Optimal adaptive control for quantum metrology with time-dependent Hamiltonians

Quantum metrology has been studied for a wide range of systems with time-independent Hamiltonians. For systems with time-dependent Hamiltonians, however, due to the complexity of dynamics, little has been known about quantum metrology. Here we investigate quantum metrology with time-dependent Hamiltonians to bridge this gap. We obtain the optimal quantum Fisher information for parameters in time-dependent Hamiltonians, and show proper Hamiltonian control is generally necessary to optimize the Fisher information. We derive the optimal Hamiltonian control, which is generally adaptive, and the measurement scheme to attain the optimal Fisher information. In a minimal example of a qubit in a rotating magnetic field, we find a surprising result that the fundamental limit of T2 time scaling of quantum Fisher information can be broken with time-dependent Hamiltonians, which reaches T4 in estimating the rotation frequency of the field. We conclude by considering level crossings in the derivatives of the Hamiltonians, and point out additional control is necessary for that case.

SUPPLEMENTARY NOTE 1. DERIVATION OF hg (T ) In this Supplementary Note, we derive h g (T ) for a time-dependent Hamiltonian H g (t), dened as where U g (0 → T ) is the unitary evolution under the time-dependent Hamiltonian H g (t), and it determines the quantum Fisher information of estimating g in the following way, where |ψ 0 is the initial state of the system. The maximum quantum Fisher information is the square of the gap between the maximum and minimum eigenvalues of h g (T ). In order to obtain h g (T ), we break the unitary evolution U g (0 → T ) for a time duration T into small time intervals ∆t, Then, U g (T − ∆t → T ) · · · U g ((k + 1)∆t → (k + 2)∆t) [∂ g U g (k∆t → (k + 1)∆t)]U g ((k − 1)∆t → k∆t) · · · U g (0 → ∆t) . (4) When the time interval ∆t is suciently small, the Hamiltonian H g (t) is approximately time-independent during each time interval k∆t ≤ t ≤ (k + 1)∆t, i.e., U g (k∆t → (k + 1)∆t) ≈ exp(−iH g (k∆t)∆t), and by expanding exp(−iH g (k∆t)∆t) to the rst order of ∆t, we have exp(−iH g (k∆t)∆t) = I − i∆tH g (k∆t) + O(∆t 2 ), (6) so, ∂ g U g (k∆t → (k + 1)∆t) ≈ −i∆t∂ g H g (k∆t) + O(∆t 2 ).

(7)
In the limit ∆t → 0, ∂ g U g (0 → T ) can be written in the following integral form, which is the exact solution to ∂ g U g (0 → T ).
When U † g (0 → T ) is multiplied to ∂ g U g (0 → T ) from the left, since h g (T ) in Eq. (1) turns out to be This gives the integral form of h g (T ) in the main text for a time-dependent Hamiltonian H g (t) at time t = T . When H g (t) is independent of time, Eq. (10) degenerates to the relevant formula in [1]. In this Supplementary Note, we derive the estimator and the measurement scheme that can gain the upper bound of Fisher information given by Eq. (7) in the main manuscript with the assistance of the optimal Hamiltonian control.
Suppose the parameter that we want to estimate is g. According to Eq. (8) of the main manuscript, the optimal control Hamiltonian is where H g (t) is the original Hamiltonian with the parameter g, |ψ k (t) are the eigenstates of ∂ g H g (t) and f k (t) are arbitrary real functions. Since we do not know the exact value of g, the parameter g in the optimal control Hamiltonian (11) should be replaced by some known estimate of g, g c , in practice, and the control Hamiltonian is actually where | ψ k (t) are also dependent on g c instead of g, i.e., | ψ k (t) are eigenstates of ∂ g H g (t)| g=gc .
It should be noted that the parameter g c in the control Hamiltonian (12) is always a constant, even when it is equal to the real value of g, while the parameter g in the original Hamiltonian H g (t) is a variable. This should be kept in mind in computing the Fisher information for g. The appearance of g in the optimal control Hamiltonian (11) just indicates what value of g c maximizes the Fisher information, and it turns out to be the real value of g.
When g c is close to g, the total Hamiltonian can be written as up to the rst order of δg, where δg = g − g c . The evolution under H tot (t) can be decomposed as and at time t, (15) Note that generally dierent orderings of the three terms at the right side of Eq. (15) will give dierent results when ∆t is nite, but when ∆t → 0, they will give the same result. The ordering chosen in (15) is for convenience of computation below.
If a system is initially in an eigenstate of ∂ g H g (t)| g=gc at t = 0, say | ψ k (0) , then according to Eq. (14) and (15), the state after an evolution of time T is Now, suppose the maximum and minimum eigenvalues of ∂ g H g (t)| g=gc at time t are µ max (t) and µ min (t), and the corresponding eigenstates are | ψ max (t) and | ψ min (t) , respectively. To achieve the maximum Fisher information given by Eq. (7) in the main manuscript, we can prepare the system in an equal superposition of | ψ max (0) and | ψ min (0) at the initial time t = 0, Then after evolving for time T , the state of the system is Note that θ max / min (t) denotes the value of θ k (t) associated with the maximum or minimum eigenvalue of ∂ g H g (t)| g=gc , but not the maximum or minimum over θ k (t).
Since ∂ g = ∂ δg , the quantum Fisher information of estimating of g by measuring |Ψ(T ) is which is exactly the upper bound of quantum Fisher information given by Eq. (7) in the main manuscript.
To attain this quantum Fisher information, we can measure the following observable which are varying with time T . It is important to note that |± are also dependent on g c , instead of g, which makes it possible to implement the measurement of O without knowing the exact value of g. It can be obtained that the expectation value and variance of O under the state |Ψ(t) are O can be considered as an estimator of g (with some local unit dierence characterized by ∂ δg O and potential systematic errors which can be eliminated by calibration) since it is dependent on g. The parameter g can be obtained from O , and the variance of the estimate [2,3] is In this Supplementary Note, we derive the evolution of a qubit in a rotating magnetic eld and the optimal quantum Fisher information for the amplitude B and the rotation frequency ω of the magnetic eld in the absence of control Hamiltonian.
Suppose the rotating magnetic eld is B(t), where the amplitude of the eld B is assumed to be constant for simplicity. The interaction Hamiltonian between a qubit and the eld is When there is no control on the Hamiltonian, the evolution of the qubit is determined by the Schrödinger equation To derive the evolution of the qubit in this case, note that We know that if a state under a general Hamiltonian H(t) is transformed by exp(iH 0 t), the eective Hamiltonian for the evolution of the new state is Therefore, the rotating Hamiltonian (26) can be perceived as the eective Hamiltonian of an original Hamiltonian

The evolution of a qubit under the Hamiltonian
, so the evolution under the rotating Hamiltonian (26) is Having obtained the evolution of the qubit, we can compute the Fisher information for B and ω.
To obtain the optimal quantum Fisher information for the eld amplitude B, we calculate The optimal quantum Fisher information is determined by the squared gap between the maximum and the minimum eigenvalues of From Eq. (2), it can be obtained that the optimal quantum Fisher information for B is Similarly, to obtain the optimal quantum Fisher information for the eld rotation frequency ω, we calculate And the h matrix for ω is And the optimal quantum Fisher information for ω is This Supplementary Note is to obtain the Fisher information for the rotation frequency ω of the magnetic eld in the presence of the control Hamiltonian with the control parameter ω c close to ω.
It has been obtained in the main manuscript that the maximum Fisher information of estimating ω with the optimal Hamiltonian control is and the optimal control Hamiltonian is However, one generally does not know the exact value of ω, so the real control that can be applied on the qubit is where ω c is a control parameter that is close to ω, and the maximum Fisher information is saturated when ω c = ω.
In this case, the total Hamiltonian is A natural question is how the deviation of ω c from ω inuences the Fisher information. Will a slight deviation of ω c from ω cause a large drop in Fisher information? This is important to the design and realization of the Hamiltonian control scheme in practice.
In this Supplementary Note, we derive the Fisher information of estimating ω when ω c is deviated from the true value of ω. We will show how to use this result to design an adaptive control scheme to approach the T 4 scaling of the Fisher information in the next Supplementary Note.
Since the maximum quantum Fisher information of estimating ω is determined by the gap between the highest and lowest levels of h ω (T ), we need to obtain h ω (T ) rst. As ω c is close to ω, we can expand h ω (T ) with ω c near ω.
The eigenvalues of h ω (T ) in Eq. (50) are so, nally, the Fisher information for ω with ω c near ω is where higher order terms of δω have been neglected. This shows that when ω c is deviated a little from the true value of ω, the drop in Fisher information is only of the order δω 2 , and the T 4 scaling is unaected, implying the reliability of the above Hamiltonian control scheme. It lays the foundation for the feedback control scheme to approach the T 4 scaling which will be introduced in the next Supplementary Note.

SUPPLEMENTARY NOTE 5. ADAPTIVE HAMILTONIAN CONTROL FOR ESTIMATING ω
Basing on the result in Supplementary Note 4, we can design schemes to approach the T 4 scaling of the Fisher information for estimating ω when the exact value of ω is unknown and an estimate of ω is used in the control Hamiltonian.
We rst give an overall analysis about why the Hamiltonian control may increase the Fisher information. In order to apply the Hamiltonian control scheme, we need an estimate of ω to be used as the control parameter ω c in the control Hamiltonian. But if the precision of the estimate of ω needed is comparable to the precision that can be gained, the Hamiltonian control scheme would be senseless. Fortunately, Eq. (52) shows that an approximate ω with a precision δω of order 1/T is sucient to produce an estimate of ω with a precision of order T −2 (i.e., the Fisher information is of order T 4 ). This guarantees that the Fisher information can be increased by the Hamiltonian control scheme.
The simplest feedback control scheme is just to rst obtain an initial estimate of ω without any Hamiltonian control, then use it in the Hamiltonian control (39) to produce a high precision estimate of ω, without any iteration of the scheme. If the initial estimate of ω without Hamiltonian control is suciently good, Eq. (52) guarantees that the nal estimate of ω with the Hamiltonian control can attain the Fisher information of order T 4 .
For the initial estimation of ω, suppose the qubit evolves for time T without any Hamiltonian control and then it is measured to estimate ω, and this procedure is repeated for N times. According to Eq. (36), the variance of the estimate is where we assumed T 1/ √ 4B 2 + ω 2 to neglect the lower order terms of T for simplicity. Then if we apply the control Hamiltonian with this estimate of ω and let the qubit evolve for the same time T , the Fisher information of the nal estimation will be (54) which reaches the T 4 scaling. The price for this scheme is that we need additional time N T to obtain an initial estimate of ω in the absence of the Hamiltonian control. But as the additional time N T is linear with T , the T 4 scaling with the Hamiltonian control can still signicantly outperform the usual T 2 scaling without the Hamiltonian control.
However, we can also feed the new estimate of ω from the estimation with the Hamiltonian control back into the control Hamiltonian so that a better estimate of ω can be produced, and this step can be iterated to further rene the estimate of ω. This will make the feedback control scheme more ecient.
In the following, we give a detailed analysis about such an adaptive feedback control scheme, and show a minimum requirement for the precision of the initial estimation of ω without Hamiltonian control to make the feedback control scheme work.
For the sake of generality, we assume the Fisher information of each measurement in the initial estimation without Hamiltonian control is I 0 , and the measurement is repeated N times, then the variance of the initial estimate of ω is Now, we apply the Hamiltonian control with this estimate of ω and choose the evolution time as T 1 = √ I 0 , then the Fisher information will be ). (57) We repeat the measurement N times, and use the new estimate of ω in the control Hamiltonian. Let the qubit evolve for time T 2 = √ I 1 , then the Fisher information becomes If we iterate this process for n rounds and choose the evolution time in the n-th round to be T n = I n−1 , the Fisher information is ).
It can be derived from this iterative relation that and In terms of T n , I n can be rewritten as ).
If N is large, I n ≈ B 2 T 4 n then. This recovers the T 4 scaling of the Fisher information in the presence of Hamiltonian control.
A question is how many rounds of feedback control it takes to reach T n = T for a desired time T . From Eq. (60), it can be obtained that It can be seen that when T is large, which is a double logarithm of T , implying n increases extremely slowly with T and very few rounds of feedback control is sucient to reach a large T . So this iterative feedback control scheme converges to the T 4 scaling very fast.
We also want to remark that there is a lower bound that I 0 must satisfy in order to increase the Fisher information by this method. For Eq. (57), I 1 is not always larger than I 0 , and if we want to increase the Fisher information, we need I 1 > I 0 , so it requires When I 0 is above this threshold, I 1 will also be larger than ) as the Fisher information increases. This again makes I 2 > I 1 . Repeating this step, we will nd that the Fisher information is always increasing, and thus the allowed T for the T 4 scaling is also increasing. On the contrary, if I 0 does not satisfy the lower bound in Eq. (65), I 1 would be smaller than I 0 , I 2 would be smaller than I 1 , and so forth. Therefore, Eq. (65) is the minimum requirement for the precision of the initial estimation of ω without Hamiltonian control in order to make the iterative feedback control scheme work.
If T is so small that Eq. (63) becomes negative, we just need to set n = 1 and the T 4 scaling can be attained. Note that the T n given in Eq. (60) is just the upper bound of T that the T 4 scaling can be achieved (up to a relative loss of 1/N ) in the n-th round, it does not mean that T must be chosen as T n in that round.
We summarize this iterative feedback control scheme as follows. First, a rough estimation of ω should be made in the absence of the Hamiltonian control with the Fisher information satisfying the threshold (65), then the estimate of ω is used to apply the Hamiltonian control and get a new estimate of ω. The new estimate of ω can be fed back to the control Hamiltonian to further update the estimate of ω. This procedure is repeated for n rounds until the desired evolution time T is reached, and the number of rounds is given by Eq. (63). The relative loss of Fisher information due to the nite number of measurements is 1 18N , where N is the number of measurements in each round of the feedback control scheme, which does not aect the T 4 scaling. SUPPLEMENTARY NOTE 6. ADDITIONAL CONTROL AT LEVEL CROSSINGS OF ∂gHg(t) In this Supplementary Note, we analyze the eect of the additional control H a (t) proposed in the Methods of the main manuscript.
We rst prove that under an arbitrary unitary transformation V (t) on the states, a Hamiltonian H(t) is transformed to Consider an arbitrary state |φ(t) evolving under H(t). It satises the Schrödinger equation Now, if we transform the state |φ(t) to |ϕ(t) = V (t)|φ(t) , then |ϕ(t) satises Therefore, the Hamiltonian for |ϕ(t) is the H (t) in Eq. (66). Now we return to our problem. In the presence of the optimal control Hamiltonian, the total Hamiltonian is where , and To see the eect of the total Hamiltonian clearly, let us transform the time-dependent eigenbasis {e −iθ k (t) |ψ k (t) } of ∂ g H g (t) to an arbitrary time-independent basis {|k }. The transformation can be written as Then the Hamiltonian in the new basis can be obtained from (66).
It is straightforward to verify that and where σ mn is the σ X -like transition operator between |m and |n , σ mn = |n m| + |m n|.
Eq. (76) indicates that the state |n will be transformed to |m in the new basis {|k } by the total Hamiltonian with the additional control, so in the original basis, |ψ n (t) will be transformed to |ψ m (t) correspondingly, which is exactly what we want.
Note that H tot (t) commutes between dierent t, so the evolution in the new basis {|k } under H tot (t) for a short time interval τ − 1 2 δt ≤ t ≤ τ + 1 2 δt is U (t) = exp − i where τ is the moment that the level crossing occurs. To drive the system exactly from |ψ n (t) to |ψ m (t) , which corresponds to driving |n to |m in the new basis {|k }, we need thus h(t) must satisfy τ+ 1 2 δt τ− 1 2 δt h(t )dt = l + 1 2 π, l ∈ Z. (80) This proves Eq. (23) in the main text. A notable point in Eq. (78) is that since sin(l + 1 2 )π = (−1) l , U (t) will introduce an additional phase (−1) l+1 i to |ψ m (t) and |ψ n (t) . This will aect the relative phase and needs to be taken into account, when the system is in a superposed state involving |ψ m (t) or |ψ n (t) .
When there are multiple crossings between the maximum/minimum eigenvalue and other eigenvalues of ∂ g H g (t) during the whole evolution process 0 ≤ t ≤ T , an additional Hamiltonian control H a (t) is needed at each level crossing.