Optimal control for Hamiltonian parameter estimation in non-commuting and bipartite quantum dynamics

interesting results

The authors proposed to apply quantum control for the improvement of the precision of parameter estimation. They demonstrated that by applying the optimal quantum control, the quantum Fisher information, which quantifies the local precision limit for the estimation, can be significantly improved. The method also shows other advantages, such as the time stability in maintaining the maximum QFI over long time spans. The results have wide applications in quantum sensing and the determination of the models for quantum systems. I thus recommend the publication.

In the current study, the authors assume perfect controls. Although controls with systematic errors can be dealt with by many existing techniques, such as robust controls and composite pulses, the authors should at least mention it and have a discussion on this issue, which is important for practical applications.

In Optimal control Hamiltonian parameter estimation [...] Qin et al. propose a method to determine what control fields to apply to a system in order to best estimate parameters of the Hamiltonian describing that system.

The concept of the paper is a creative application of optimal control and leads to a counterintuitive result. I have a concern over the validity of the results related to taking the effect of errors in the controlling fields into account (detailed below). Until this concern is addressed I cannot recommend this paper for publication.

After introducing the method, the authors work through two examples, the estimation of the magnetic field (frequency) of a single-qubit system, and the estimation of the coupling strength (both XX and ZZ) of a two-qubit system.

In the single-qubit example, the authors show the counterintuitive result that placing the qubit in the $|0\rangle$ state during most of the experiment and applying the $u_x$ and $u_y$ fields near the end of the estimation period results in a better estimate of the eigenfrequency of the system than placing the system in the $|+\rangle$ state during the entire estimation period as is commonly done in conventional Ramsey type estimation of the frequency.

One assumption the authors seem to make implicitly is that the controls $u_x$, $u_y$ and $u_z$ are ideal. In practice, the quantity that is being estimated (the frequency) is exactly the same quantity that is used to determine the phases $x$ and $y$ of the control fields $u_x$ and $u_y$. As such, I would expect the method to work only because the phase tracking that is normally done by the qubit state in the $|+\rangle$ state to now be done by the "ideal" control fields.

If the authors could explain intuitively how this is not the case, and the error in the control would not affect the estimate it would improve the manuscript significantly.

Although the 2-qubit example is not directly about estimating the qubit frequencies but rather the coupling, it is not clear that this estimate is not also affected by frequency errors in the control fields.

Some minor points.

When reading the abstract and introduction it was not immediately clear that this manuscript is about using optimal control to determine what fields to apply to acquire better estimates rather than using optimal control to determine what driving fields would optimally implement a certain operation. This is mostly because finding optimal control pulses is a very common application these days. Being a bit more explicit could help the casual reader identify what this paper is about quicker.

The images of the driving fields could be improved by adding a few images of the Bloch sphere depicting the (ideal) state of the qubit during different parts of the protocol. This would also help the intuitive understanding.

In the examples it is not clear to me if there is a discrete control pulse near the end of the estimation period $T$ or if the fields are brought to oscillate during the entire estimation period. The textual description seems to suggest the former while the figures (typically figure b) seem to suggest the latter.

Unexpected and interesting results

Hand wavy explanations on the numerical findings
Poor presentation of the results

The authors explored the problem of maximizing the quantum Fisher information under decoherence. Their findings are quite interesting and maybe useful in practice. However, I cannot recommend it for publication in SciPost, at least in the current form.

The authors found that the use of quantum control drastically improved the maximum attainable QFI with T1 decoherence, in comparison to the case of T2. This is unexpected, and one would normally expect the opposite. The reason is that T1 washes away information contained in the Pauli-Z expectation value in addition to Pauli- X and Y. Therefore, it is harder to hide quantum information from T1 decoherence with control. I found the authors’ explanation on this surprising result unsatisfactory.

It will be useful to show that the scheme is robust to the unknown parameters in the drive Hamiltonian. Do small changes in B cause big changes in the controls?

Finally, I have a few suggestions on presentations: 1. Consider plotting the Bloch vectors (and their derivatives with respect to B) as functions of time. 2. A reader might be curious to know the maximum attainable QFI as a function of the decoherence rates. 3. Include some arguments on why one should use quantum control instead of just repeating the experiments for more times with reduced evolution times.