Digital quantum simulation of bosonic systems and quantum complementarity

Dear Editor, thank you for your e-mail regarding our manuscript entitled ``Digital quantum simulation of bosonic systems and quantum complementarity''. We would also like to thank you for the opportunity to do a major revision. Following Referee #2's comments, we realized that our manuscript could perhaps be improved, as it was, and that it consequently reflects the points raised by Referee #1. With the contributions of both Referees, we believe our work has been improved. Therefore, we thank both Referees for the points raised. Please find below our response to the Referees' comments. Finally, a list of the changes made in the manuscript is included.


Our response to Referee #1: In addition to what we have already answered, it follows that: The above considerations are preliminary responses before Referee #2's review is released. Referee #2's opinion touches on the points raised by Referee #1, but in a more comprehensive manner. Based on both opinions, we introduce additional elements that we believe address Referee #1's questions in the same way. Therefore, we invite Referee #1 to follow the response to Referee #2.


Our response to Referee #2: We appreciate the Referee's willingness to carefully review our manuscript. We consider the review to be comprehensive and to have prompted a number of inquiries that require further elucidation. These inquiries, we are confident, will certainly contribute to enhancing and refining our manuscript. It is our intention to address the inquiries raised as they are of significant relevance.

Regarding item 1: We agreed with the Referee that we are not using more than one quanton which goes in a different line from the intention of the formalism used by Mohan et al. to simulate the HOM effect. Although that formalism was developed with a different purpose in mind, our extension builds upon it to enable the analysis and treatment of more intricate and sophisticated optical setups, such as those employed in the experiment by Pessoa Júnior. Even though our implementation does not involve multiple quantons, the extension of the formalism we develop remains equally suitable for scenarios where they are present. Additionally, to our knowledge, implementing the simulations is feasible for the standard versions of the interferometers, but becomes challenging with the addition of blockers. As the formalism separates the problem into modes, with single photons the same state that represents the arm where the blocker is inserted is packed in the qubit. An initial approach discussed in this manuscript involves focusing on the single qubit until it enters the blocker region (discarding other components), disregarding statistics for cases not passing through the blocker, and recording the data. In a subsequent experiment, we must manually set the post-blocker state and proceed with the experiment as if the ``photon'' traversed the blocker region. We will then collect data from before and after the blocker to present the experimental conditions. It seems feasible to us with the aid of an auxiliary qubit for correlation, but we failed to find a better version than the one presented in the manuscript. Therefore, it was precisely this scenario that directed us to Mohan et al.'s and other digital simulation papers and ultimately to the present manuscript. Thus, with just a single circuit, we can somewhat simulate the experiment as a whole without needing to make the kind of considerations we mentioned earlier. Personally, we find this approach less artificial than the previous one, which requires two distinct configurations and the input of a manufactured state. In our approach, we simply record the measurement in the blocker and reset the qubit's mode to show it has been ``absorbed'' by the blocker. However, even in cases where it is not ``absorbed'', the experiment remains valid and captures the scenario where the photon is not ``absorbed'', which seems to be much more similar to what happens in a bench experiment.
We include a few sentences along the text to emphasize this feature of the experiment. Despite that and building on your questioning, we introduce an initial state in the modified Unruh's experiment that occupies both modes and follow with a discussion similar to that presented by Pessoa Júnior. Therefore we include section E ``Modified Unruh's experiment for the two-photon case'', which is truly innovative in this regard and simultaneously addresses Referee \#1's questions.

Regarding item 2: At the time the demonstrations on quantum hardware were conducted, we only had access to IBM's open ``Eagle'' processors, which are not the best in terms of noise and gate error. However, we recently earned credits from that company to be spent on any of the available hardware, including the ``Heron'' type, which so far has proved to be substantially better than earlier processors in the demonstrations we have conducted for other works. With that in mind, we ran some demonstrations on better hardware and observed a significant improvement in the results, and because of that, we carried out the demonstrative results again. Error bars have been included in the new version. We can see that the results have been significantly improved with the use of higher-quality hardware.

Regarding item 3: We agreed with the Referee that some elements are missing to enrich the section. A significant missing component in the theoretical discourse is the omission of the critique on retro-inference methods utilized by Pessoa Júnior. Although this is mentioned within the manuscript, it is not revisited in this section to develop a more comprehensive critique. The QCP framework brings an updated view of this type of experimental setup, which is, in some aspects, new in the literature. Pessoa Júnior extrapolates the most well-known phase variation to infer the wave behavior (also particle behavior) in the quantitative Bohr's complementarity principle. In addition to the QCP framework being similarly utilized in this scenario, the analysis is innovative and sheds light on a fundamental discussion within quantum mechanics, but was clearly not well employed, as noted by the Referee. To fill this gap, we discuss a new version of the delayed-choice experiment that arises when we apply the extrapolation methods used by Pessoa Júnior. This setup, besides being a new experiment, increases the discussion on retro-inference and further emphasizes the lack of a formal definition of Bohr's complementarity principle. Additionally, we explore the QCP analysis for the two-photon case. This analysis also goes beyond what we have previously discussed on the subject. Therefore, we believe this result also greatly enriches the manuscript and closes the gap highlighted by the Referee in this section. It is worth noting that we performed quantum state tomography by reconstructing $\rho$ using the mean values of the Pauli basis observables. Even using classical simulation, our results were not as good for predictability as they were for coherence; the results can be seen in Fig. 15. However, the results in the classical simulation were only relatively good when we used $2^{23}$ shots in $100$ Trotter steps. Unfortunately, it is completely unfeasible to perform this type of experiment on current quantum hardware. As we saw, in our simplest case, we already had problems obtaining the results.

Regarding minor issues: 1) The sentence in the manuscript contained an error and has been corrected to display as $D_B$ ($D_A$) where it was previously written as $D_A$ ($D_B$). 2) In the graphs in Fig.~7 we have used $\phi_{\text{E}} = \pi/2$ and $\phi_{\text{H}} = 0$ and we have included a sentence in the subsequent text to make everything more consistent. 3) Thank you for noticing this mistake too, the right version is $\left\vert {B_{1}}\right\rangle =-\frac{R}{2}\left\vert 01000\right\rangle_{\text{DLMQR}}$ and it was also corrected. 4) In fact, we uploaded an unintended image and we have corrected this issue as well. Thank you for the circuit version. Although we do not use it, we will try to explain this point better based on the updated version. The intended version has some markings on the unitary block, labeled $``0"$ and $``1"$, to highlight which qubits are involved in the operation and is the usual form used by Qiskit.