I'm stumped by these questions in Chuang's book.
If I have a state $|ψ\rangle=1/2\sqrt2(1+M_0)(1+M_1)(1+M_2)|0\rangle_7$, where $$M_0=X_0X_4X_5X_6; M_1=X_1X_3X_5X_6; M_2=X_2X_3X_4X_6$$
I rewrite its superposition expression without operators on the computational basis:
I am not sure if it is right. Hope you guys can give me some suggestions.
Also, I am not sure $X$ or $Z$ operations are fault-tolerant. I learned from Chuang's book that a complete set of gates consisting of $H$-gate, phase-gate, C-NOT, and $T$ gates can be constructed using fault-tolerant procedures. In my opinion, $X$ is not fault-tolerant and $Z$ is fault-tolerant, since $Z$ is one of the phase gates.
Am I wrong? Hope someone can help me with my problems!