The Hadamard gate

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:

$H=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right].$

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

$H|0⟩=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right],$

$H|1⟩=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}\end{array}\right],$

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:

$HH=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I,$

where $I$ is the identity matrix.

See also

• Bloch Sphere - Visualize qubits on a Bloch sphere.
• Quantum Random Number Generator - Generate random numbers using quantum computing.
• Quantum Gates Quiz - Test your knowledge of quantum logic gates.
• Say "Hello, World!" with Q# - Creating a simple Q# console application.
• Q# Editor - Edit Q# files online.
• Awesome quantum computing - A curated list of awesome resources on quantum computing.
• Quantum Computing Crossword - A daily crossword puzzle about quantum computing.
• Quantum computing pronunciation guide - A pronunciation guide for quantum computing terms.