The Hitchhiker’s Guide to the Surface Code

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript "The Hitchhiker's Guide to the Surface Code' describes the problem of quantum error correction codes, focusing on the surface code method. It is very well written, with an almost pedagogical introduction to the standard techniques. The scientific problem is clearly stated, the presentation sound and complete. I recommend the publication of this paper in an unaltered form.

Author Response

Thanks to the reviewer for appreciating the manuscript.

Reviewer 2 Report

Comments and Suggestions for Authors

As entitled as Hitchhiker’s Guide, I supposed that this manuscript is to help innocent readers to understand or picture the surface codes in plain language. However, it is not self-consistent and not easy to go through this manuscript. Some but not all comments are listed below.

 

(i) This manuscript needs many examples for the need of quick understanding. For example, the authors should take Shor’s 9-qubit codes as the simplest case, explaining the terms “Z/X stabilizers” “row/column-like” geometrically and depicting logical states explicitly. Another example is the connection or formulation between the logical state and the stabilizing operators. In addition, some lines stabilizer-based quantum error correction codes should be put in the introduction section.

 

(ii) On the other hand, the caption of Figure 1 is very vague and confusing. For example, the reason why surface code logical operators are drawn using different colored lines in Fig. 1 (b) and Fig. (d). The explanation of Fig. 1 (c) in lines 371-377 is difficult to understand. In fact, captions of many figures like Figures 3, 4 and gate teleportation are very unclear and hard to understand. On the other hand, many quantum circuits in first ten pages are unnecessary. In addition, the flow chart or figure should be added for better understanding of the lattice surgery.

 

(iii) There are many terms need explanation. For example, what are inner code and outer code? What does the term ”data qubit” mean? A more essential term is surface code. Why surface code matters? What is pro and con of surface codes? In addition, the Gottesman-Knill theorem is mentioned but the context and the importance of this theorem is untold.

 

To sum up, much effort must be done to make this manuscript friendly enough to the  reader even with background of quantum information science.

 

Author Response

Comments 1: As entitled as Hitchhiker’s Guide, I supposed that this manuscript is to help innocent readers to understand or picture the surface codes in plain language. However, it is not self-consistent and not easy to go through this manuscript. Some but not all comments are listed below.
Response 1: "Helping innocent readers" is indeed what I am striving for with this manuscript. I did realize, while writing this manuscript, that even "the fundamentals of surface codes" cover more topics than I have imagined. Thanks for going through my manuscript and pointing out the problems; I have revised the manuscript accordingly, as I will detail below.

Comments 2: (i) This manuscript needs many examples for the need of quick understanding. For example, the authors should take Shor’s 9-qubit codes as the simplest case, explaining the terms “Z/X stabilizers” “row/column-like” geometrically and depicting logical states explicitly. Another example is the connection or formulation between the logical state and the stabilizing operators.
Response 2: I have added figures depicting the construction of the 9-qubit code, and the stabilizers and logical operators of the quantum repetition code and the 9-qubit code. In general I believe it is not worth it to write down the *logical states* explicitly (as state vectors) for any code more complicated than the repetition code, as I have already done in Equation (2).

Comments 3: In addition, some lines stabilizer-based quantum error correction codes should be put in the introduction section.
Response 3: I am intentionally delaying the introduction of the stabilizer formalism to Section 3. Even though my target readers are people with some background of quantum information science, I opt to keep the introduction section as non-technical as possible, to allow the reader to have a physical intuition before beginning working with formal stuff.

Comments 4: (ii) On the other hand, the caption of Figure 1 is very vague and confusing. For example, the reason why surface code logical operators are drawn using different colored lines in Fig. 1 (b) and Fig. (d).
Response 4: You are right that several captions needs improvement. I have expanded the caption of the original Figure 1 (now Figure 4), and it should also help understanding that the newly added previous figures are using the same graphical representation.

Comments 5: The explanation of Fig. 1 (c) in lines 371-377 is difficult to understand.
Response 5: In addition to revising this paragraph (now lines 394-402), I have also revised the original Figure 1 (now Figure 4), adding diagrams that visually demonstrate the logic of Pauli string intersection. I have also added some sentences in the previous paragraph to verbally discuss this logic.

Comments 6: In fact, captions of many figures like Figures 3, 4 and gate teleportation are very unclear and hard to understand.
Response 6: I have expanded the caption of these figures (now Figures 6 and 7), and added a detailed explanation of the gate teleportation circuit (lines 1014-1036).

Comments 7: On the other hand, many quantum circuits in first ten pages are unnecessary.
Response 7: I recognize that some quantum circuit diagrams do not play an essential role, but I still want to keep them, in the hope that readers who are familiar with quantum circuits will find them helpful and reassuring.

Comments 8: In addition, the flow chart or figure should be added for better understanding of the lattice surgery.
Response 8: I do not think a flow chart is a good representation of the lattice surgery procedure. The appropriate representation is 3D defect diagrams, like the ones I have presented the the current Figures 7, 8, and 9, which give a more holistic view of the lattice surgery procedure.

Comments 9: (iii) There are many terms need explanation. For example, what are inner code and outer code?
Response 9: I have revised the introduction to the Shor code and defined those terms (lines 56-57).

Comments 10: What does the term ”data qubit” mean?
Response 10: I have added the definition on line 567.

Comments 11: A more essential term is surface code. Why surface code matters? What is pro and con of surface codes?
Response 11: I have discussed some advantages of the surface code in the introduction (lines 103-127) and at the beginning of Section 4 (lines 362-366). A more in-depth discussion of the pros and cons of surface codes would require comparison with other state-of-the-art codes such as BB codes, which is outside the scope of this manuscript.

Comments 12: In addition, the Gottesman-Knill theorem is mentioned but the context and the importance of this theorem is untold.
Response 12: I have revised Section 3 (the stabilizer formalism) to define stabilizer circuits and the conclusion of the Gottesman-Knill theorem explicitly. In the context of quantum error correction, the importance of the theorem is that it gives a powerful method to completely characterize the behaviors of stabilizer circuits (lines 194-195).

Comments 13: To sum up, much effort must be done to make this manuscript friendly enough to the  reader even with background of quantum information science.
Response 13: I thank the reviewer again for their detailed comments, and hope that my revisions have addressed the issues.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Since the revision follows the given comments with well presentation, I recommend the publication of this manuscript.