H gate

The Hadamard gate: matrix, effect on |0⟩ and |1⟩, the h() call in Python and JS, and what it becomes on IonQ hardware.

The H gate, the Hadamard, turns $\vert 0 \rangle$ into an equal superposition of $\vert 0 \rangle$ and $\vert 1 \rangle$ , and turns $\vert 1 \rangle$ into the same superposition with the sign of $\vert 1 \rangle$ flipped. It acts on one qubit and takes no parameters. The Circuit panel draws it as H. 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.h(qubit)`
JS	`circuit.h(qubit)`
Qiskit class	`HGate`

Matrix

H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

On the Bloch sphere this is a half-turn about the axis midway between X and Z, which exchanges the X and Z axes.

Effect on basis states

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

Measured straight after H, a qubit that started in $\vert 0 \rangle$ or $\vert 1 \rangle$ reads 0 or 1 with equal probability. The sign difference between $\vert + \rangle$ and $\vert - \rangle$ does not show in a measurement; it decides how later gates interfere.

Inverse

H is its own inverse: $HH = I$ . Two H gates in a row on the same qubit cancel.

Usage

from qiskit import QuantumCircuit

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

import { QuantumCircuit } from 'qiskit';

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

Over many shots the counts split close to evenly between 0 and 1.

On IonQ hardware

H is not a native gate. Up to a global phase it is a Y rotation by π/2 applied after a Z rotation by π, so one GPi2 pulse is enough, with the Z rotation carried as a virtual Z that costs no physical gate. What runs after IonQ's optimiser can still differ, because the compiler rewrites the whole circuit rather than one gate at a time. See How your circuit is compiled.

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