I am reading Quantum Computing 1st Edition By Parag Lala, this book says

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It seemed that the Z gate swapped the complex amplitudes $\alpha$ and $\beta$.

Can Z gate implement that, or are there any errata? Because

$$ \begin{pmatrix} \alpha \\ -\beta \end{pmatrix} = \alpha\begin{pmatrix} 1 \\ 0 \end{pmatrix} - \beta\begin{pmatrix} 0 \\ 1 \end{pmatrix} \neq \alpha\begin{pmatrix} 0 \\ 1 \end{pmatrix} + \beta\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \alpha|1\rangle + \beta|0\rangle $$

And, is it true that Z Gate merely add $\pi$ to the relative phase $\phi$ of a superposition $|q\rangle$?

$$ |q\rangle = \alpha|0\rangle + e^{i\phi}\beta|1\rangle $$ $$ Z|q\rangle = \alpha|0\rangle + e^{i(\phi+\pi)}\beta|1\rangle $$

  • 5
    $\begingroup$ that's probably just a typo. The $Z$ gate only changes the sign of the amplitude of $|1\rangle$ $\endgroup$
    – glS
    May 26, 2021 at 9:19

1 Answer 1


This is wrong for sure.

And according to the book reviews on Amazon, this book is "unreliable", "riddled with errors", and "someone studying for the first time will get confused"


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