Random Networks with Quantum Boolean Functions

Round 1

Reviewer 1 Report

In this manuscript entitled "Random Boolean Network with Unitary Dynamics", M. Franco et al. developed a theoretical framework of a novel model  similar to Boolean Networks. The proposed model is based on previously developed quantum Boolean functions, and has the properties of being reversible in contrast to random Boolean networks which are irreversible.  BNs and its generalizations such as probabilistic Boolean networks, reversible BNs, etc, are widely used specially in computational Biology to study the behavior and properties of intracellular interactions such as Gene regulatory networks, etc. BNs have also been used in other areas of mathematical modeling. In my opinion this manuscript will be relevant to future research activities of computational modeling. I recommend accepting this manuscript after minor revisions. 

The suggestion are itemized in the following list:

• It is not clear how the title is related to the manuscript, how unitary dynamics is related to the quantum formulation you described? it is not very evident from the text.
• Could you expand the advantages of using a new paradigm of Boolean networks, could you identify potential problems for which the proposed quantum BNs will offer advantages over the traditional BNs?
• Line 43: It is not clear how the need to account for all the parts of a system is related to the reversible aspects of physics?
• Line 88: I don't think the update scheme is neglected by the BNs users, in fact there are many groups that use an asynchronous update scheme. 
• Could you add a reference to reversible BNs in the subsection 2.3 (line 95) ?
• Line 100: Please clarify what do you mean by “all attractors behave  equally in conventional Boolean networks”.
• Subsection 3.1: Could you state how a state is defined in Quantum Boolean Networks, for someone that is familiar with BNs it is not easy to compare the notion of state in BNs with those of quantum BNs.
• In equation (3), how the permutations matrices Pi enters into the dynamics described in equation (3)?
• Subsection 3.1. Each U_i is a quantum Boolean function of k_i qubits; but  do the U_i s have the same dimension?  
• Line 143: Could you explain why “this” set is still not useful from a practical point of view?
• Line 143 Could expand on why you selected the Pauli and Hadamard operators? Are there some special properties of those operators?
• Line 158: For those not familiar with RevBNs could you expand on dynamical properties of the RevBN so we can better understand the comparison between quantum BNs and RevBNs?
• PAragraph in line 160: Is there a way to summarize these results? Are those reflected in Figure 3? This figure is not referenced in the main text.
• Line 167: “to states” should be “two states”
• Line 178: what are configurations in quantum BNs? Are those different to states?
• Figure 5 and 4 are not referenced in the main text.

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The authors propose a quantum version of random Boolean networks and explore its properties by numerical simulations. The model has reversible dynamics reminiscent of the classical reversible Boolean networks. The authors discuss the difference in behaviors between them in terms of cycle length and entropy. 

The results reported in this paper seem to remain in the budding stage. However, they contain interesting insights into the behavior of the proposed model. In addition, the paper is written clearly and is readable. The paper could be published after the following issues are considered. 

1. Sec.2.3: It would be helpful for readers to explain the reason why the update scheme shown above Line 96 yields reversible dynamics.

2. Line 103 (5th line from the bottom on Page 5): Should "its" be "is"?

3. Sec.3.2: It seems that Fig.3 is not mentioned anywhere in the main text. Please refer to it at the appropriate places. Similar flaws occur for Fig.4 and Fig.5. 

4. Figure 2: Do the four cases correspond to the four experiments mentioned in Line 146? If so, please make it explicit. 

5. In the caption of Fig.2, the authors wrote "we see that QBNs are capable of both ordered and chaotic dynamics". However, no further evidence for the dynamical regimes (e.g., assessment of stability) is given other than the state amplitude patterns shown in Fig.2. So, it is better to avoid using the word "ordered" and "chaotic". They could be replaced by, for example, "regular" and "irregular", respectively. The reviewer prefers words that are merely descriptive terms for the patterns. Please also reconsider the use of "order" and "chaos" in Sec.3.4.

6. Line 161: Is "at" missing before "the next timestep"?

7. Line 167: Should "to states" be "two states"?

8. Line 219: Should "... QBN. Especially because" be "... QBN, especially because"?

Author Response

Please see the attachment.

Author Response File: Author Response.pdf