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enter image description hereI'm learning quantum error correction. I also had a hard time understanding the following stabilizer form. I read this paper, which gives two circuits for Z-Stabilizer, I can understand a but I don't know why b can achieve the same effect: use ancilla qubits to control the original state will that change the state of original qubit?

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This follows easily from the observation that $$(I\otimes H)CZ(I\otimes H)=CNOT$$ see for instance this post and the answers within. Then, since $H^2=I$ simply put an $H\cdot H$ in between all the CZ (controlled-$Z$) gates to turn them into CNOTs, so (b) and (a) are equivalent circuits.

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