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Optimal Control Strategies for Parameter Estimation of Quantum Systems
by Quentin Ansel, Etienne Dionis, Dominique Sugny
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Submission summary
| Authors (as registered SciPost users): | Quentin Ansel |
| Submission information | |
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| Preprint Link: | scipost_202306_00025v1 (pdf) |
| Date submitted: | 2023-06-19 09:33 |
| Submitted by: | Ansel, Quentin |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Optimal control theory is an effective tool to improve parameter estimation of quantum systems. Different methods can be employed for the design of the control protocol. They can be based either on Quantum Fischer Information (QFI) maximization or selective control processes. We describe the similarities, differences, and advantages of these two approaches. A detailed comparative study is presented for estimating the parameters of a spin$-\tfrac{1}{2}$ system coupled to a bosonic bath. We show that the control mechanisms are generally equivalent, except when the decoherence is not negligible or when the experimental setup is not adapted to the QFI. In this latter case, the precision achieved with selective controls can be several orders of magnitude better than that given by the QFI.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-9-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202306_00025v1, delivered 2023-09-05, doi: 10.21468/SciPost.Report.7762
Strengths
Interesting connections between selectivity and QFI
Weaknesses
the connections are mainly analyzed with examples, which lacks generality.
Report
attached
Report #1 by Jing Liu (Referee 1) on 2023-8-17 (Invited Report)
- Cite as: Jing Liu, Report on arXiv:scipost_202306_00025v1, delivered 2023-08-17, doi: 10.21468/SciPost.Report.7662
Report
The authors studied the performance of selective control processes in quantum parameter estimation and compared it with the direct maximization of the QFI/CFI. The topic is interesting and provides a useful angle for practical quantum metrology since different control algorithms or different controls may present different performances in different scenarios and a fair comparison would help the community better understand how to wisely chose the control amplitude or control itself in practice. I think this manuscript deserves to be published. My comments are as follows.
1) The cost function in Eq. (21) chooses a uniform distribution of the three functions F1, F2, and F12, is this uniform distribution optimal in this case or there exists a better distribution? In some cases, the optimal distribution might be learned via some machine learning techniques.
2) A typing error: in the caption of Figure 3, "d) Same as panel b)" should be "c) Same as panel b)".
3) Why does the performance between the optimal selective solution and the direct maximization of QFI/CFI show such a significant difference when the QFI is replaced with the CFI? In Fig. 3(b) the performance of the orange line is even a little worse than the dashed-blue line at the final time yet in Fig. 3(c), the orange line is significantly larger than the dashed-blue line at the final time.
4) Does the performance of Fig. 3(c) rely on the choice of POVM? Would it vanish when a different POVM is chosen?
5) The control amplitudes in Figs. 3 and 4 are discrete, and more importantly, the control amplitude number is small [in Fig. 3(d) there exist only 3 amplitudes for each control]. Do there exist some assumptions here or these controls are "globally" optimal? If there exist some assumptions on the amplitude number, the next question does the current results hold when the amplitude number is large? For example, the amplitude number is 100 instead of 3 in Fig. 3(d).
6) In the second paragraph of the introduction "we can mention the maximization of Quantum Fisher Information (QFI)", personally I think a related reference [Phys. Rev. A 96, 012117 (2017)] is missed.
Author: Quentin Ansel on 2023-09-18 [id 3988]
(in reply to Report 1 by Jing Liu on 2023-08-17)
We are grateful for this reviewing. We address in the following the points raised in the same order as they appear in the report. We have attached a version of the manuscript with changes written with colors.
1) The cost function in Eq. (21) chooses a uniform distribution of the three functions F1, F2, and F12, is this uniform distribution optimal in this case or there exists a better distribution? In some cases, the optimal distribution might be learned via some machine learning techniques.
Reply: We thank the Referee for this interesting comment. We consider in this section the QFI as a figure of merit. To study the properties of the control mechanism, we then show that this latter can be written as the sum of three different contributions denoted F1, F2 and F12. This expression of the cost functional C is given in Eq. (21) which is equivalent to the maximization of the QFI. As mentioned by the Referee, this second expression could be used as a new figure of merit (if the gradients of the different terms can be calculated analytically) by playing with the weights of the different contributions in order to enhance e.g. either the initialization or the stabilization processes. A different choice of cost functional will lead to a different control protocol. With non equal weights we loose the interpretation that C is the QFI a the final time, and thus, it is not easy to know for estimation parameter what the optimal distribution between the three functions could be. This point goes beyond the scope of this paper but could be an interesting direction to follow.
2) A typing error: in the caption of Figure 3, "d) Same as panel b)" should be "c) Same as panel b)".
The manuscript has been changed accordingly.
3) Why does the performance between the optimal selective solution and the direct maximization of QFI/CFI show such a significant difference when the QFI is replaced with the CFI? In Fig. 3(b) the performance of the orange line is even a little worse than the dashed-blue line at the final time yet in Fig. 3(c), the orange line is significantly larger than the dashed-blue line at the final time.
Reply:
- Concerning Fig 3(b): The optimal selective control has a slightly lower final QFI value because the spin state must leave the equator of the Bloch sphere, the equator being the ensemble of states maximizing the QFI (see paragraph below Eq. (26)). The QFI then increases with a sub-optimal rate at the end of the process. -Concerning Fig 3(c): The CFI is the quantity with the most important difference. Here, the optimal control of the QFI leads to a constantly zero CFI, which is due to the orthogonality of the optimal measure basis for $\Delta$ and the basis $|\uparrow\rangle,|\downarrow \rangle$. However, with the optimal selective control, we reach the maximum value of the CFI at the very end of the control process, because the target states correspond to the measurement basis." These points are discussed in p. 13. We have modified slightly the text to emphasize certain key points.
4) Does the performance of Fig. 3(c) rely on the choice of POVM? Would it vanish when a different POVM is chosen?
Reply:
We confirm the Referee’s intuition on the fact that the choice of POVM has an impact on the time evolution of the CFI. We could have chosen a POVM such that the dashed blue curve in Fig. 3c is different from zero. For example, in the example of section 5.3, a single curve is represented in each plot because the results are similar in all cases. This observation is due to the fact that the POVM is well adapted to the different situations under study.
5) The control amplitudes in Figs. 3 and 4 are discrete, and more importantly, the control amplitude number is small [in Fig. 3(d) there exist only 3 amplitudes for each control]. Do there exist some assumptions here or these controls are "globally" optimal? If there exist some assumptions on the amplitude number, the next question does the current results hold when the amplitude number is large? For example, the amplitude number is 100 instead of 3 in Fig. 3(d).
Reply: In the case of Figure 3, the controls are globally optimal. This is discussed in the lower part of p. 12 and in the upper part of p. 13. We do not proceed to an exhaustive justification of the global optimality because most of the key points have been already discussed in the literature, in a different context. We refer to [45,46,32] for the technical details. In Ref. [33], some of the authors provide an exhaustive analysis of the optimality of the selective control field of Fig 3(d) in the case without relaxation. Since the control fields are globally optimal, increasing the number of pulses cannot further increase the efficiency of the control processes. In Fig. 4, we agree with the Referee that a specific constraint has been added to the control search. We assume that the control is a piecewise constant function with five-time steps in which the phase, the amplitude and the duration of the time step are optimized. As mentioned in the paper, this choice leads to a good compromise between computational time and control efficiency. Numerical optimization methods with smooth controls could also be used to solve this control problem, but with final fidelities of the same order of magnitude.
6) In the second paragraph of the introduction "we can mention the maximization of Quantum Fisher Information (QFI)", personally I think a related reference [Phys. Rev. A 96, 012117 (2017)] is missed.
Reply: We thank the Referee for pointing out this reference which has been added to the bibliography of the paper.
Author: Quentin Ansel on 2023-09-18 [id 3989]
(in reply to Report 2 on 2023-09-05)We are grateful for this reviewing. We address in the following the points raised in the same order as they appear in the report. We have attached a version of the manuscript with changes written with colors.
1) The connection between QFI and the Bures distance is only valid locally for neighboring states where D(ρx, ρx+δx) << 1, using Eq.(16) in the regime D2 = 2 for the connection to selective control is a bit problematic.
Reply. We disagree with this comment of the Referee. In Eq. (16), we introduce \mathcal{F}_{fd}, which is a finite difference approximation of the QFI. The QFI is reached only when the difference between the two density matrices goes to zero. This approximation is precisely used to avoid mathematical problems. With this finite difference version of the QFI, it is straightforward to see that the QFI is necessarily infinite if the goal is to generate orthogonal states for two infinitesimally close system parameters. With the assumption that the QFI increases monotonically with time, this situation can only be achieved in infinite time.
2) In the case of fixed POVM, the comparison of QFI based optimization and the selective control seems unfair. Here the relevant quantity is CFI, which may be connected to the selectivity of the probability distributions
Reply. We completely agree with this comment of the Referee mentioning that the relevant quantity is the CFI. We have commented on our choice in detail in several paragraphs (see e.g. p 6 above Def. 3, in Sec. 5.5, and in the conclusion). We have focused our attention on the QFI, for the following reasons: - Many different studies (see the literature presented in the introduction) investigated the optimal control of the QFI, and it was interesting to revisit this point and to show the limits of this approach. - The optimization of the CFI with a control field is in general a very arduous numerical problem, due to numerical instabilities and control traps, while the maximization of the QFI is easier. It is therefore very interesting to consider the second problem rather than the first and to check a posteriori whether the solution is indeed relevant experimentally. The complete analysis of the connection between CFI and selectivity goes beyond the scope of this paper and will be done in a forthcoming paper.
3) The performance of the selective control for parameter estimation depends on the choices of the target states. It is not clear how these states should be chosen. Even for the specific qubit example for the estimation of γ presented in the manuscript, the choice of the target state as the completely mixed state is not well explained. It is not clear why the other pole is not used as the target state. Even the steered state cannot reach it, why it is not chosen so the steered state can be made as closer to it as possible?
Reply. We confirm the comment of the Referee in the sense that there is no recipe for the choice of target state, this is left to the intuition of the physicist based on knowledge of system dynamics. This is one of the main difficulties of this approach, as outlined in the conclusion (the solution is not intrinsic to the system). For the estimation of γ, we have clarified this point at the end of page 16. We consider the north pole as one of the target states because it is the attractor of the relaxation process, ensuring that a final state close to this target can be reached. For the second state, we do not take the opposite pole because we know that it cannot be reached (due to relaxation effects, the reachable set corresponds only to a subspace of the Bloch ball. An interesting reference on the subject is: PhysRevA.88.033407). Instead, we consider an intermediate position with the center of the Bloch ball. We could have used other target states for which the result would have been very similar. Here, the positions of the target states are not a key factor because orthogonal states cannot be generated, due to relaxation process.
4) As briefly mentioned in the conclusion, the QFI has a closer connection to the selective control when the target states are not fixed, but directly maximizing the distance of the final states. With fixed target states, it seems the two methods are equivalent only when D(ρ0, ρtarget,0) +D(ρ1, ρtarget,1) = D(ρ0, ρ1) where ρ0 and ρ1 are the final states. For the examples where the selectivity defers from the QFI, I would suggest the authors to consider checking this condition and if possible choose a different set of target states that satisfy this condition for a further comparison.
Reply. We agree with this comment of the Referee, but we must be careful when comparing the two variants of the selectivity problem. If the target states are fixed, we minimize a cost functional of the form D(ρ0, ρtarget,0) +D(ρ1, ρtarget,1), while if the targets are free, we maximize D(ρ0, ρ1). After optimization, the first cost functional is ideally equal to 0. In this second case, the results using a selective control are expected to be closer to an optimization with the QFI.
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