Nov 30, 2020

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:

$H=[2 1 2 1 2 1 −2 1 ].$

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:

$HH=[2 1 2 1 2 1 −2 1 ][2 1 2 1 2 1 −2 1 ]=[10 01 ]=I,$

where $I$ is the identity matrix.

See also

Say "Hello, World!" with Q#

Creating a simple Q# console application.

On the nature of the wave function

The wave function is an abstract mathematical concept and cannot be "measured" directly. So what is it then?

Dirac notation for quantum states

How to read the bra–ket notation?

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