Phase Gates as Z-Rotations

The Z gate, the S gate, and the T gate all leave $|0\rangle$ unchanged and multiply the amplitude of $|1\rangle$ by a phase factor. On the Bloch sphere each of them is a pure rotation about the $z$ -axis — they differ only in how far they turn.

Phase gates as diagonal matrices

All three gates share the same structure:

Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}, \qquad T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}.

Because the $|0\rangle$ amplitude is always unchanged and $|1\rangle$ picks up a phase, none of these gates can ever shift probability from $|0\rangle$ to $|1\rangle$ or back. They are phase-only operations.

The rotation angle

The general $R_z(\lambda)$ gate is defined as

R_z(\lambda) = \begin{pmatrix} e^{-i\lambda/2} & 0 \\ 0 & e^{i\lambda/2} \end{pmatrix}.

Up to a global phase (which is physically unobservable) we can match each phase gate to an $R_z$ rotation:

$Z$ rotates by $\lambda = \pi$ (a half-turn): the $|1\rangle$ amplitude picks up $e^{i\pi} = -1$ .
$S$ rotates by $\lambda = \pi/2$ (a quarter-turn): the $|1\rangle$ amplitude picks up $e^{i\pi/2} = i$ .
$T$ rotates by $\lambda = \pi/4$ (an eighth-turn): the $|1\rangle$ amplitude picks up $e^{i\pi/4}$ .

On the Bloch sphere, the north pole $|0\rangle$ stays fixed and every equatorial state traces a circular arc. Starting from $|+\rangle$ on the $+x$ axis, a T gate nudges the Bloch vector $45°$ toward the $+y$ axis; an S gate advances it $90°$ ; a Z gate flips it $180°$ to $-x$ .

Gate relationships

Because the rotations compose by adding angles, the hierarchy is clear:

T^2 = S, \qquad S^2 = Z, \qquad T^4 = Z.

Each identity is just $\frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2}$ (or doubling the angle again for the next step). You can verify the first one by multiplying:

T^2 = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}^2 = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} = S.

Try it

Apply two T gates in sequence to verify the identity $T^2 = S$ . The grader compares the full unitary matrix, so leaving the circuit empty will not pass.

Run your code to see the quantum state.