Hadamard

The '''Hadamard transform''' ('''Hadamard transformation''', also known as the '''Walsh-Hadamard transformation''') is an example of a generalized class of [[Fourier transform]]s. It is named for the [[France|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 $|0\; \backslash rangle$ and $|1\; \backslash rangle$. Usually the phases are chosen so that we have :$\backslash frac\{|0\backslash rangle+|1\backslash rangle\}\{\backslash sqrt\{2\}\}\backslash langle0|+\backslash frac\{|0\backslash rangle-|1\backslash rangle\}\{\backslash sqrt\{2\}\}\backslash langle1|$ in [[Dirac notation]]. This corresponds to the [[transformation matrix]] : in the $|0\; \backslash rangle\; ,\; |1\; \backslash rangle$ basis. Many [[quantum algorithm]]s 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 $|0\; \backslash rangle\; ,\; |1\; \backslash rangle$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]]. ==See also== * [[Hadamard matrix]] {{FromWikipedia}} [[Category:Evolutions and Operations]]