Hadamard transform

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.

Related Topics:
Fourier transform - French - Mathematician - Jacques Hadamard

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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 angle and |1 angle . Usually the phases are chosen so that we have

Related Topics:
Quantum information processing - Quantum gate - Qubit - Rotation

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:rac{|0 angle+|1 angle}{sqrt{2}}langle0|+rac{|0 angle-|1 angle}{sqrt{2}}langle1|

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in Dirac notation. This corresponds to the transformation matrix

Related Topics:
Dirac notation - Transformation matrix

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:H=rac{1}{sqrt{2}}egin{pmatrix} 1 & 1 \ 1 & -1 end{pmatrix}

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in the |0 angle , |1 angle basis.

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

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

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See also: Hadamard matrix.

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