The Hadamard gate

The definition of the Hadamard gate and some of its properties.

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 = [\frac{1}{2} \frac{1}{2} \frac{1}{2} - \frac{1}{2}] .

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

H ∣ 0 ⟩ = [\frac{1}{2} \frac{1}{2} \frac{1}{2} - \frac{1}{2}] [10] = [\frac{1}{2} \frac{1}{2}],

H ∣ 1 ⟩ = [\frac{1}{2} \frac{1}{2} \frac{1}{2} - \frac{1}{2}] [01] = [\frac{1}{2} - \frac{1}{2}],

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:

H H = [\frac{1}{2} \frac{1}{2} \frac{1}{2} - \frac{1}{2}] [\frac{1}{2} \frac{1}{2} \frac{1}{2} - \frac{1}{2}] = [1001] = I,

where $I$ is the identity matrix.