CSS Code and Fault-tolerant Problem in Nielsen and Isaac Chuang‘s book

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.

• please edit the title to something describing the actual problem – glS Nov 28 '20 at 12:27
• Thanks, bro! I have revised it. – user13341 Nov 28 '20 at 13:30

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 $$\bar X|1\rangle_L=|0\rangle_L$$.
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).
• 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$|±\rangle$), – user13341 Nov 28 '20 at 13:25