PONE-D-23-25182Novel real number representations in Ising machines and performance evaluation: Combinatorial random number sum and constant divisionPLOS ONE

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   This work was supported by JSPS KAKENHI (Grant Numbers JP21K03391, JP23H05447) and JST Grant Number JPMJPF2221. The Human Biology-Microbiome-Quantum Research Center (Bio2Q) is supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan. 

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This work was supported by JSPS KAKENHI (Grant Numbers JP21K03391, JP23H05447) and JST Grant Number JPMJPF2221. The Human Biology-Microbiome-Quantum Research Center (Bio2Q) is supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan.

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This work was supported by JSPS KAKENHI (Grant Numbers JP21K03391, JP23H05447) and JST Grant Number JPMJPF2221. The Human Biology-Microbiome-Quantum Research Center (Bio2Q) is supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan.

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Reviewer #1: Yes

Reviewer #2: Partly

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: No

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Reviewer #1: Yes

Reviewer #2: No

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Reviewer #2: No

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5. Review Comments to the Author

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Reviewer #1: Changes in representation for search problems has a history going back to the 1980s. The 'bumps' in the binary representation referred to by the authors, for instance, are known in this literature as 'Hamming cliffs'. The authors make no mention of this prior work.

A few key results in this literature are:

1. The 'no free lunch theorem' [Wolpert and Macready, IEEE Trans. Evol. Comp., 1(1), 67–82, 1997] and special-case corollaries [Whitley, Proc. 1st Ann. Conf. Gen. and Evo. Comp., 726–733, 1999].

2. A link between the advantage of the incremental shifts in a Gray or real representation and local correlations in the objective function ('schema similarity') [Caruana and Schaffer, Mach. Learn. Proc., 153–161, 1988].

3. Identification that the behaviour of a function is often highly scale dependent [Goldberg, Complex Systems, 5, 139–167, 1991], with search (often) rapidly approaching a regime in which Gray-type encoding is profitable, and the associated use of hybrid schemes involving dynamic representations, multi-step evolution, and redundant bits.

The representations proposed by the authors, including the randomised encodings, are similar to those considered in [Cheong et al., IEEE Symp. Found. Comp. Int., 251–258, 2007].

Now, with the above caveats, those aspects of the representation problem specific to the QUBO restriction and so to quantum annealing do remain, to my knowledge, largely unexplored. For instance, do Gray codes necessarily introduce higher order non-linearities that are disallowed in this setting? There appears to be room for discussion here, and I believe the results of the authors could serve to open this discussion, but some acknowledgement of prior, related work is necessary.

Otherwise, I believe the analysis in the paper is understandable and credible. The authors might consider slight revision to clarify their error analysis and to introduce some estimates of sample deviation in their plots.

Reviewer #2: Overall, this manuscript has many significant flaws, both in the presentation of the information, and the actual experimental details. This manuscript requires extremely extensive revision and justification of the proposed methods to be suitable for publication, and without significant improvement I would recommend rejection of the paper.

# General Notes

1. Figures 2, 3, 4, 5 do not have axis labels! They absolutely need axis labels for the figures to be understandable.

2. The primary focus of the paper is around what the authors refer to as "representation of real numbers". I find this particular phrase extremely confusing. The real numbers that are of interest here are the cost function values for discrete combinatorial optimization problems. First, they do not need to be high precision real numbers - they can be integers. But more importantly, I do not think that the proposed methods are representing real numbers in a different way, I think they are representing cost functions in a different way. Which is fine, assuming the cost functions are correct, but this term is very confusing.

3. Following from the previous point, lines 38-41 state that "we noticed that when real numbers are represented in binary number, there are numbers that can only be reached by inverting several bits simultaneously under the restriction of not increasing a given Hamiltonian, which makes the optimization very difficult.". I do not understand what is being said here -- what does it mean that real numbers are represented in "binary number". For optimization, there is a cost function that defines a mapping from a set of variables to some cost output, which is a real number. But, that does not mean that the real numbers are represented in binary, rather the binary variables have some cost defined by them depending on what cost function you care about. Also, why is the restriction of fewer bitflips relevant? Especially for quantum annealing, early on the anneal when the transverse field is dominating the states of the qubits are flipping very often. Additionally, bit assignments to get to a lower energy state do not necesarely need to be inverted all at the "same time".

4. The authors state that experiments were performed on D-Wave quantum annealers, and "Fixstars Amplify Annealing Engine". First, the exact names (e.g. chip ids in the case of the D-Wave quantum annealers) needs to be specified. Second, the technology of the devices needs to be described. For example, were any hybrid quantum annealing solvers used? What what is the technology behind "Fixstars Amplify Annealing Engine"? Much of the paper descrbed quantum annealing technology, but is "Fixstars Amplify Annealing Engine" a digital annealer, simulated annealing, some other type of classical optimization technique (and who is the manufacturer)? Third, the solver parameters used need to be very explicitely stated for all machines - for example, for the D-Wave quantum annealers was minor-embedding used, what annealing time was used, if minor embeddings were used what was the structure and number of variables, and what chain break resolution algorithm was used?

5. Line 151-152 states that "Since the longer the computation time, the better the expected solution, the timeout time". Well, this is not strictly true for all problem types in the case of quantum annealing.

6. Something to note is that D-Wave quantum annealers have limited precision for programming the coefficient weights on the hardware, so the uniform random coefficient choices will mean that the quantum annealers are unlikely to find the global ground state, especially if minor embedding was used (I assume it was).

7. Is equation 5 applied iteratively in order to compute better and better x solutions?

8. What is the matrix size that corresponds to the varying number of variables (line 237)? I assume it is the square root of N

9. Is q the decision variable in equations 3 and 4?

10. Line 147 and equation 5; I would like to see a more explicit example of one of these QUBOs -- I think that almost always QUBOs constructed for matrix factorization will yield higher order terms.

11. What are the "real variables x" in line 144? Are these the decision variables? or is x the coefficient of the decision variables?

12. i is not defined in equation 5

13. Line 139; what is {F_i, r} ? Are these two distinct real numbers being multiplied?

14. Critically, the central procedure that this paper describes is completely not clear. Equation 4 appears to be a polynomial containing only linear terms, with decision variables q. It is not clear how the four different "real number representations" are actually applied to the problem QUBO. To me, Equations 7, 8, 9, and 10 appear to just be four different ways of generating coefficients for the linear terms in Equation 4 -- it is completely unclear whether the generated QUBOs (all of which are, it seems, general integer matrix factorization QUBOs) using these four "real number representations" are actually equivalent (e.g. do the QUBOs/ Ising models have the same eigenspectrum, or at least the same ground states). This comparison only works (the central argument of the paper) if the four "real number representation" methods are actually producing equivalent QUBOs.

15. Line 166; If I understand correctly, the selected matrix coefficients are positive random, real numbers (up to some precision). If that is correct, then I am very curious what the exact QUBO formulation is (e.g. item 10), because these generated QUBOs would, I think, be very complex in terms of range of coefficients and total number of variables. And, it seems like some matrix entries are also chosen which are negative - which makes me even more curious what the exact QUBO formulation is, since I am not sure that matrix factorization with negative coefficients can be formulated as easily as is shown in eq. 5.

16. I suppose I would consider this a small comment, but it would be of interest to sample the problem QUBOs using classical heuristics as well, such as simulated annealing, steepest gradient descent, etc.

# Grammar / Spelling / clarification Notes

1. There are numerous grammatical errors throughout the maniscript that would need to be corrected before the paper is suitable for publication. Here are a few examples (this list is not comprehensive, and there are many more errors that need to be corrected):

1. Line 32. A potential fix would be to pluralize the phrase "quantum annealing machine"

2. Line 39. A potential fix would be "represented in their binary form"

3. Lines 56, 57, 58 do not make any sense - what does it mean to replace the objective function with the energy of the system? In quantum annealing specifically, the energy is the expectation value of the objective function, encoded using analog computation.

4. Lines 60, 61, 62 state that: "The principle of operation is to find a solution to the variables by adjusting quantum fluctuations so that the objective function eventually becomes the global minimum". This is not accurate, and also does not make sense - what does it mean for the objective function to become the global minimum? The objective function just tells us what the cost value, or energy, or expectation value is of the objective function for a specific set of spin (or variable) states.

5. Line 66 states that "Many differential equations are derived from the energy functional based on the principle of least action" - what does differential equations implicately say about why quantum annealing makes sense as an analog computation for physical systems? This reasoning about why quantum annealing may be useful as an analog computation is not a correct summarization of how quantum annealing works.

6. Lines 62 and 63: "Since the Ising model is used as the basis, binary variables are used" - this is incorrect, an ising model corresponds to spin variable states not binary variables.

7. Line 151: "the minimum Hamiltonian is explored using an Ising machine" I think the authors meant to say that the minimum variable assignment is computed using an Ising machine.

# Literature references

There are several papers relevant to the topic of solving (or obtaining good solutions) types of linear algebra problems using quantum algorithms, including quantum annealing, that the authors do not cite or discuss.

1. https://arxiv.org/abs/2206.10576

2. https://arxiv.org/abs/1704.01605

3. https://arxiv.org/abs/2007.05565

4. https://arxiv.org/abs/2205.00645

Note that the generalization of matrix factorization is tensor factorization, the boolean version of which has also been investigated in the context of quantum annealing:

1. https://arxiv.org/abs/2107.13659

2. https://arxiv.org/abs/2103.07399

3. https://ieeexplore.ieee.org/document/9325388

The authors make some numerous observations about bit-flipping of some variable states on the problem Hamiltonians, including simultaneous bit flipping. This paper which investigates this topic in quantum annealing may be of interest: https://arxiv.org/abs/2210.16513

Lastly, it seems to me that this manuscript falls somewhat into the category of problem Hamiltonian encoding, and there are a couple of papers on ideas for other types of encodings:

1. https://arxiv.org/abs/1903.05068

2. https://ieeexplore.ieee.org/abstract/document/9951263/

3. https://arxiv.org/abs/2108.12004

4. https://arxiv.org/abs/2102.12224

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Reviewer #1: No

Reviewer #2: Yes: Elijah Pelofske

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PONE-D-23-25182R1Novel real number representations in Ising machines and performance evaluation: Combinatorial random number sum and constant divisionPLOS ONE

Dear Dr. Muramatsu,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

 One of the reviewers has indicated that while addressing most of their comments further revision is required. 

Please submit your revised manuscript by Apr 18 2024 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

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Reviewers' comments:

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Reviewer #2: (No Response)

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The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #2: Partly

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Reviewer #2: Yes

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Reviewer #2: Yes

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Reviewer #2: No

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6. Review Comments to the Author

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Reviewer #2: # Follow up from my previous comments

I believe that the authors have, with great detail, addressed the majority of the questions and comments -- and I think that the current manuscript is significantly clearer. The clarifications on the real number representations make much more sense now, and the descriptions of quantum annealing theory and implementation are much clearer.

- For the literature references I suggested, to be clear the reason I mentioned matrix factorization papers is because they are solving types of linear equations, granted typically the underlying decision variables are spins or boolean variables.

- I do not think the authors need to perform such simulations, but I mentioned steepest gradient descent as a possible additional classical heuristic - and it is true that gradient descent does not actually apply for these problems, but there does exist discrete steepest descent solvers: https://docs.ocean.dwavesys.com/projects/greedy/en/latest/

- Additionally, now that the authors have clarified that the GPU based simulator was in fact running simulated annealing, I am satisfied with this comparison.

# Comments on the revised manuscript

1. An overall comment on the motivation and experimental description of the manuscript is that simulated annealing, and in general classical heuristics, are not constrained to perform only Hamming distance 1 (e.g. single bit-flip) local neighborhood searches. In particular, statements such as Line 351 where a single bit flip example is used, or Line 420: "bit-flip-based search, which mimics the process of simulated annealing" are not true that this type of search would mimic simulated annealing.

2. Line 84: "For example, consider the case where a single real number is optimized in a classical simulator": This example is unclear and should to be clarified -- what is being optimized? Is the real number the cost function?

3. Line 637: "The error bars denote the standard error of the mean errors." I do not understand this measure -- do the authors intend to say that the errors bars denote the standard deviation of the distribution of the error measure for the repeated trials?

4. Line 357: "Then, we search for the state that is closest to the target value, limiting the path to only repeated transitions to states with decreasing distances that can be reached by the flipping of bits up to the": What is the distance being used here? Is it hamming distance?

5. Line 395: "Fig 5 shows the results of the task, where each subplot in the figure corresponds to a different condition"; is this referring to each subplot showing results for a different condition number? Or, different parameter combinations? From looking at Figure 5, it does not appear that the condition number is mentioned in the plots.

6. There are still a few grammatical errors throughout the text that should be fixed.

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Reviewer #2: No

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Novel real number representations in Ising machines and performance evaluation: Combinatorial random number sum and constant division

PONE-D-23-25182R2

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