S phase gate

The $S$ gate (phase gate) is $S=\begin{pmatrix}1&0\\0&i\end{pmatrix}$, applying a $\pi/2$ phase to $|1\rangle$: $S|0\rangle=|0\rangle$, $S|1\rangle=i|1\rangle$. Squaring the matrix gives $S^2=\begin{pmatrix}1&0\\0&-1\end{pmatrix}=Z$, and applying it four times returns the identity, $S^4=I$ — exactly the power relations this record verifies by direct computation. $S$ is unitary but not Hermitian ($S^\dagger=S^{-1}=S^3\neq S$), and it is a Clifford gate: conjugating Pauli $X$ by $S$ gives $SXS^\dagger=Y$ up to sign convention, so $S$ maps the $X$/$Z$ stabilizer picture onto $Y$. Because $S=\sqrt Z$, it is the natural bridge between the Pauli group and finer phase rotations; composed with Hadamard it builds a $Y$-basis change of basis and, together with $T=\sqrt S$, generates approximately universal single-qubit rotations.