Hadamard transform

The Hadamard transform (Hadamard transformation, also known as 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 <math>|0 \rangle <math> and <math>|1 \rangle <math>. Most times the phases are chosen so that we have

<math>\frac{|0\rangle+|1\rangle}{\sqrt{2}}\langle0|+\frac{|0\rangle-|1\rangle}{\sqrt{2}}\langle1|<math>

in Dirac notation. This corresponds to the transformation matrix

<math>H=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}<math>

in the <math>|0 \rangle , |1 \rangle <math> basis.

Quantum algorithms use the Hadamard transform as initial step, since it maps n qubits initialized with |0› to a superposition of all 2ⁿ orthogonal states in the <math> |0 \rangle , |1 \rangle <math>basis with equal weight.

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

See also: Hadamard matrix.

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