Hands-on Introduction to Randomized Benchmarking

Dear Editor,

Thank you for the opportunity to revise our manuscript entitled “Hands-on Introduction to Randomized Benchmarking” to be considered for publication in SciPost Lecture Notes. We would also like to thank the reviewer for taking the time to carefully read our manuscript and for their detailed comments and valuable suggestions for improvement.

A point-by-point response to the reviewer comments has been made and the revised manuscript is provided, along with an extra PDF where every change in the text has been tracked in blue color. We have also changed the template to the corresponding SciPost Lecture Notes template and we have added all the DOIs for the references listed in our bibliography.

Thank you for your consideration.

Yours sincerely,
Ana Silva and Eliska Greplova



==== Response to Reviewer #1 ====

We greatly thank the reviewer for their careful reading of our manuscript and for their very helpful remarks. We are glad that the reviewer finds our tutorial relevant and pedagogical. We will address the remarks of the reviewer point by point below.

"It would benefit the reader to include a concise comparative summary table that lists the four main RB variants (standard, simultaneous, correlated, interleaved) and highlights their key features. Such a table could include each protocol’s primary purpose (e.g., “measure average error rate of a gate set” for standard RB, “detect crosstalk errors” for simultaneous RB, etc.), the main assumptions (e.g., gate-independent errors, etc.), and the figure of merit it produces (average error rate r, crosstalk error metric, specific gate error rate, etc.). This would serve as a handy reference for readers to quickly recall differences and use-cases. It can be placed at the end of the tutorial or at the beginning of the conclusion section as a capstone. Given the rich content presented, a summary table would reinforce understanding by allowing a side-by-side comparison of the protocols."

We thank the reviewer for this important suggestion. It is indeed helpful to include a summary table that allows a quick comparison between the different RB protocols. This not only reinforces the message of each chapter and improves the clarity of the manuscript, but also allows the reader to more quickly browse the document and refresh their memory on the topic.
We have added a comparison table at the end of the lecture notes (Chapter 6). This table provides a side-by-side comparison of the four RB protocols discussed in the lecture notes: standard RB, simultaneous RB, correlated RB and interleaved RB. The table allows the reader to quickly recall the main purpose of each protocol, the main assumptions under which each protocol operates, and the figure of merit that each protocol aims to estimate.


"Some of the more advanced parts (particularly the correlated RB chapter and the gate-set shadow protocol chapter) introduce complex ideas that might be challenging for a novice reader. The authors might consider adding a few more sentences of intuitive explanation or a simple example to those sections. For instance, when introducing the correlators in correlated RB or the sequence correlation functions in the shadow protocol, a short intuitive description of what these quantities mean physically would complement the mathematical definition. The text as written is correct, but an extra bit of intuition (perhaps in the Discussion subsections) could help readers who are less mathematically inclined. Essentially, ensure that for every new parameter or function introduced, the reader has a mental picture of what it represents in terms of errors or circuits."


We thank the reviewer for raising this issue. We agree that some of the more technical aspects might be more difficult for a first-time reader. We have tried to make all the steps involved in the mathematical derivations clear, so that the reader can follow them line by line if they wish. We also note that one of the purposes of the supplementary Python notebooks is to provide clear examples of the protocols, so that the reader has a good sense of how the mathematical formulae enter the protocol in practice.
However, we agree with the reviewer that our previous version of the manuscript lacked a proper pedagogical introduction to correlation functions. The correlation functions are key mathematical objects for both the correlated RB and the shadow protocol. We have now added more explanation onto the meaning of correlation functions, particularly in Section 3.1, where we give a general description of the correlated RB protocol. We also refer to the introduction of correlated functions in Section 3.1 later when we introduce the shadow protocol in Section 5.1, and provide more clarity on what the correlation functions are in the shadow protocol.


"The tutorial currently relies on simulated data (via the notebooks) to illustrate the protocols. It could be inspiring to include or discuss briefly an example of real experimental RB data from the literature. For example, when talking about standard RB, the authors could reference a specific experiment (perhaps from Refs. [7,8] which they cite as uses of RB in practice) and mention the typical values or outcomes (e.g., “a two-qubit device achieving an average error per gate of X%”). Similarly, for interleaved RB, referencing a real benchmarking of a particular gate (like a CNOT gate fidelity from a superconducting qubit experiment) would show how the theory translates to practice. Even without adding new data, a short description or figure showing an actual experimental decay curve and how it fits to extract r would connect the tutorial to hands-on lab work. This addition would underscore the “hands-on” aspect by demonstrating real-world relevance and could motivate readers by showing actual results achieved with these methods."

We thank the reviewer for this suggestion. We have now added a new section to the standard RB protocol chapter, providing an example, with real experimental data, of how to use the standard RB protocol to benchmark single-qubit operations in a real quantum device. We guide the reader through the main steps of the fitting procedure and provide actual experimental decay curves for the single-qubit fidelities. We also put the estimated values for the fidelities in the context of previously documented single-qubit fidelities for the same device. We also provide an accompanying Python notebook (see here: https://gitlab.com/QMAI/papers/rb-tutorial/-/blob/
main/StandardRB/Benchmarking_a_real_device.ipynb ), so that the readers can try out the fitting procedure for themselves on the same data.

"As mentioned, there are a few minor typos (e.g., “randomzied”→“randomized” page 41) and formatting inconsistencies that should be corrected."

We thank the reviewer for drawing our attention to the presence of typographical errors in the manuscript. We have made every effort to correct all typographical errors in the revised version of our manuscript.

"The paper contains many derivations and equations. It might help readers if the authors highlight the most important formulas in some way (either by numbering them and referencing them in the text or by explicitly stating in words that “this equation is the central result of the protocol”). For example, the final expression relating the average sequence fidelity to the depolarizing parameter p in standard RB, or the equation giving the interleaved error rate in terms of two decay parameters, are crucial takeaways. Making sure these stand out – perhaps by referencing them in the Conclusions or Discussion as the “key results” – can aid a reader doing a quick review of the material. In a tutorial context, explicitly summarizing “what you should remember” is very useful. The authors do much of this in discussions, but a little more emphasis on formula labeling could help."


We thank the reviewer for this suggestion. We have now added a summary table to each chapter of the manuscript, which lists the key findings for each protocol. In addition, we have made it clearer which formulae are actually the most important in each protocol by explicitly stating this in the text when these formulae appear for the first time.

To summarise the changes:

In section 1.3: lines 286-287.
Added new section 1.4.
In section 2.3: lines 582-583, lines 593-596, lines 661-662.
Added new section 2.5.
In section 3.1: lines 793-836, lines 890-892.
In section 3.2: lines 903-904.
Added new section 3.4.
In section 4.1: lines 1016-1017.
Added new section 4.3.
In section 5.1: lines 1096-1098, lines 1103-1119, lines 1160-1162.
Added new section 5.4.
Added new table in chapter 6.