√X† gate

The √X† gate, the inverse of √X: matrix, effect on the basis states, the sxdg() call in Python and JS, and its IonQ native-gate form.

The √X† gate, SXdg, undoes the √X gate. Two of them in a row also make one X, turning the other way. It acts on one qubit and takes no parameters. The Circuit panel draws it as √X†. Measured straight after it, a qubit that started as 0 reads 0 or 1 with equal probability.

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.sxdg(qubit)`
JS	`circuit.sxdg(qubit)`
Qiskit class	`SXdgGate`

Matrix

\sqrt{X}^\dagger = \frac{1}{2} \begin{pmatrix} 1 - i & 1 + i \\ 1 + i & 1 - i \end{pmatrix}

On the Bloch sphere this is a quarter-turn about the X axis in the opposite direction to √X. It is the same operation as rx(-π/2) up to a global phase.

Effect on basis states

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

Measured straight after √X†, a qubit that started in $\vert 0 \rangle$ or $\vert 1 \rangle$ reads 0 or 1 with equal probability.

Inverse

The inverse of √X† is √X. A √X followed by a √X† on the same qubit is the identity.

Usage

from qiskit import QuantumCircuit

qc = QuantumCircuit(1, 1)
qc.sx(0)
qc.sxdg(0)
qc.measure(0, 0)

import { QuantumCircuit } from 'qiskit';

const circuit = QuantumCircuit(1, 1);
circuit.sx(0);
circuit.sxdg(0);
circuit.measure(0, 0);

Every shot reads 0: the pair cancels.

On IonQ hardware

√X† is GPi2(0.5) up to a global phase, a native gate with its phase half a turn from √X, so it costs one physical pulse. See How your circuit is compiled.

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