The Hadamard gate is a type of single-qubit quantum gate—a
basic operation that can be applied to a qubit. Like all quantum
gates, the Hadamard gate is a unitary transformation on a Hilbert
space, and it is defined as follows:

$H=[2 1 2 1 2 1 −2 1 ].$

When acted on either of the basis states, the Hadamard gate produces
an even superposition of $∣0⟩$ and $∣1⟩$:

$H∣0⟩=[2 1 2 1 2 1 −2 1 ][10 ]=[2 1 2 1 ],$

$H∣1⟩=[2 1 2 1 2 1 −2 1 ][01 ]=[2 1 −2 1 ],$

with either of the measurement outcomes being equally likely.

Additionally, the Hadamard operation is its own inverse; applying it
twice returns a qubit to its original state: