Large-Scale Simulation of Shor’s Quantum Factoring Algorithm

Round 1

Reviewer 1 Report

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Author Response

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Reviewer 2 Report

The manuscript under review presents a large-scale simulation of Shor’s algorithm using 2048GPU-based supercomputers, which simulate a 40-qubit quantum computer. By investigating the “lucky” case and the post-processing procedure, the size of the factoring problem using the simulation software is far beyond what has been achieved previously. The results presented in this paper are interesting and could inspire confidence in the quantum factoring procedure. The paper is well-written in a detailed and accessible style.

I just have some suggestions:

1.     Fig1 in page 2, the “k/r” in the figure, I don’t find any definition about “k”, and I suggest the author define it properly.

I think the main idea of “shorgpu” is to simulate a quantum computer using thousands of classical GPUs, and then it can be used to deal with factoring problem through Shor’s algorithm. I’m curious about the capability of other classical factoring algorithms using the same amount of GPUs. Could the author make a comparison?

Author Response

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Reviewer 3 Report

Review (Round 1) on the manuscript "Large-scale simulation of Shor's quantum factoring algorithm" with ID mathematics-2601999

In general, the manuscript is interesting. Please see the following items. 

1. It is recommended to start the section "1. Introduction" with some short comment about some classical (non-quantum) algorithms for the problem of natural numbers factorization, e.g., Euler's factorization algorithm, etc. In this regard, for example, the following references are recommended to add.

(a) Sherman Lehman, R. Factoring Large Integers. {\em Math. Comput.}. {\bf 1974}, {\em 28}, 637--646. \url{https://doi.org/10.2307/2005940}

(b) Bressoud, D.M. {\em Factorization and Primality Testing}; Springer: New York, NY, 1989. \url{https://doi.org/10.1007/978-1-4612-4544-5}

2. It is recommended to take into account in References and Introduction the following article:

 

Li, Jun; Peng, Xinhua; Du, Jiangfeng; Suter, Dieter. An efficient exact quantum algorithm for the integer square-free decomposition problem. {\em Sci. Rep.} {\bf 2012}, {\em 2}, 260. \url{https://doi.org/10.1038/srep00260} 

3. The manuscript (including References) needs to be formatted according to the style of Mathematics. See some examples from https://www.mdpi.com/journal/mathematics and the latex template from  https://www.mdpi.com/authors/latex

4. It is recommended to make some list of abbreviations (including GPU, gcd, RSA, NISQ, etc.) above References.  See the latex template in this regard.

5. In a number of places in your manuscript, including the 8th page, I see "2048 GPUs". At the same time, the 8th page contains "a GPU cluster with 3744 NVIDIA A100 Tensor Core GPUs". In this regard, it is recommended to write clearly in the manuscript, why firstly you write about 2048 GPUs and after that you write about 3744 GPUs. Moreover, it is recommended to write in this manuscript which characteristics have these 2048 GPUs.  

6. The 85th item in References contains the short URL https://doi.org/10.1142/8334 and the URL  https://www.worldscientific.com/doi/pdf/10.1142/8334 . It is recommended to use the shorter URL https://doi.org/10.1142/8334 . Moreover, at the end of the 85th item, I see such the hyperlink that its text is https://www.worldscientific.com/doi/pdf/10.1142/8334 and its URL is https://arxiv.org/abs/https://www.worldscientific.com/doi/pdf/10.1142/8334 which is something wrong.  In this regard, check References.

Author Response

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Round 2

Reviewer 3 Report

Review (Round 2) on the manuscript "Large-scale simulation of Shor's quantum factoring algorithm" with ID mathematics-2601999

1. The 23rd item of References represents the book "Quantum Computation and Quantum Information: 10th Anniversary Edition" by Nielsen, M.A. and Chuang, I.L. If you know the following books, then it is recommended to take them also into account in References and above.

(a) Holevo, A.S. {\em Quantum Systems, Channels, Information: A~mathematical Introduction}; 2nd Rev. and Expanded Ed.: De Gruyter, Berlin, Boston, 2019. \url{https://doi.org/10.1515/9783110642490}

(b) Wilde, M.M. {\em Quantum Information Theory}, 2nd Ed.; Cambridge: Cambridge Univ. Press, 2017. \url{https://doi.org/10.1017/9781316809976} 

2. Page 12, line 392. See the fragment "than expected: In Figure".

Page 14. See the caption for the Fig. 5. This caption contains both the fragments "Shor’s algorithm" and "Shor algorithm". Write them uniformly. In the line 433 (page 14), see "of Shor’s algorithm. For the conventional Shor algorithm". Please check in this regard the whole manuscript.

Author Response

Reply to Referee #3 (Round 2):

1. The 23rd item of References represents the book "Quantum Computation and Quantum Information: 10th Anniversary Edition" by Nielsen, M.A. and Chuang, I.L. If you know the following books, then it is recommended to take them also into account in References and above.

(a) Holevo, A.S. {\em Quantum Systems, Channels, Information: A~mathematical Introduction}; 2nd Rev. and Expanded Ed.: De Gruyter, Berlin, Boston, 2019. \url{https://doi.org/10.1515/9783110642490}

(b) Wilde, M.M. {\em Quantum Information Theory}, 2nd Ed.; Cambridge: Cambridge Univ. Press, 2017. \url{https://doi.org/10.1017/9781316809976} 

 

We thank the Referee for providing these additional references. We have added them to the part in Section 2.8.2 in which we define quantum channels and quantum operations.

 

2. Page 12, line 392. See the fragment "than expected: In Figure".

Page 14. See the caption for the Fig. 5. This caption contains both the fragments "Shor’s algorithm" and "Shor algorithm". Write them uniformly. In the line 433 (page 14), see "of Shor’s algorithm. For the conventional Shor algorithm". Please check in this regard the whole manuscript.

 

We see the Referee’s point. Regarding the two forms "Shor’s algorithm" and "the <adjective> Shor algorithm", we have carefully checked the whole manuscript and have consistently chosen the appropriate form in the appropriate context. In particular, the possesive form with apostrophe and without definite article is used to refer to factoring algorithms by Peter Shor in a general context. The second form with definite article and an additional adjective, "the <adjective> Shor algorithm", refers to one of two particular implementations of the algorithm, specified by the adjective "conventional" or "iterative".

Author Response File: Author Response.pdf