RZ and Phase

Every qubit state can be written as

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,$

where $\alpha$ and $\beta$ are complex numbers. The relative phase between $|0\rangle$ and $|1\rangle$ is the argument of the ratio $\beta/\alpha$ . Although global phase is unobservable, relative phase is physically meaningful — it determines how states interfere and shapes every quantum algorithm.

The RZ gate

$RZ(\theta)$ rotates the Bloch sphere about the $z$ -axis by angle $\theta$ . Its matrix is

$RZ(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}.$

Acting on a general state $\alpha|0\rangle + \beta|1\rangle$ , the gate advances the phase of $|1\rangle$ by $\theta$ relative to $|0\rangle$ :

$RZ(\theta)\big(\alpha|0\rangle + \beta|1\rangle\big) = e^{-i\theta/2}\alpha|0\rangle + e^{i\theta/2}\beta|1\rangle.$

The common factor $e^{-i\theta/2}$ is a global phase and can be dropped; what remains is that $|1\rangle$ picks up a factor of $e^{i\theta}$ relative to $|0\rangle$ .

A concrete example: RZ(π) on |+⟩

Start with the equal superposition $|+\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ produced by a Hadamard gate. Applying $RZ(\pi)$ gives

$RZ(\pi)|+\rangle = \frac{1}{\sqrt{2}}\big(e^{-i\pi/2}|0\rangle + e^{i\pi/2}|1\rangle\big) = \frac{e^{-i\pi/2}}{\sqrt{2}}\big(|0\rangle - |1\rangle\big).$

Dropping the global factor $e^{-i\pi/2}$ , the result is $|{-}\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$ . The magnitudes of both amplitudes are unchanged — measurement probabilities remain 50/50 — but the sign (phase) of the $|1\rangle$ component has flipped.

Connection to Z and S gates

The $Z$ gate is a special case: $Z = RZ(\pi)$ up to global phase. Similarly, $S = RZ(\pi/2)$ and $T = RZ(\pi/4)$ . The parameterized family $RZ(\theta)$ generalizes all of them to arbitrary angles, giving you precise control over the phase between $|0\rangle$ and $|1\rangle$ .

Try it

Prepare the $|+\rangle$ state with a Hadamard, then apply $RZ(\pi)$ to flip the relative phase and reach $|{-}\rangle$ . The grader checks the full state vector.

Run your code to see the quantum state.

After running, inspect the amplitudes: both should have magnitude $\approx 0.707$ . The simulator displays the raw statevector $\tfrac{1}{\sqrt{2}}(-i|0\rangle + i|1\rangle)$ , which equals $|{-}\rangle$ up to the global factor $e^{-i\pi/2} = -i$ . The relative phase between the two amplitudes is $\pi$ — that is the relative phase in action.