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Could someone help me with $Problem-10.52$ in Nielsen and Isaac Chuang‘s book?

The screenshot is shown below. I have no idea about that.

Hope someone can give me some suggestions.

By the way, if I want to know whether an operator is fault-tolerant, what should I DO to check it? For example, the $Z$ operator (as shown below) is a fault-tolerant operator in the encoding procedure, or not. I only know the fault-tolerant operator only allows 1-bit error to happen.

enter image description here

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    $\begingroup$ please edit the title to something describing the actual problem $\endgroup$
    – glS
    Nov 28 '20 at 12:27
  • $\begingroup$ Thanks, bro! I have revised it. $\endgroup$
    – user13341
    Nov 28 '20 at 13:30
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I think you just get stuck, this is an easy job. For stabilizer codes, the logical state $|0\rangle_L$ and $|1\rangle_L$ are just superposition of states(maintaining the same state under stabilizer operation).

To verify $\bar Z$ and $\bar X$ are logical $Z$ and $X$ operators, just notice that $\bar Z|0\rangle_L=|0\rangle_L,\ \bar Z|1\rangle_L=-|1\rangle_L,\ \bar X|0\rangle_L=|1\rangle_L,$ and $\bar X|1\rangle_L=|0\rangle_L$.

These two are the logical states for Steane code(or see the original paper):

enter image description here

You can see that all superposition states of $|0\rangle_L$ contains even number of $|1\rangle's$, so $\bar Z|0\rangle_L=|0\rangle_L$, while for $|1\rangle_L$ the number of $|1\rangle$ is odd-definite. I'll leave the $\bar X$ part of work for you(it's quite tedious).

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  • $\begingroup$ Hi Yitian @Yitian Wang! Thank you for your comment. Yes, I have solved this exercise a few hours ago. But I am still confused about how to judge whether an operator is a fault-tolerant operator. Just, for example, $$Z=Z_7Z_6Z _5Z_4Z_3Z_2Z_1$$. My idea is whether Z is a fault-tolerant is based on that basis that we may choose ($|0/1\rangle$ or$|±\rangle$), $\endgroup$
    – user13341
    Nov 28 '20 at 13:25
  • $\begingroup$ This paper(journals.aps.org/pra/abstract/10.1103/PhysRevA.86.032324) can enlighten your question. The key point is that only legal operation can maintain the state of the ancillae, while environmental (or other) noise will change them. So after the computation, we know what our ancillae we know what the measurement result will come. If it differs, we can detect the error and do some post-handling to fix(hmmm quite like what every QEC should be?). $\endgroup$ Nov 29 '20 at 2:35

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