a fundamental quantum gate

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

the one-qubit version of the quantum fourier transform

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

a rotation of about the axis at the Bloch sphere

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

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 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 1-qubit rotation

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 .

a well-known gate in quantum computing that achieves this

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

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.

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

The H gate

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

a strictly quantum structure that does not map into classic logic gates

A Hadamard gate, or H-gate, is a strictly quantum structure that does not map into classic logic gates.

a rotation of $\pi$ about the vector

If by rotations, you refer to the Bloch sphere representation, the Hadamard gate is a rotation of $\pi$ about the vector $\hatx+\hatz$, not $\pi/4$ about the vector $\hatx$ as you stated.

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.

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.

a basic logic gate that maps $|0\rangle

For a simple example, the Hadamard gate is a basic logic gate that maps $|0\rangle

a unitary that sends the basis $\{\vert 0\rangle , \

Specifically, the Hadamard gate is a unitary that sends the basis $\{\vert 0\rangle , \...

one example of rotations on the sphere

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

the single-qubit version of the Fourier Transform*,

The Hadamard gate is the single-qubit version of the Fourier Transform*, and the Fourier Transform is very useful.

nothing more that the quantum fourier transform over the simplest of all groups

The Hadamard gate is nothing more that the quantum fourier transform over the simplest of all groups, the cyclic group with two elements, Z_2.

\ ( {\mathcal H} \)

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