# Hadamard

The **Hadamard transform** (**Hadamard transformation**, also known as the **Walsh-Hadamard transformation**) is an example of a generalized class of Fourier transforms. It is named for the French mathematician Jacques Hadamard.

In quantum information processing the Hadamard transformation, more often called **Hadamard gate** in this context (cf. quantum gate), is a one-qubit rotation, mapping the qubit-basis states |0› and |1› to two superposition states with equal weight of the computational basis states ∣0⟩ and ∣1⟩. Usually the phases are chosen so that we have

$$\frac{|0\rangle+|1\rangle}{\sqrt{2}}\langle0|+\frac{|0\rangle-|1\rangle}{\sqrt{2}}\langle1|$$ in Dirac notation. This corresponds to the transformation matrix

$$H=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$

in the ∣0⟩, ∣1⟩ basis.

Many quantum algorithms use the Hadamard transform as an initial step, since it maps *n* qubits initialized with |0› to a superposition of all 2*n* orthogonal states in the ∣0⟩, ∣1⟩basis with equal weight.

The Hadamard matrix can also be regarded as the Fourier transform on the two-element *additive* group of **Z**/(2).

The Hadamard transform is used in many signal processing, and data compression algorithms.