Sdg gate

The Sdg gate, the inverse of S: matrix, effect on the basis states, the sdg() call in Python and JS, and why it costs nothing on IonQ hardware.

The Sdg gate, S dagger or $S^\dagger$ , undoes the S gate: it multiplies the $\vert 1 \rangle$ amplitude by $-i$ and leaves $\vert 0 \rangle$ alone. It acts on one qubit and takes no parameters. The Circuit panel draws it as Sdg. On its own it changes nothing a measurement can see.

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.sdg(qubit)`
JS	`circuit.sdg(qubit)`
Qiskit class	`SdgGate`

Matrix

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

On the Bloch sphere this is a quarter-turn about the Z axis in the opposite direction to S. It is the special case p(-π/2) of the P gate.

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 Sdg shows no change on $\vert 0 \rangle$ or $\vert 1 \rangle$ .

Inverse

The inverse of Sdg is S. An S followed by an Sdg on the same qubit is the identity.

Usage

from qiskit import QuantumCircuit

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

import { QuantumCircuit } from 'qiskit';

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

Every shot reads 0: the pair cancels.

On IonQ hardware

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

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