Does Z gate swap complex amplitudes of $|0\rangle$ and $|1\rangle$?

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

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 $$|q\rangle$$?

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

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