S

The S gate leaves \( |0\rangle \) alone and multiplies \( |1\rangle \) by \( i \) - a quarter-turn phase shift. It's also called the √Z gate, since applying it twice is the same as one Z gate.

Qubits	1
Parameters	None
Inverse	S† (not self-inverse)
Also called	√Z, phase gate; equal to P(π/2)

What it does

S adds a relative phase of \( \pi/2 \) (90°) between the \( |1\rangle \) and \( |0\rangle \) components of a state, without changing measurement probabilities in the computational basis. \( S^2 = Z \), which is why it's read as "square root of Z."

Matrix

\[ S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} \]

Action on basis states

• \( |0\rangle \rightarrow |0\rangle \)
• \( |1\rangle \rightarrow i\,|1\rangle \)

Phase and Bloch-sphere effect

S rotates the Bloch vector by \( \pi/2 \) about the Z-axis. On a state that's already a superposition, this is exactly the kind of phase change visualized by phase disks, the Q-Sphere, and the Statevector chart - see the worked H → S → T example on the phase disks page.

Example

H then S on q[0] produces \( \dfrac{1}{\sqrt{2}}(|0\rangle + i|1\rangle) \): the same 50/50 measurement split as plain H, but the \( |1\rangle \) bar in the Statevector chart is now colored a quarter-turn around the Phase wheel from the \( |0\rangle \) bar.

Other notations

Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.

Language	Typical form
OpenQASM 2.0	s q[0];
Qiskit	circuit.s[q[0]]
Cirq	cirq.S(q[0])
Q#	S(q[0]);

Related

• S† - the inverse of S
• T - a smaller, π/4 phase step
• P - the general phase gate; S = P(π/2)
• Phase disks