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Currently, I reading the toric code section in the beginning of the paper Topological Quantum Memory. Here are a couple of sections that are somewhat confusing to me.

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This quote below is under a small section that gives general remarks about surface codes.

These codes are especially well suited for fault-tolerant implementation, because the procedure for measuring the error syndrome is highly local.

Question: Why is this locality important? My guess is local errors do not destroy the quantum information if the information is distributed throughout the lattice, but I am not sure if this is what the author had in mind.

(3) Question: Throughout the section on the toric code, I see the words "preserves the code subspace," but I am not sure what the meaning is. What is a code subspace and what does it mean to preserve it?

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    $\begingroup$ Hi. I recommend that you focus on one particular question from among the three here. Although they are all clearly related, it is important to have a specific focus on only one question per posting. You can post several at once (just don’t go overboard). $\endgroup$ Mar 18, 2022 at 1:36
  • $\begingroup$ @MarkS Thanks for letting me know! $\endgroup$
    – Debbie
    Mar 18, 2022 at 10:30

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Regarding your first question:

These codes are especially well suited for fault-tolerant implementation, because the procedure for measuring the error syndrome is highly local.

What is meant here is probably "locality" in the sense of 1. the fact taht stabilizers measure only adjacent qubits, and 2. the relatively low weight of the stabilizers, which is 4. This leads to robustness against so-called "hook errors", that can be as large as $\lfloor w/2 \rfloor$ where $w$ is the weight of each stabilizer in the code. In this paper, these hook errors are referred to as "horitontal hooks" and are discussed in p. 26. The code preserves well against this type of error because their weight is at most 2, and they are very localized in space so the odds of them causing a homologically non-trivial error is small. BTW, there are places where they are described more visually and in a simpler way, e.g in fig. 11 here.

for the rest of the questions, you can open separate threads, since they're quite distinct from this question.

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  • $\begingroup$ Thank you very much. $\endgroup$
    – Debbie
    Mar 18, 2022 at 10:32

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