 ; Unden2016 ; Sekatski2017 ; Rafal2017 ; Zhou2018 ; Layden2018 ; Layden2019 ; Zhou2019 ; Layden2020 . For the correlated parameterization, however, the noise changes with the parameter. A fixed QEC only corrects the noise corresponds to a specific value of the parameter. When the parameter changes, the QEC needs to be adjusted, i.e., the QEC needs to be designed according to the specific value of the parameter and such adaptive QEC only corrects the noise completely at the specific value of the parameter. The dynamics remains noisy when the parameter takes other values. From Eq.(1), it can be seen that the QFI is determined by the distance between the neighboring states, evolved under the neighboring dynamics. If we denote the noisy dynamics as $\frac{d\rho}{dt}=L_{x}(\rho)$ with $L_{x}$ as the super-operator that governs the dynamics, the neighboring dynamics is then $\frac{d\rho}{dt}=L_{x+dx}(\rho)$. Under the correlated parametrization the noises in $L_{x}$ and $L_{x+dx}$ are different since the noises change with the parameter. The adaptive QEC corrects the noise in $L_{x}$ which makes the corrected dynamics at $x$, denoted as $\frac{d\rho}{dt}=L_{x}^{C}(\rho)$ (here $C$ means with the adaptive QEC), effectively unitary. However, it does not correct the noise in $L_{x+dx}$ completely, i.e., the dynamics with QEC at $x+dx$, denoted as $\frac{d\rho}{dt}=L_{x+dx}^{C}(\rho)$ (note that $L_{x+dx}^{C}$ is obtained under the same QEC designed according to the estimated value $\hat{x}$), is still noisy and the noise in it can make the dynamics more different from $L_{x}^{C}$ than a purely unitary evolution. Intuitively, the adaptive QEC corrects the part of the noise in $L_{x+dx}$ that is common to the noise in $L_{x}$, but keeps the different part of the noise in $L_{x+dx}$. By eliminating the common part of the noises, the adaptive QEC reduces the negative effect of the noise, as common noises decreases the distance between the neighboring states. And by keeping the difference of the noise, the adaptive QEC preserves the positive effect of the noise, as the difference of the noise can make the neighboring dynamics more divergent. Take the qubits for example, the fidelity between the neighboring states can be decomposed into two parts asHUBNER1992239

$$F(\rho_{x},\rho_{x+dx})=Tr(\rho_{x}\rho_{x+dx})+2\sqrt{det(\rho_{x})det(\rho_{x+dx})}.$$

(10)

In the correlated parametrization, different noises in the neighboring dynamics make $\rho_{x}$ and $\rho_{x+dx}$ more different which has the effect of reducing the first term thus increase the Bures distance $d_{Bures}^{2}=2-2F(\rho_{x},\rho_{x+dx})$. However, if there are no control strategies, the states will become mixed and the second term in Eq.(10) increases which can override the positive contributions of the noise. By applying the adaptive QEC which corrects the common part of the noises in $\rho_{x}$ and $\rho_{x+dx}$, we are able to keep the positive contribution of the noise while reducing the negative effect. In the asymptotic limit when the estimated value $\hat{x}\rightarrow x$, under the adaptive QEC that is designed according to $\hat{x}$, $\rho_{x}$ will become a pure state, the second term in Eq.(10) vanishes, i.e., the negative effect is eliminated. But the positive contribution remains as the noise in $\rho_{x+dx}$ is not completely eliminated. Only the part that is common with the noise in $\rho_{x}$ is eliminated but the part that is different remains which helps reduce the first term in Eq.(10) as it makes $\rho_{x+dx}$ more different from $\rho_{x}$ (similar effects holds in higher dimensional space although a general formula to decompose the fidelity this way has not been known). Thus the noise in the correlated parametrization can make the neighboring states further apart than the states from purely unitary dynamics, as illustrated in Fig.3. We emphasize that the positive contribution comes from the fact that the noise changes with the parameter in the correlated parametrization, the adaptive QEC is just a tool to keep the positive contribution of the noise from being smeared by the negative mixing effect. The adaptive QEC itself is not the source of the outperformance over the unitary dynamics. Without the correlated parametrization, the QEC can at most recover the performance of the unitary evolution as in previous studies with QECPlenio2000 ; Dur2014 ; Arrad2014 ; Kessler2014 ; Ozeri2013 ; Unden2016 ; Sekatski2017 ; Rafal2017 ; Zhou2018 ; Layden2018 ; Layden2019 ; Zhou2019 ; Layden2020 .

We now provide an explicit protocol with the adaptive QEC. With an acillary spin (unless specified, an operator, $A$, will be understood as $A\otimes I$ with $A$ on the probe spin and Identity on the acillary spin), we choose the code space as $\{|C_{0}\rangle=|+_{\hat{\theta}}\rangle|+_{\hat{\theta}}\rangle,|C_{1}\rangle=|-_{\hat{\theta}}\rangle|-_{\hat{\theta}}\rangle\}$, where $|+_{\hat{\theta}}\rangle=-\cos\frac{\hat{\theta}}{2}|0\rangle+\sin\frac{\hat{\theta}}{2}|1\rangle$, $|-_{\hat{\theta}}\rangle=\sin\frac{\hat{\theta}}{2}|0\rangle+\cos\frac{\hat{\theta}}{2}|1\rangle$ are the eigenvectors of $\frac{\partial H(\hat{\theta})}{\partial\theta}=B(-\sin\hat{\theta}\sigma_{1}+\cos\hat{\theta}\sigma_{3})$. Here $\hat{\theta}$ is the estimation of $\theta$ obtained from the prior knowledge and the accumulated measurement data. Together with $|C_{2}\rangle=|-_{\hat{\theta}}\rangle|+_{\hat{\theta}}\rangle$ and $|C_{3}\rangle=|+_{\hat{\theta}}\rangle|-_{\hat{\theta}}\rangle$, they form a basis for the space of two spins. It is easy to check that $\sigma_{\vec{n}(\hat{\theta})}|C_{0}\rangle=|C_{2}\rangle$ and $\sigma_{\vec{n}(\hat{\theta})}|C_{1}\rangle=|C_{3}\rangle$. If we prepare the initial probe state as $\frac{|C_{0}\rangle+|C_{1}\rangle}{\sqrt{2}}$, which is just the maximally entangled state $\frac{|00\rangle+|11\rangle}{\sqrt{2}}$, both the Hamiltonian and the noisy operator will drive the state out of the code space, however, by applying the QEC, which steers the state back to the code space, we can make the evolution as the Identity in the asymptotic limit when $\hat{\theta}$ converges to $\theta$.

Specifically, suppose $\rho_{C}(0)$ is in the code space, then after a small period $dt$, the state evolves to $\rho(dt)=$

$$\rho_{C}(0)-i[B\sigma_{\vec{n}(\theta)},\rho_{C}(0)]dt+\gamma(\sigma_{\vec{n}(\theta)}\rho_{C}(0)\sigma_{\vec{n}(\theta)}-\rho_{C}(0))dt,$$

(11)

which can be out of the code space. A recovery operation, which consists of two Kraus operators, $\{\Pi_{C},\Pi_{C}\sigma_{\vec{n}(\hat{\theta})}\}$, is then applied on the state, here $\Pi_{C}=|C_{0}\rangle\langle C_{0}|+|C_{1}\rangle\langle C_{1}|$ and $\sigma_{\vec{n}(\hat{\theta})}=\cos\hat{\theta}\sigma_{1}+\sin\hat{\theta}\sigma_{3}$. For any $\rho_{C}(0)$ in the code space, we have $\Pi_{C}\rho_{C}(0)\Pi_{C}=\rho_{C}(0)$ and $\Pi_{C}\sigma_{\vec{n}(\hat{\theta})}\rho_{C}(0)=\rho_{C}(0)\sigma_{\vec{n}(\hat{\theta})}\Pi_{C}=0$. The state after the recovery operation can then be obtained as

$\displaystyle\begin{aligned} &\rho_{C}(dt)\\ =&\Pi_{C}\rho(dt)\Pi_{C}+\Pi_{C}\sigma_{\vec{n}(\hat{\theta})}\rho(dt)\sigma_{\vec{n}(\hat{\theta})}\Pi_{C}\\ =&\rho_{C}(0)-i[B\Pi_{C}\sigma_{\vec{n}(\theta)}\Pi_{C},\rho_{C}(0)]\\ &+\gamma[\Pi_{C}\sigma_{\vec{n}(\theta)}\Pi_{C}\rho_{C}(0)\Pi_{C}\sigma_{\vec{n}(\theta)}\Pi_{C}\\ &+\Pi_{C}\sigma_{\vec{n}(\hat{\theta})}\sigma_{\vec{n}(\theta)}\Pi_{C}\rho_{C}(0)\Pi_{C}\sigma_{\vec{n}(\theta)}\sigma_{\vec{n}(\hat{\theta})}\Pi_{C}-\rho_{C}(0)]dt,\end{aligned}$

(12)

where $\Pi_{C}\sigma_{\vec{n}(\theta)}\Pi_{C}=\sin(\theta-\hat{\theta})(|C_{0}\rangle\langle C_{0}|-|C_{1}\rangle\langle C_{1}|)$, $\Pi_{C}\sigma_{\vec{n}(\hat{\theta})}\sigma_{\vec{n}(\theta)}\Pi_{C}=\cos(\theta-\hat{\theta})(|C_{0}\rangle\langle C_{0}|+|C_{1}\rangle\langle C_{1}|)$. Denote $\sigma_{3}^{C}=|C_{0}\rangle\langle C_{0}|-|C_{1}\rangle\langle C_{1}|$, we then obtain the dynamics of the recovered state as

$\displaystyle\begin{aligned} \frac{d\rho_{C}}{dt}=&-i[B\sin(\theta-\hat{\theta})\sigma_{3}^{C},\rho_{C}]\\ &+\gamma\sin^{2}(\theta-\hat{\theta})[\sigma_{3}^{C}\rho_{C}\sigma_{3}^{C}-\rho_{C}].\end{aligned}$

(13)

If the initial state is taken as $\frac{|C_{0}\rangle+|C_{1}\rangle}{\sqrt{2}}$, the final state can be easily obtained as $\rho_{C}(t)=\frac{1}{2}(|C_{0}\rangle\langle C_{0}|+|C_{1}\rangle\langle C_{1}|+e^{-gt}|C_{0}\rangle\langle C_{1}|+e^{-g^{*}t}|C_{1}\rangle\langle C_{0}|)$ with $g=2iB\sin(\theta-\hat{\theta})+2\gamma\sin^{2}(\theta-\hat{\theta})$. The QFI is $J_{Q}^{\theta}(t)=$

			(14)
	$\displaystyle 4t^{2}\cos^{2}(\theta-\hat{\theta})\frac{B^{2}(1-e^{-4\gamma t\sin^{2}(\theta-\hat{\theta})})+4\gamma^{2}\sin^{2}(\theta-\hat{\theta})}{e^{4\gamma t\sin^{2}(\theta-\hat{\theta})}-1},$

which, expanded to the second order of $d\theta=\hat{\theta}-\theta$, is

	$\displaystyle J_{Q}^{\theta}(t)=$	$\displaystyle 4B^{2}t^{2}+4\gamma t$		(15)
		$\displaystyle-4B^{2}d\theta^{2}t^{2}(1+2\frac{\gamma^{2}}{B^{2}}+\frac{\gamma}{B^{2}t}+4\gamma t)+o(d\theta^{4}).$

It is now easy to see that the QFI achieves the maximal value, $4B^{2}t^{2}+4\gamma t$, at the asymptotic limit when $d\theta=0$. This is also exactly the value predicted from the heuristic argumentnote2 . In the finite regime where $\hat{\theta}$ is not equal to $\theta$, the $t^{2}$ scaling is maintained within the time order of $(Bd\theta)^{-1}$. We note that this is not much of a restriction, the evolution time should be restricted to the same order even under the unitary evolution in order to avoid the phase ambiguities(see Supplement Material). As shown in Fig.5(a), in the finite regime under the adaptive QEC, with quite an amount of estimation error the QFI surpasses the highest value achievable under the unitary dynamics. And in this case the stronger the fluctuation, the more the improvement. The highest precision achievable under the unitary dynamics thus does not play the role of a fundamental bound.

Consider a fluctuating field that rotates in the $XY$-plane with a frequency $\Omega$ and the dynamics of a spin is then described by

$$\frac{d|\psi\rangle}{dt}=-i[B+\xi(t)]\sigma_{\vec{n}}(\Omega,t)|\psi\rangle,$$

(16)

here $\xi(t)$ is the fluctuation, $\sigma_{\vec{n}}(\Omega,t)=\cos(\Omega t)\sigma_{1}-\sin(\Omega t)\sigma_{2}$ with $\Omega$ as the frequency to be estimated. If the fluctuation has a faster time scale than the rotation, the dynamics can be equivalently described by the master equation

$$\frac{d\rho}{dt}=-i[B\sigma_{\vec{n}}(\Omega,t),\rho]+\gamma[\sigma_{\vec{n}}(\Omega,t)\rho\sigma_{\vec{n}}(\Omega,t)-\rho].$$

(17)

The estimation of the frequency of a rotating field is a fundamental problem in quantum metrology and has been extensively studied Pang2017 ; Schmitt832 ; Boss837 ; Naghiloo2017 ; Glenn2018 . The highest QFI is believed to be achieved under the optimally controlled unitary dynamics, which takes the value $B^{2}t^{4}$ in the asymptotic limitPang2017 . This has been widely regarded as the ultimate limit for the estimation of the frequency. We now show that this can be surpassed at the presence of the fluctuation.

Again we first give a heuristic argument. For each trajectory we have $h_{\Omega}=$

$\displaystyle\begin{aligned} &\int_{0}^{t}U^{\dagger}(\tau)\frac{\partial H}{\partial\Omega}U(\tau)d\tau\\ =&\int_{0}^{t}U^{\dagger}(\tau)\{-\tau[B+\xi(\tau)](\sin\Omega\tau\sigma_{1}+\cos\Omega\tau\sigma_{2})\}U(\tau)d\tau\\ =&\int_{0}^{t}U^{\dagger}(\tau)\{-\tau[B+\xi(\tau)]e^{-i\frac{\Omega}{2}\sigma_{3}}\sigma_{2}e^{i\frac{\Omega}{2}\sigma_{3}}\}U(\tau)d\tau.\end{aligned}$

If we can make $U(\tau)=e^{-i\frac{\Omega}{2}\sigma_{3}}$, then $h_{\Omega}=\int_{0}^{t}-\tau[B+\xi(\tau)]\sigma_{2}d\tau=-[\frac{1}{2}Bt^{2}+\int_{0}^{t}\tau\xi(\tau)d\tau]\sigma_{2}$, the average variance of $h_{\Omega}$ is then given by $E[(\frac{1}{2}Bt^{2}+\int_{0}^{t}\tau\xi(\tau)d\tau)^{2}]=\frac{1}{4}B^{2}t^{4}+\frac{1}{3}\gamma t^{3}$. The QFI can thus potentially reach $B^{2}t^{4}+\frac{4}{3}\gamma t^{3}$, which is higher than the highest value achievable under the unitary dynamics. We show explicitly how this can be achieved with the adaptive QEC.

As the noise in this case not only changes with the parameter, but also changes with time, the code space of the adaptive QEC will also be time-dependent. We choose the basis of the code space at time $t$ as $\{|C_{0}(t)\rangle=|+(\hat{\Omega},t)\rangle|0\rangle,|C_{1}(t)\rangle=|-(\hat{\Omega},t)\rangle|1\rangle\}$, where $|\pm(\hat{\Omega},t)\rangle$ are the eigenvectors of $h_{\hat{\Omega}}(t)=\frac{\partial H(\hat{\Omega},t)}{\partial\Omega}=-Bt(\sin\hat{\Omega}t\sigma_{1}+\cos\hat{\Omega}t\sigma_{2})=-Bte^{-i\hat{\Omega}t\sigma_{3}}\sigma_{2}e^{i\hat{\Omega}t\sigma_{3}}$. Here $\hat{\Omega}$ is the estimated value of $\Omega$ obtained from previous data.

Suppose at time $t$ the probe state, denoted as $\rho_{C}(t)$, is in the code space, then after a period of $dt$, it evolves to

$\displaystyle\begin{aligned} \rho(t+dt)=&\rho_{C}(t)-i[B\sigma_{\vec{n}}(\Omega,t),\rho_{C}(t)]dt\\ &+\gamma[\sigma_{\vec{n}}(\Omega,t)\rho_{C}(t)\sigma_{\vec{n}}(\Omega,t)-\rho_{C}(t)]dt.\end{aligned}$

(18)

We then apply a recovery operation, which consists of two Kraus operators, $\{\Pi_{C}(t),\Pi_{C}(t)\sigma_{\vec{n}}(\hat{\Omega},t)\}$, on the state, where $\Pi_{C}(t)=|C_{0}(t)\rangle\langle C_{0}(t)|+|C_{1}(t)\rangle\langle C_{1}(t)|$ and $\sigma_{\vec{n}}(\hat{\Omega},t)=\cos(\hat{\Omega}t)\sigma_{1}-\sin(\hat{\Omega}t)\sigma_{2}$. The state after the recovery operation can be obtained as

$\displaystyle\begin{aligned} \tilde{\rho}_{C}(t+dt)&=\Pi_{C}(t)\rho(t+dt)\Pi_{C}(t)\\ &+\Pi_{C}(t)\sigma_{\vec{n}}(\hat{\Omega},t)\rho(t+dt)\sigma_{\vec{n}}(\hat{\Omega},t)\Pi_{C}(t)\\ &=\rho_{C}(t)-i[B\sin(\Omega-\hat{\Omega})t\sigma_{3}^{C}(t),\rho_{C}(t)]dt\\ &+\gamma\sin^{2}(\Omega-\hat{\Omega})t[\sigma_{3}^{C}(t)\rho_{C}(t)\sigma_{3}^{C}(t)-\rho_{C}(t)]dt,\end{aligned}$

(19)

where $\sigma_{3}^{C}(t)=|C_{0}(t)\rangle\langle C_{0}(t)|-|C_{1}(t)\rangle\langle C_{1}(t)|$. After the recovery operation, we apply another unitary operation, $\tilde{U}(dt)$, on the state. This unitary operation is to rotate the code space at $t$ to the code space at $t+dt$, i.e., $\tilde{U}(dt)\Pi_{C}(t)\tilde{U}^{\dagger}(dt)=\Pi_{C}(t+dt)$. As $|C_{0}(t)\rangle,|C_{1}(t)\rangle$ are the eigenvectors of $h_{\hat{\Omega}}(t)=-Bte^{-i\hat{\Omega}t\sigma_{3}}\sigma_{2}e^{i\hat{\Omega}t\sigma_{3}}$, we have $|C_{0}(t)\rangle=e^{i\frac{\hat{\Omega}}{2}t\sigma_{3}}|C_{0}(0)\rangle$ and $|C_{1}(t)\rangle=e^{i\frac{\hat{\Omega}}{2}t\sigma_{3}}|C_{1}(0)\rangle$, then $\frac{d}{dt}\Pi_{C}(t)=i[\frac{\hat{\Omega}}{2}\sigma_{3},\Pi_{C}(t)]$, thus $\tilde{U}(dt)=e^{i\frac{\hat{\Omega}}{2}\sigma_{3}dt}$. After applying the unitary $\tilde{U}(dt)$, the state becomes

$\displaystyle\begin{aligned} &\rho_{C}(t+dt)\\ =&\tilde{\rho}_{C}(t+dt)+i[\frac{\hat{\Omega}}{2}\sigma_{3},\tilde{\rho}_{C}(t+dt)]dt\\ =&\rho_{C}(t)+i[\frac{\hat{\Omega}}{2}\sigma_{3},\rho_{C}(t)]dt\\ &-i[B\sin(\Omega-\hat{\Omega})t\sigma_{3}^{C}(t),\rho_{C}(t)]dt\\ &+\gamma\sin^{2}(\Omega-\hat{\Omega})t[\sigma_{3}^{C}(t)\rho_{C}(t)\sigma_{3}^{C}(t)-\rho_{C}(t)]dt.\end{aligned}$

(20)

The dynamic of the corrected state can then be obtained as

$\displaystyle\begin{aligned} \frac{d\rho_{C}(t)}{dt}=&i[\frac{\hat{\Omega}}{2}\sigma_{3}-B\sin(\Omega-\hat{\Omega})t\sigma_{3}^{C}(t),\rho_{C}(t)]\\ &+\gamma\sin^{2}(\Omega-\hat{\Omega})t[\sigma_{3}^{C}(t)\rho_{C}(t)\sigma_{3}^{C}(t)-\rho_{C}(t)].\end{aligned}$

(21)

To compute the final state under this dynamics, we move to the rotating frame with $\rho_{R}(t)=e^{-i\frac{\hat{\Omega}}{2}\sigma_{3}t}\rho_{C}(t)e^{i\frac{\hat{\Omega}}{2}\sigma_{3}t}$, then

$\displaystyle\begin{aligned} \frac{d\rho_{R}(t)}{dt}=&-i[B\sin(\Omega-\hat{\Omega})t\sigma_{3}^{C}(0),\rho_{R}(t)]\\ &+\gamma\sin^{2}(\Omega-\hat{\Omega})t[\sigma_{3}^{C}(0)\rho_{R}(t)\sigma_{3}^{C}(0)-\rho_{R}(t)],\end{aligned}$

(22)

where we have used the fact that $e^{-i\frac{\hat{\Omega}}{2}\sigma_{3}t}\sigma_{3}^{C}(t)e^{i\frac{\hat{\Omega}}{2}\sigma_{3}t}=\sigma_{3}^{C}(0)$. Note that $\rho_{R}(t)$ has the same QFI as $\rho_{C}(t)$ since $e^{-i\frac{\hat{\Omega}}{2}\sigma_{3}t}$ only depends on $\hat{\Omega}$. When the initial state is taken as $\frac{|C_{0}(0)\rangle+|C_{1}(0)\rangle}{\sqrt{2}}$, we have

$\displaystyle\begin{aligned} \rho_{R}(t)=\frac{1}{2}(&|C_{0}(0)\rangle\langle C_{0}(0)|+|C_{1}(0)\rangle\langle C_{1}(0)|\\ +&e^{-\int_{0}^{t}g(\tau)d\tau}|C_{0}(0)\rangle\langle C_{1}(0)|\\ +&e^{-\int_{0}^{t}g^{*}(\tau)d\tau}|C_{1}(0)\rangle\langle C_{0}(0)|),\end{aligned}$

(23)

where $g=2\sin(\Omega-\hat{\Omega})t[iB+\gamma\sin(\Omega-\hat{\Omega})t]$. Up to the second order of $d\Omega=\hat{\Omega}-\Omega$, the state’s QFI is

$\displaystyle\begin{aligned} J_{Q}^{\Omega}(t)=&B^{2}t^{4}+\frac{4}{3}\gamma t^{3}\\ &-(\frac{1}{2}B^{2}t^{3}+\frac{4}{5}\gamma t^{2}+\frac{4}{3}B^{2}\gamma t^{4}+\frac{8}{9}\gamma^{2}t^{3})d\Omega^{2}t^{3},\end{aligned}$

(24)

which achieves the maximal value $B^{2}t^{4}+\frac{4}{3}\gamma t^{3}$ in the asymptotic limit when $d\Omega\rightarrow 0$, the same as predicted from the heuristic argument. In Fig.5(b) we simulated the exact QFI with some finite $d\Omega$, it can be seen that with quite an amount of estimation error the QFI surpasses the highest value achievable under the unitary dynamics.