From the journal Digital Discovery Peer review history

23-Dec-2022

Dear Dr Yanai:

Manuscript ID: DD-ART-09-2022-000093
TITLE: Artificial Neural Network Encoding of Molecular Wavefunctions for Quantum Computing

Thank you for your submission to Digital Discovery, published by the Royal Society of Chemistry. I sent your manuscript to reviewers and I have now received their reports which are copied below.

I have carefully evaluated your manuscript and the reviewers’ reports, and the reports indicate that major revisions are necessary.

Please submit a revised manuscript which addresses all of the reviewers’ comments. Further peer review of your revised manuscript may be needed. When you submit your revised manuscript please include a point by point response to the reviewers’ comments and highlight the changes you have made. Full details of the files you need to submit are listed at the end of this email.

Digital Discovery strongly encourages authors of research articles to include an ‘Author contributions’ section in their manuscript, for publication in the final article. This should appear immediately above the ‘Conflict of interest’ and ‘Acknowledgement’ sections. I strongly recommend you use CRediT (the Contributor Roles Taxonomy, https://credit.niso.org/) for standardised contribution descriptions. All authors should have agreed to their individual contributions ahead of submission and these should accurately reflect contributions to the work. Please refer to our general author guidelines https://www.rsc.org/journals-books-databases/author-and-reviewer-hub/authors-information/responsibilities/ for more information.

Please submit your revised manuscript as soon as possible using this link:

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You should submit your revised manuscript as soon as possible; please note you will receive a series of automatic reminders. If your revisions will take a significant length of time, please contact me. If I do not hear from you, I may withdraw your manuscript from consideration and you will have to resubmit. Any resubmission will receive a new submission date.

The Royal Society of Chemistry requires all submitting authors to provide their ORCID iD when they submit a revised manuscript. This is quick and easy to do as part of the revised manuscript submission process.   We will publish this information with the article, and you may choose to have your ORCID record updated automatically with details of the publication.

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I look forward to receiving your revised manuscript.

Yours sincerely,
Dr Kedar Hippalgaonkar
Associate Editor, Digital Discovery
Royal Society of Chemistry

************

Reviewer 1

In this work by Masaya Hagai et al. the authors present a "quantum-native" implementation of the neural network quantum state (NQS) approach of Carleo and Troyer (Science 2017).
The neural networks used in these types of approaches are often, but not exclusively, based on the (restricted) Boltzmann machine (BM) architecture; specifically, three architectures of restricted and non-restricted BMs are used in this work. By considering the specific form of the marginal probability distribution associated with Boltzmann machines, the authors propose an implementation that relies on the preparation of Gibbs states on a quantum computer. The parameters associated with the neural network are then optimised with an approach analogous to the variational quantum eigensolver (VQE).
In practice the method is equivalent to a VQE algorithm with a NQS ansatz for the wavefunction. Instead of using variational Monte Carlo to optimize the model parameters, a mixed quantum-classical approach is used that measures energy and gradients on the quantum computer and performs the optimization steps on classical hardware.

By considering the significant impact of NQS in the simulation of model systems and (more recently) realistic molecules it is certainly of great interest to exploit and to enhance this approach also in the context of quantum computing. This work represents a major effort in this direction and for this reason I support its publication in Digital Discovery.

On the practical side the preparation of Gibbs states on quantum computers is non-trivial and this affects the efficiency of this NQS-based approach as compared to traditional VQE ansatzes. The authors propose an algorithm to prepare the Gibbs state based on quantum phase estimation (QPE), which involves a significant overhead in terms of register and ancilla qubits and circuit depth. As a result, the overall methodology inherits some of the issues of VQE (e.g., the scaling with respect to the number of measurements) and the non-NISQ features of QPE. As mentioned in some of the detailed points below, the manuscript could be improved by extending the discussion on the Gibbs state preparation and by considering alternative approaches.

To improve the presentation in the manuscript, the authors should address these points:
-In a recent paper A. Warren and collaborators (https://arxiv.org/abs/2203.12757) proposed an approach to prepare Gibbs states that could be more suitable for short term devices. This algorithm is based on an ADAPT-type ansatz, and, if used in the context of the present work by Masaya Hagai et al., this might influence the expressivity of the quantum Boltzmann machine. Still, it would be interesting to discuss in the manuscript if variational approaches to prepare Gibbs state could help to make the NQS approach more NISQ friendly.
-In a similar way it would be interesting to discuss the pros and cons of other techniques to prepare Gibbs states, for example based on block encoding (see the work of Y. Tong et al. Phys. Rev. A 104, 032422).
-Optimization of the variational quantum eigensolver is known to be a challenging problem, as it could be affected by issues such as Barren plateaus. The authors should discuss if they expect the NQS ansatz to be more efficient than traditional approaches from the point of view of the parameter optimization.

Minor points:
-The part of the sentence “ …but is primarily weighted towards uncovering a direct route to connect the ML inspired theory and quantum computing.” seems misleading since ML and quantum computing are already connected in the QML field. To better express what the authors actually mean “ML inspired theory” could be replaced with “RBM-based NQS approach” or simply “NQS approach”.
-In the rather comprehensive reference list about ML approaches for wavefunctions of realistic systems in chemistry and materials science the authors have missed the recent work by B. Herzog et. al. https://arxiv.org/abs/2208.06708

Reviewer 2

I enjoyed reading this paper by Yanai and co-workers. This paper attempts to prepare an artificial neural network (ANN) quantum state on a quantum computer and optimize the ansatz by computing gradients from the quantum computer.

I have two main issues regarding the proposed approach, and I recommend rejection based on these two issues. If these can be addressed properly on resubmission, I expect this paper can be published:
(1) I don't know why we need to sample the wavefunction on the quantum computer to perform optimization of the boltzman machine ansatze. They can be efficiently optimized classically already, so what is the quantum computer offering here? Am I missing something?
(2) I think that without looking at the impact of noise this paper lacks urgency and impact. Quantum computing research for chemistry nowdays is the most impactful if one can show some noise-robustness of the algorithm.
The authors state that they are not testing the impact of noise in this paper, and this greatly reduces the impact of the paper.
I believe this alone makes this paper not yet publishable in this journal.

Reviewer 3

The paper has dealt with a challenging adaptation of artificial neural networks based on Boltzmann machine architectures to the approach for multiconfigurational many-electron wavefunctions in modern quantum chemical calculations. Indeed, this work has presented an extended formalism based on the authors’ cutting-edge work, as recently published in JCTC, to a quantum algorithm enabling the preparation of neural quantum states by exploiting quantum gates executable on a quantum computer, but as the authors clearly stated, their approach described in the paper is free from classical random sampling and can afford to form neural quantum states via quantum circuits and to compute relevant energies. Finding any quantum-native algorithm highly relevant to the suitability for machine learning based neural quantum state theories is practically significant in an emerging stage of quantum chemical calculations on quantum computers.

The MS has been written in an elaborate and practical manner, including detailed discussions on computational and intrinsic error analyses and those of the trained neural quantum states. Subsections 2.1 and 2.2 seem to be the highlights of the paper for readers of Digital Discovery, as well as Section 4, which gives the detailed results of the numerical simulations carried out for three carefully chosen chemical entities, scrutinizing the authors’ approach in comparison with those of QPE and UCCSD.

In the conclusion section, the authors have stated some concerns in their neural quantum state preparation algorithm, which they believe are not in the scope of the current research, and at the same time they rather importantly scrutinized relevant points already described in their previous work. The author should emphasize practical or intrinsic importance of their findings described in this work along with that of their achievements carried out in the previous work.

The MS has been intended to be prepared in American English, but there are some typos in terms of the language rule. And “behavior” is uncountable; convergence behaviors?

Referee: 1
Comments to the Author
In this work by Masaya Hagai et al. the authors present a "quantum-native" implementation of the neural network quantum state (NQS) approach of Carleo and Troyer (Science 2017).
The neural networks used in these types of approaches are often, but not exclusively, based on the (restricted) Boltzmann machine (BM) architecture; specifically, three architectures of restricted and non-restricted BMs are used in this work. By considering the specific form of the marginal probability distribution associated with Boltzmann machines, the authors propose an implementation that relies on the preparation of Gibbs states on a quantum computer. The parameters associated with the neural network are then optimised with an approach analogous to the variational quantum eigensolver (VQE).
In practice the method is equivalent to a VQE algorithm with a NQS ansatz for the wavefunction. Instead of using variational Monte Carlo to optimize the model parameters, a mixed quantum-classical approach is used that measures energy and gradients on the quantum computer and performs the optimization steps on classical hardware.

By considering the significant impact of NQS in the simulation of model systems and (more recently) realistic molecules it is certainly of great interest to exploit and to enhance this approach also in the context of quantum computing. This work represents a major effort in this direction and for this reason I support its publication in Digital Discovery.


[Authors’ reply]
Thank you very much for carefully reading our manuscript and understanding the value of our work.
-----


On the practical side the preparation of Gibbs states on quantum computers is non-trivial and this affects the efficiency of this NQS-based approach as compared to traditional VQE ansatzes. The authors propose an algorithm to prepare the Gibbs state based on quantum phase estimation (QPE), which involves a significant overhead in terms of register and ancilla qubits and circuit depth. As a result, the overall methodology inherits some of the issues of VQE (e.g., the scaling with respect to the number of measurements) and the non-NISQ features of QPE. As mentioned in some of the detailed points below, the manuscript could be improved by extending the discussion on the Gibbs state preparation and by considering alternative approaches.


[Authors’ reply]
We are highly grateful for your valuable and critical comments. We have added a line for acknowledging the reviewers’ constructive comments in the Acknowledgements section. They were indeed useful for improving the quality of our manuscript.
-----


To improve the presentation in the manuscript, the authors should address these points:
-In a recent paper A. Warren and collaborators (https://arxiv.org/abs/2203.12757) proposed an approach to prepare Gibbs states that could be more suitable for short term devices. This algorithm is based on an ADAPT-type ansatz, and, if used in the context of the present work by Masaya Hagai et al., this might influence the expressivity of the quantum Boltzmann machine. Still, it would be interesting to discuss in the manuscript if variational approaches to prepare Gibbs state could help to make the NQS approach more NISQ friendly.
-In a similar way it would be interesting to discuss the pros and cons of other techniques to prepare Gibbs states, for example based on block encoding (see the work of Y. Tong et al. Phys. Rev. A 104, 032422).


[Authors’ reply]
Many thanks for your insightful concerns about the relation to the studies of Warren et al. (arXiv:2203.12757) and Tong et al. (Phys. Rev. A 104, 032422). Considering this feedback, we have added a new section (Sec. 2.3) for detailed discussions on these studies and related work on Gibbs state preparation for quantum computation.

As mentioned in Sec. 2.3., our algorithm processes the preparation of a superposition state with the Gibbs distribution factor based on the classical function E(v;θ) of the BM model. On the other hand, the studies of Warren et al. [Ref. 61] and Tong et al. [Ref. 75] as well as many others [Refs. 31,32,53-60,62-70] deal with Gibbs state ρ=exp(-βH) for the Hamiltonian H of a quantum system for statistical simulation of the physical system in thermal equilibrium.

Note: We have rephrased the way of calling our prepared state from “Gibbs state” to “Gibbs distribution state” in order to distinguish it from the Gibbs state studied by Warren et al. and Tong et al. as well as several others [Refs. 31,32,53-70] for simulating quantum systems. (The corresponding revision was made in the first paragraph of Sec. 2.2.3)

Because in our Gibbs distribution model, there is no operator form of H directly representing the classical BM energy function, the use of the variational technique shown by Warren et al. and the block encoding by Tong et al. for our approach appear to be a nontrivial task. This aspect has been discussed in detail in Sec 2.3.

In the meantime, related to the variational treatment mentioned by the reviewer, it may be interesting to explore the adaptation of the variational Quantum Boltzmann machine (QBM) [Refs.68, 69] to our NQS-based electronic structure computation for future work. These algorithms are based on the variational quantum imaginary time evolution scheme [Refs. 65, 66], and reportedly compatible with the NISQ computers. We detailed the studies of the variational QBM, mentioning the possibility of using it as an underlying Gibbs state preparation algorithm for deriving an alternative molecular wavefunction solver.
-----

-Optimization of the variational quantum eigensolver is known to be a challenging problem, as it could be affected by issues such as Barren plateaus. The authors should discuss if they expect the NQS ansatz to be more efficient than traditional approaches from the point of view of the parameter optimization.


[Authors’ reply]
In our experience, the use of the NQS still suffers from convergence issues analogous to the problem of the Barren plateau in the VQE optimization. These issues were already observed in our previous work [Yang et al J. Chem. Theory Comput. 16, 3513 (2002)]. Our previous study mitigated this issue to a certain degree via the two-step optimization: The first hundred learning iterations only optimize the phase parameters τ with the amplitude parameters θ fixed with the initial random values, and the rest of the iterations undergo the optimization of both θ and τ. We again used this scheme in the present work. In addition, it is important to check the convergence using different random seeds. We have cited Refs. 87 and 88, which studied Barren plateaus for the QBM.

The above descriptions have been added in the second paragraph of Sec. 4.2.
-----


Minor points:
-The part of the sentence “ …but is primarily weighted towards uncovering a direct route to connect the ML inspired theory and quantum computing.” seems misleading since ML and quantum computing are already connected in the QML field. To better express what the authors actually mean “ML inspired theory” could be replaced with “RBM-based NQS approach” or simply “NQS approach”.


[Authors’ reply]
Thank you for pointing out the misleading expression. We have accordingly rephrased the corresponding sentence.
-----


-In the rather comprehensive reference list about ML approaches for wavefunctions of realistic systems in chemistry and materials science the authors have missed the recent work by B. Herzog et. al. https://arxiv.org/abs/2208.06708


[Authors’ reply]
Thanks for bringing this reference to our attention. We accordingly cited recent work by B. Herzog et al. https://arxiv.org/abs/2208.06708 (Ref. 24) as a study relevant to developing ML-inspired wavefunction algorithms.
-----



Referee: 2
Comments to the Author
I enjoyed reading this paper by Yanai and co-workers. This paper attempts to prepare an artificial neural network (ANN) quantum state on a quantum computer and optimize the ansatz by computing gradients from the quantum computer.


[Authors’ reply]
Thank you very much for reading our paper and enjoying it. We have considered your critical concerns very seriously, putting significant effort into addressing them.
-----


I have two main issues regarding the proposed approach, and I recommend rejection based on these two issues. If these can be addressed properly on resubmission, I expect this paper can be published:
(1) I don't know why we need to sample the wavefunction on the quantum computer to perform optimization of the boltzman machine ansatze. They can be efficiently optimized classically already, so what is the quantum computer offering here? Am I missing something?


[Authors’ reply]
We thank you for pointing out what was unclear in our paper. In the introduction of the previous revision, we attempted to highlight the importance of resorting to the quantum algorithm-based sampling of the BM wavefunction. Our interest is in using the BM ansatz to describe the multireference (MR) (or strongly-correlated/multiconfigurational) wavefunctions of quantum chemical systems. Our approach is based on the CAS-CI framework, which is a typical approach to describe the MR wavefunction in which the multiple electronic configurations that should be considered in the wavefunction construction play a critical role on an equal footing. What is critical is that the dimension of the configuration (or Hilbert) space grows exponentially; for example, CAS(2e,2o) has 16 (=24) configurations, CAS(4e,4o), CAS(6e,6o), CAS(8e,8o) amount to 256, 4096, 65536 configurations, respectively; in general, the dimension of CAS(ke,ko) is 2^{2k}.

As is well known, this exponential-growing complexity needs to be faced when studying many important chemical phenomena; for example, multiple bonding-breaking (e.g., N2 bond breaking as a simplest case shown in our previous study [Yang et al, JCTC 2020]), multi-radical systems (e.g., the emergence of diradical demonstrated in Sec. 4.3), transition metal complexes.

In the classical approach to BM-based NQS, as was also done in our previous study [Yang et al, JCTC 2020], the sampling for this exponentially-growing Hilbert space is performed using the Markkov chain Monte Carlo (MCMC) technique. In our experience, this classical MCMC-based integration was found to be the most computationally demanding due to the aforementioned intrinsic quantum complexity of the sampled CAS-CI configurations. This quantum complexity towards the sampling to cover a large number of CAS-CI configurations is generally known to pose a fundamentally hard problem.

The sophisticated GPGPU-based implementation may be a promising classical approach to accelerate the MCMC step [Ref 16-19] by a notable speedup factor. However, this factor delivered by the classical treatment cannot be exponential. Ref. 16-18 studied bicyclobutane and cyclobutadiene as the largest cases, whose full valence π active-space treatments correspond to CAS(6e,6o) and CAS(4e,4o). The molecules studied in Ref 19 are systems with a minor static correlation effect. These studies may focus on the computation of dynamic (or weak/perturbative) correlation associated with high-energy virtual orbital space; note that the complexity of perturbative electron correlation is considered not exponential but polynomial.

Therefore, it is still important to propose an alternative approach to sample the BM-based NQS wavefunction with a focus on the case where the BM is used to describe the MR (or static) electron correlation. In this study, we proposed a fully-quantum algorithm to prepare (or sample) the BM state to possibly offer an efficient route to calculate the MR electronic state with this BM model. This is our answer to address the reviewer’s question “what is the quantum computer offering here?”. Our aim is the exponential speedup offered by the anticipated quantum advantage of a quantum computer, but of course this quantum advantage or supremacy relies on the existence of quantum computers, as noted by a review article by Biamonte et al Nature (2017). In an exactly similar spirit, the adaptation of the BM-based NQS to the quantum algorithm was studied by Ref. 14 [Xia and Kais, Nature Comm. (2018)] and Ref. 50 [Torlai et al. Phys. Rev. Res. (2020)], which also aimed at addressing the quantum complexity issue for MR systems. As stated in this introduction, Refs 14 and 50 handle part of the processes via the classical sampling of the intermediate states prepared on a quantum computer.

To more clearly describe what is offered using the quantum computer in this study, we have added the following lines in the Introduction:

- [the 3rd paragraph] In our anstaz, the number of electron configurations considered in the CAS-CI scheme is formally written as 2^{2k}, where k is the number of spin-orbitals; in the MR electronic structure cases, these numerous configurations can individually play major roles in the wavefunction construction.

- [the 4th paragraph] With advanced implementations, Ref. 16-18 studied bicyclobutane and cyclobutadiene as the largest cases, which involve six π electrons in six π orbitals and four π electrons in four π orbitals. The molecules studied in Ref 19 are systems with a minor static correlation effect. All of these studies are thought of as focusing on the computation of dynamic (or weak/perturbative) correlation using high-energy virtual orbital space associated with dynamic correlation effect; note that the complexity of perturbative electron correlation is considered not exponential but polynomial.

-----


(2) I think that without looking at the impact of noise this paper lacks urgency and impact. Quantum computing research for chemistry nowdays is the most impactful if one can show some noise-robustness of the algorithm.
The authors state that they are not testing the impact of noise in this paper, and this greatly reduces the impact of the paper.
I believe this alone makes this paper not yet publishable in this journal.


[Authors’ reply]
Thank the reviewer for this comment. We have added an appendix to discuss an additional test result to elucidate the impact of the noise on the state preparation. Ref. 65 conducted noise resilience tests using a 10^{-4} error rate per gate and 10^2 to 10^6 shot noise. We performed tests with similar noise settings, showing that the average error in the predicted E with the prepared state is robust against the shot noise. In addition, the behavior of the standard deviation of the noise errors was analyzed with respect to various shot and gate noises.
-----


Referee: 3
Comments to the Author
The paper has dealt with a challenging adaptation of artificial neural networks based on Boltzmann machine architectures to the approach for multiconfigurational many-electron wavefunctions in modern quantum chemical calculations. Indeed, this work has presented an extended formalism based on the authors’ cutting-edge work, as recently published in JCTC, to a quantum algorithm enabling the preparation of neural quantum states by exploiting quantum gates executable on a quantum computer, but as the authors clearly stated, their approach described in the paper is free from classical random sampling and can afford to form neural quantum states via quantum circuits and to compute relevant energies. Finding any quantum-native algorithm highly relevant to the suitability for machine learning based neural quantum state theories is practically significant in an emerging stage of quantum chemical calculations on quantum computers.

The MS has been written in an elaborate and practical manner, including detailed discussions on computational and intrinsic error analyses and those of the trained neural quantum states. Subsections 2.1 and 2.2 seem to be the highlights of the paper for readers of Digital Discovery, as well as Section 4, which gives the detailed results of the numerical simulations carried out for three carefully chosen chemical entities, scrutinizing the authors’ approach in comparison with those of QPE and UCCSD.

In the conclusion section, the authors have stated some concerns in their neural quantum state preparation algorithm, which they believe are not in the scope of the current research, and at the same time they rather importantly scrutinized relevant points already described in their previous work.


[Authors’ reply]
We thank the reviewer for carefully reading our manuscript and giving us positive and encouraging comments. We appreciate that the reviewer shares our viewpoint in terms of our attitude to scrutinize the presented algorithm prudently.
-----


The author should emphasize practical or intrinsic importance of their findings described in this work along with that of their achievements carried out in the previous work.


[Authors’ reply]
Thank you for the encouraging suggestion. We have accordingly modified the last paragraph of the conclusion section slightly to put more emphasis on the importance of our findings.
-----


The MS has been intended to be prepared in American English, but there are some typos in terms of the language rule. And “behavior” is uncountable; convergence behaviors?


[Authors’ reply]
Thank you for this comment. We have replaced “behaviors” with “behavior.” The English editing service proofread our manuscript but seemed to overlook this error. We again checked the misspellings with great care.
-----

09-Feb-2023

Dear Dr Yanai:

Manuscript ID: DD-ART-09-2022-000093.R1
TITLE: Artificial Neural Network Encoding of Molecular Wavefunctions for Quantum Computing

Thank you for your submission to Digital Discovery, published by the Royal Society of Chemistry. I sent your manuscript to reviewers and I have now received their reports which are copied below.

I have carefully evaluated your manuscript and the reviewers’ reports, and the reports indicate that major revisions addressing the first reviewer's queries are necessary. In addition, the third reviewer had this comment, which I am inclined to agree with - if this is helpful for your revision, please include the references in the manuscript:

"I disagree with a comment of the second reviewer: "I think that without looking at the impact of noise this paper lacks urgency and impact. Quantum computing research for chemistry nowdays is the most impactful if one can show some noise- robustness of the algorithm." I would like to point out that there is a growing interest in fault tolerant quantum algorithms also in quantum chemistry. This is also related to the growing concern that quantum advantage in chemistry or other fields will not be possible with NISQ era devices. See for example:
-https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.041015
-https://journals.aps.org/prxquantum/pdf/10.1103/PRXQuantum.2.030305
These papers are very impactful and many more are published in the field of fault tolerant simulations for quantum chemistry."

Please submit a revised manuscript which addresses the reviewers’ comments. Further peer review of your revised manuscript may be needed. When you submit your revised manuscript please include a point by point response to the reviewers’ comments and highlight the changes you have made. Full details of the files you need to submit are listed at the end of this email.

Digital Discovery strongly encourages authors of research articles to include an ‘Author contributions’ section in their manuscript, for publication in the final article. This should appear immediately above the ‘Conflict of interest’ and ‘Acknowledgement’ sections. I strongly recommend you use CRediT (the Contributor Roles Taxonomy, https://credit.niso.org/) for standardised contribution descriptions. All authors should have agreed to their individual contributions ahead of submission and these should accurately reflect contributions to the work. Please refer to our general author guidelines https://www.rsc.org/journals-books-databases/author-and-reviewer-hub/authors-information/responsibilities/ for more information.

Please submit your revised manuscript as soon as possible using this link:

*** PLEASE NOTE: This is a two-step process. After clicking on the link, you will be directed to a webpage to confirm. ***

https://mc.manuscriptcentral.com/dd?link_removed

(This link goes straight to your account, without the need to log on to the system. For your account security you should not share this link with others.)

Alternatively, you can login to your account (https://mc.manuscriptcentral.com/dd) where you will need your case-sensitive USER ID and password.

You should submit your revised manuscript as soon as possible; please note you will receive a series of automatic reminders. If your revisions will take a significant length of time, please contact me. If I do not hear from you, I may withdraw your manuscript from consideration and you will have to resubmit. Any resubmission will receive a new submission date.

The Royal Society of Chemistry requires all submitting authors to provide their ORCID iD when they submit a revised manuscript. This is quick and easy to do as part of the revised manuscript submission process.   We will publish this information with the article, and you may choose to have your ORCID record updated automatically with details of the publication.

Please also encourage your co-authors to sign up for their own ORCID account and associate it with their account on our manuscript submission system. For further information see: https://www.rsc.org/journals-books-databases/journal-authors-reviewers/processes-policies/#attribution-id

I look forward to receiving your revised manuscript.

Yours sincerely,
Dr Kedar Hippalgaonkar
Associate Editor, Digital Discovery
Royal Society of Chemistry

************

Reviewer 2

(1) I had to skim Kais's Nat. Comm. and Wiebe's paper on training RBM on the quantum computer to see the actual benefit of why one would train RBM on the quantum computer.

I don't think the speedup is exponential, although the authors seem to imply it in their response. Can the authors clarify the benefit of training the RBM with the quantum?

I think that the very point of this research article is not clear to me, so perhaps this is an important comment for the authors to address.

(2) The authors added "In our anstaz, the number of electron configurations considered
in the CAS-CI scheme is formally written $2^{2k}$, where k is the number of spin-orbitals"
It should be 2^k since k is the number of spin orbitals, not spatial orbitals.

(3) Based on the Appendix, the gate error rate the authors need to achieve ~ 1mH accuracy is 10^-5 or better. This is hardly NISQ-friendly, and therefore, I don't think we should call this a "NISQ" algorithm. For instance, besides many places in the manuscript, Table 3 should not have "NISQ" listed for BM2-based NISQ.

(4) I think the discussion of noise should be put up front in the main text. This is an issue for the method, and it is unrealistic to expect good accuracy without further error mitigation. I would suggest that the authors perform some error mitigation if the authors want to present this approach as NISQ-friendly.

Reviewer 3

The authors have responded to the comments and revised their previous MS in a scrutinizing manner except the language rules of American English, in spite of the authors’ statement, “The English editing service proofread our manuscript but seemed to overlook this error. We again checked the misspellings with great care”. There are still some typos, e.g. towards in the British English spelling. They will be corrected during editing procedures.

Reviewer 1

The authors well replied to the criticism raised by the other reviewers and myself. The revised version of the manuscript is ready for publication.

Dear Professor Hippalgaonkar,

We are most thankful to you for handling our manuscript (Manuscript ID: DD-ART-09-2022-000093) entitled “Artificial Neural Network Encoding of Molecular Wavefunctions for Quantum Computing” and for giving us an opportunity for revision. We would also like to thank the reviewers for checking the previous revision.

We have revised the manuscript according to the comments and suggestions from Referee 2. The detailed answers to reviewers’ comments have been included in the 'Your Response' text box. We also uploaded a supplementary file highlighting all the changes in the manuscript in red (DiffsMS.pdf).

We hope you find the revised manuscript suitable for publication in Digital Discovery and are looking forward to your reply.

Yours sincerely,
Takeshi Yanai

-------------------------------------------------------

Referee: 2

Comments to the Author
(1) I had to skim Kais's Nat. Comm. and Wiebe's paper on training RBM on the quantum computer to see the actual benefit of why one would train RBM on the quantum computer.

I don't think the speedup is exponential, although the authors seem to imply it in their response. Can the authors clarify the benefit of training the RBM with the quantum?

I think that the very point of this research article is not clear to me, so perhaps this is an important comment for the authors to address.

[Authors’ reply]
-----
Thank you very much. Our approach is a hybrid quantum-classical approach. The training of RBM -- optimizing neural network parameters -- is done classically with a polynomial complexity as similarly done in conventional BM computation, while with quantum, our quantum step carries out the evaluation of the energy and gradients via the state preparation of the NQS. Because our quantum process step is fully formulated with the quantum circuit, it allows for the evaluation of the energy and gradients -- which requires the Hilbert space sampling -- using a direct quantum representation of the NQS. This quantum process for the ensemble sampling may thus offer a speedup, which is a potentially strong benefit compared to the classical sampling approach, as anticipated in quantum computing research for quantum chemistry calculation.
-----


(2) The authors added "In our anstaz, the number of electron configurations considered
in the CAS-CI scheme is formally written $2^{2k}$, where k is the number of spin-orbitals"
It should be 2^k since k is the number of spin orbitals, not spatial orbitals.

[Authors’ reply]
-----
Thanks. The error has been corrected accordingly.
-----


(3) Based on the Appendix, the gate error rate the authors need to achieve ~ 1mH accuracy is 10^-5 or better. This is hardly NISQ-friendly, and therefore, I don't think we should call this a "NISQ" algorithm. For instance, besides many places in the manuscript, Table 3 should not have "NISQ" listed for BM2-based NISQ.

[Authors’ reply]
-----
We have eliminated the row of Table 3 regarding "Target device" to avoid indicating that BM2 is NISQ-friendly.
In addition, the line "If N^QAA is sufficiently small, the BM2 may be comparable to or even advantageous over the UCCSD in terms of complexity." has been removed.
Also, we have mentioned a recently-highlighted trend of the research on fault-tolerant quantum computation for molecular electronic structure calculation (Refs 50 and 51). This new trend indicates that the advent of fault-tolerant technologies is expected to address noise errors. We have mentioned recent research on fault-tolerant quantum computation for quantum chemistry in Introduction and Section 4.2.
-----


(4) I think the discussion of noise should be put up front in the main text. This is an issue for the method, and it is unrealistic to expect good accuracy without further error mitigation. I would suggest that the authors perform some error mitigation if the authors want to present this approach as NISQ-friendly.

[Authors’ reply]
-----
The discussion of noise has been moved from Appendix to the new Section 4.2 in the main text.
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Referee: 3

Comments to the Author
The authors have responded to the comments and revised their previous MS in a scrutinizing manner except the language rules of American English, in spite of the authors’ statement, “The English editing service proofread our manuscript but seemed to overlook this error. We again checked the misspellings with great care”. There are still some typos, e.g. towards in the British English spelling. They will be corrected during editing procedures.

[Authors’ reply]
-----
We appreciate your comments.
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Referee: 1

Comments to the Author
The authors well replied to the criticism raised by the other reviewers and myself. The revised version of the manuscript is ready for publication.

[Authors’ reply]
-----
Thank you very much.
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09-Mar-2023

Dear Dr Yanai:

Manuscript ID: DD-ART-09-2022-000093.R2
TITLE: Artificial Neural Network Encoding of Molecular Wavefunctions for Quantum Computing

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Reviewer 2

I thank the authors for addressing my comments. I am happy with the changes.