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

Dear Editor,

We have revised the manuscript "Optimal control for Hamiltonian parameter estimation in non-commuting and bipartite quantum dynamics" in the light of the reviewers' comments. We believe we have been able to respond to all such comments and to produce an improved version of the paper, which we hereby resubmit.

Let us list the changes in this new version below, along with the comments that prompted them, when relevant:

• All reports point out that perfect controls are assumed and that this should be discussed or at least mentioned. In our study, robustness has been dealt with through the (d) parts of our plots, where the performance of controls optimised for wrong estimates of the parameter is depicted, yielding tolerances between 5 and 20% of the parameter, depending on specific cases. In the revised version of the manuscript, we have linked the application of control optimised for wrong parameters explicitly to the robustness of the scheme by carrying out further analysis and obtaining direct information on the tolerance in terms of amplitude and timing mismatches. Hence, we have much expanded the discussion of the robustness plots with dedicated paragraphs at the end of sections 3.1 and within section 4.1, and with a comprehensive final discussion in the conclusions on page 16.

• Reviewer 2 asks whether, in single-qubit frequency estimation, the advantage of having the qubits dwell around the |0> rather than |+> state might be an artefact due to assuming perfect control fields. This is not the case, as discussed in our revised discussion of robustness already reported above. However, we have endeavoured to better explain our results by adding a detailed descriptions of the controlled dynamics and of their advantages, for both dephasing and relaxation: "In this case, as discussed in [42], the optimal controls exploit the decoherence free subspace of the states diagonal in the Pauli-z eigenbasis (i.e., the z-axis of the Bloch sphere)." and "Under relaxation, the states are first steered towards the (decoherence free) ground state and then, after a few precessions around the z-axis, towards the x-y plane. Although the protection from decoherence achieved through this trajectory is not as complete as in the case of dephasing, where a decoherence free subspace is used, states with different parameters separate faster than in the dephasing free subspace, resulting in a better distinguishability and thus a higher maximum QFI value (the reader should bear in mind that the QFI reflects the cumulative effect of dynamics upon the evolving state, rather than the information in the initial state, whose preservation is immaterial here)."

• As both reviewer 1 and 2 requested, the arguments above are now supported by including a visualisation of the controlled Bloch vectors, which we have included also for the non-commuting case and 2-qubit cases by introducing the new Figs. 2, 5 and 12. This should greatly boost the accessibility and clarity of our study.

• Reviewer 1 finds the better performance achievable under relaxation (over dephasing) surprising. This aspect has been addressed in detail through the additional explanations reported at the previous point.

• The maximal QFI as a function of decoherence rates, requested in report 1, was already addressed in Basilewitsch et al.'s paper (Ref.[42]), to which the reader is now explicitly referred at the beginning of page 6.

• Reviewer 2 points out that it is not clear at first sight whether the controls are optimised for estimation or to achieve controlled operations: the addition of "in the estimation of Hamiltonian parameters" in the abstract should dispel this confusion.

• The introductory section has been expanded, and 10 additional references have been included.

• Reviewer 2 is asking whether a final discrete pulse is applied: it is not, as the reviewer correctly surmises. We believe this confusion is down to the old description in the relaxation section, which was confusing. We have adapted the part to not mention pi/2 pulses, which should clarify the issue.

• Reviewer 1 asks to comment as to the respective advantage of control vs mere repetition. To this aim, we have added the following sentence in the conclusions: "Notice also that, in extracting information about the parameter, the advantage of controlled dynamics over repeated uncontrolled ones would ultimately depend on the latter's cost in specific circumstances (although of course controls and repetition can be combined, attaining better information rates)."

Sincerely,
Shushen Qin, Marcus Cramer, Christiane P. Koch and Alessio Serafini