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I am completely new to quantum computing and playing QuantAttack. While playing, I found something strange: SZS turns into Z.

According to the Wikipedia article that the game recommends reading, SZS must disappear because:

$$ SZS = \begin{pmatrix}1 & 0 \\ 0 & i\end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & i\end{pmatrix} = \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} = I $$

Is there some weird quantum thing I missed, or is it just a game bug?

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SZS is not equal to Z, it is equal to identity matrix as mentioned in the article. SZ$S^{\dagger}$ is equal to Z ,where $S^{\dagger}$ is the conjugate transpose of S gate.

This can be proved by matrix multiplication $$ SZS^{\dagger}=\begin{pmatrix}1 & 0 \\ 0 & i\end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & -i\end{pmatrix} = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix} = Z $$

You can check the below link for more info about S and S dagger gates https://www.quantum-inspire.com/kbase/s-gate/

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    $\begingroup$ Thank you! I asked the game developer and they said it's indeed a game bug. $\endgroup$
    – Nayuta Ito
    Commented May 4, 2023 at 0:11

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