The S (Phase) Gate

Every quantum gate acts on amplitudes, not just probabilities. The S gate — sometimes called the phase gate — leaves $|0\rangle$ unchanged and multiplies the $|1\rangle$ amplitude by $i$:

Its matrix is diagonal:

Because $i = e^{i\pi/2}$, the S gate introduces a relative phase of 90° (a quarter turn) between the two basis amplitudes.

Why relative phase matters

A global phase — a factor $e^{i\theta}$ in front of the entire state — is physically unobservable. A relative phase between $|0\rangle$ and $|1\rangle$ is a different story: it changes the state genuinely and is observable when the state is later interfered with other gates.

To see this, start from the equal superposition

and apply $S$. Because $S$ only touches the $|1\rangle$ amplitude,

The probabilities are still 50/50 — the magnitudes $|1/\sqrt{2}|^2$ and $|i/\sqrt{2}|^2$ are both $\tfrac{1}{2}$. But the relative phase has shifted from $0$ to $90°$, moving the Bloch vector from the $+x$ axis to the $+y$ axis. This state $\tfrac{1}{\sqrt{2}}(|0\rangle + i|1\rangle)$ is the $+y$ eigenstate, often written $|{+i}\rangle$ or $|y\rangle$.

Try it

Apply H and then S to the single qubit. After pressing Run, look at the amplitudes: the $|0\rangle$ amplitude should be $\approx 0.707$ (real), and the $|1\rangle$ amplitude should be $\approx 0.707i$ (purely imaginary), confirming the 90° relative phase.

Notice that the Probabilities tab still shows a 50/50 split — the phase is hidden from measurements in the $Z$ basis. It would only become visible if you applied another gate that causes interference, such as a second Hadamard.

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