I'm trying to understand how syndrome extraction in bit-flip code works. I've been reading Quantum Error Correction: An Introductory Guide, and I was wondering how the syndrome extraction part in this circuit works:
For example, if the state before the syndrome extraction part is $|++⟩|100⟩$, then $Z_2⊗Z_1$ is applied and the state is still $|++⟩|100⟩$ and then $Z_3⊗Z_2$ is applied, which turns the state to $-|++⟩|100⟩$, then Hadamard gates are applied and I think the state should be $-|00⟩|100⟩$, which means first qubit from left is $0$, but it's $1$. I used Quirk and it's also $1$ there:
How does it work? Why is the controlled qubit affected?