the one-qubit version of the quantum fourier transform

The Hadamard gate is the one-qubit version of the quantum fourier transform.

a rotation that swaps the x-axis and z-axis of a given qubit

A Hadamard gate is a rotation that swaps the x-axis and z-axis of a given qubit.

a fundamental quantum gate

The Hadamard gate (H-gate) is a fundamental quantum gate.

a type of gate that acts on a single qubit

The Hadamard gate is a type of gate that acts on a single qubit: a single qubit generates a superposition of the basis states.

a rotation

Hadamard (H) gate represents a rotation of about the axis at the Bloch sphere.

a 1-qubit rotation, mapping the qubit-basis states and to two superposition states with equal weight of the computational basis states and

The Hadamard gate is a 1-qubit rotation, mapping the qubit-basis states and to two superposition states with equal weight of the computational basis states and .

the combination of two rotations, about the Z-axis followed by about the Y-axis

Equivalently, The Hadamard gate is the combination of two rotations, about the Z-axis followed by about the Y-axis.

single qubit

the single qubit Hadamard gate can be also described as the gate which maps to .

some of the elementary quantum gates that are identified in the stock market structure

Hadamard gate, Pauli gates or controlled-Z gate are some of the elementary quantum gates that are identified in the stock market structure.

a unitary operation that transforms a qubit from the computational basis state |0

The Hadamard gate is a unitary operation that transforms a qubit from the computational basis state |0

a universal gate set for quantum computing

The Toffoli gate with the Hadamard gate is a universal gate set for quantum computing.

to a rotation of the Bloch sphere

To correct the description, note that a Hadamard gate corresponds to a rotation of the Bloch sphere by 90 degrees about the axis, followed by a degree rotation about the axis.

a well-known gate in quantum computing that achieves this

The Hadamard Gate is a well-known gate in quantum computing that achieves this.

one example of rotations on the sphere

The hadamard gate is one example of rotations on the sphere.

The H gate

The H gate is named after the mathematician Jacques Hadamard and hence often called Hadamard gate.

a 180 degree rotation around the $(\hat{x}+\hat{z})/\sqrt{2}$ axis

Just to add to the answers above: the Hadamard gate is simply a 180 degree rotation around the $(\hat{x}+\hat{z})/\sqrt{2}$ axis.

creates an equally weighted superposition of the and states

The Hadamard gate, normally denoted as H, creates an equally weighted superposition of the and states.

the simplest method for generating a random bit in a qubit

The Hadamard gate is the simplest method for generating a random bit in a qubit.

the 2×2 unitary matrix , which takes a single qubit from the upper input and produces a single qubit output

The Hadamard gate (the H box) is the 2×2 unitary matrix , which takes a single qubit from the upper input and produces a single qubit output.

\ ( {\mathcal H} \)

( {\mathcal H} \), where \( {\mathcal H} \) is the Hadamard gate that acts on the target qubit R.