S gate

The S gate, a quarter-turn phase: matrix, effect on the basis states, the s() call in Python and JS, and why it costs nothing on IonQ hardware.

The S gate multiplies the $\vert 1 \rangle$ amplitude by $i$ , a quarter turn of phase, and leaves $\vert 0 \rangle$ alone. It acts on one qubit and takes no parameters. The Circuit panel draws it as S. On its own it changes nothing a measurement can see; the phase shows once the qubit is in superposition.

Kets such as $\vert 0 \rangle$ name the basis states and rotations are described on the Bloch sphere; the qubits lesson introduces both.

Call

Where	Call
Python	`qc.s(qubit)`
JS	`circuit.s(qubit)`
Qiskit class	`SGate`

Matrix

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

On the Bloch sphere this is a quarter-turn about the Z axis. S is the square root of Z: two S gates make one Z. It is the special case p(π/2) of the P gate, and two T gates make one S.

Effect on basis states

Input	Output
$\vert 0 \rangle$	$\vert 0 \rangle$
$\vert 1 \rangle$	$i\vert 1 \rangle$
$\vert + \rangle$	$\tfrac{1}{\sqrt{2}}(\vert 0 \rangle + i\vert 1 \rangle)$

A measurement straight after S shows no change on $\vert 0 \rangle$ or $\vert 1 \rangle$ . The phase shows once the qubit is in superposition.

Inverse

The inverse of S is Sdg, sdg(qubit). S is not its own inverse: two S gates make a Z, not an identity.

Usage

from qiskit import QuantumCircuit

qc = QuantumCircuit(1, 1)
qc.h(0)
qc.s(0)
qc.s(0)
qc.h(0)
qc.measure(0, 0)

import { QuantumCircuit } from 'qiskit';

const circuit = QuantumCircuit(1, 1);
circuit.h(0);
circuit.s(0);
circuit.s(0);
circuit.h(0);
circuit.measure(0, 0);

Every shot reads 1: the two S gates make a Z, and H Z H is an X.

On IonQ hardware

S is a virtual Z. The compiler folds the quarter-turn into the phase of the next pulse, so S costs no gate and no time on the machine. See How your circuit is compiled.

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