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: enter image description here

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:

enter image description here

How does it work? Why is the controlled qubit affected?


1 Answer 1


The state is not $-\vert ++\rangle\vert 100\rangle$ after the second set of gates. That would be true if you applied $Z$ gates but you are applying controlled $Z$ gates.

Let's just look at the controlled $Z_3$ gate since that's where the bit flip error is and ignore the remaining registers. This picks up a minus sign if both the control and the target qubit are $\vert 1\rangle$. Omitting normalization constants of $\frac{1}{\sqrt{2}}$, you get

$$\vert +\rangle\vert 1\rangle = \vert 0\rangle\vert 1\rangle + \vert 1\rangle\vert 1\rangle\xrightarrow{CZ} \vert 0\rangle\vert 1\rangle - \vert 1\rangle\vert 1\rangle = \vert - \rangle\vert 1\rangle.$$

The next Hadamard on the ancilla converts $\vert -\rangle \xrightarrow{H} \vert 1\rangle$ and gives you the measurement outcome you see.


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