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In Aaronson's paper about the efficient simulation of a stabilizer circuit (https://journals.aps.org/pra/pdf/10.1103/PhysRevA.70.052328), I have a problem with finding the reason why the following statement holds. Restating the statement in the end of the 4th page of the paper, Let $\{R_{h+n}\}$ with $h=1...n$ are the Pauli stabilizer of a state $|{\psi}\rangle$. If $Z_a$ commutes with all of $\{R_{h+n}\}$, then $$\sum_{h=1}^n c_hR_{n+h} = \pm Z_a$$ for a unique choice of $c_1 \dots c_n \in {0,1}$.

But for $|\psi\rangle=|11\rangle$, we have stabilizers of ${II, ZZ}$ only, and its combination can make neither of single-qubit $Z$ gate.

If I have misunderstood, I will be very appreciative if you can correct it. Thank you in advance.

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The terms $II$ and $ZZ$ do not uniquely specify the state $|11\rangle$ because you could equally have the state $|00\rangle$. Indeed, you should not include the identity term in your stabilizer. Thus, you need to add a second term, which could be either $-ZI$ or $-IZ$. Either way, you can easily see how to make a product $-ZI$ out of your stabilizers.

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  • $\begingroup$ Thank you! I needed to check the sign of the operators. That is a good point. However, I still wonder why the equation in my question holds. Can you give proof of it, please? $\endgroup$ – Gwonhak Lee Jul 20 at 7:07

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