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Hadamard


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Not to be confused with Walsh matrix.

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It is named for the French mathematician Jacques Solomon Hadamard, the German-American mathematician Hans Adolph Rademacher, and the American mathematician Joseph Leonard Walsh. It performs an orthogonal, symmetric, involutional, linear operation on 2^m real numbers (or complex numbers, although the Hadamard matrices themselves are purely real).
The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size . It decomposes an arbitrary input vector into a superposition of Walsh functions.

Contents [hide] <LI class="toclevel-1 tocsection-1">1 Definition <LI class="toclevel-1 tocsection-2">2 Computational complexity <LI class="toclevel-1 tocsection-3">3 Quantum computing applications <LI class="toclevel-1 tocsection-4">4 Other applications <LI class="toclevel-1 tocsection-5">5 See also 6 External links

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[edit] Definition


The Hadamard transform H_m is a 2^m × 2^m matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2^m real numbers x_n into 2^m real numbers X_k. The Hadamard transform can be defined in two ways: recursively, or by using the binary (base-2) representation of the indices n and k.
Recursively, we define the 1 × 1 Hadamard transform H₀ by the identity H₀ = 1, and then define H_m for m > 0 by:


where the 1/√2 is a normalization that is sometimes omitted. Thus, other than this normalization factor, the Hadamard matrices are made up entirely of 1 and −1.
Equivalently, we can define the Hadamard matrix by its (k, n)-th entry by writing


and


where the k_j and n_j are the binary digits (0 or 1) of n and k, respectively. In this case, we have:


This is exactly the multidimensional DFT, normalized to be unitary, if the inputs and outputs are regarded as multidimensional arrays indexed by the n_j and k_j, respectively.
Some examples of the Hadamard matrices follow.




(This H₁ is precisely the size-2 DFT. It can also be regarded as the Fourier transform on the two-element additive group of Z/(2).)






where is the bitwise dot product of the binary representations of the numbers i and j. For example, , agreeing with the above (ignoring the overall constant). Note that the first row, first column of the matrix is denoted by H₀₀
The rows of the Hadamard matrices are the Walsh functions.
[edit] Computational complexity


The Hadamard transform can be computed in m log m operations, using the fast Hadamard transform algorithm.
[edit] Quantum computing applications


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 and to two superposition states with equal weight of the computational basis states and . Usually the phases are chosen so that we have


in Dirac notation. This corresponds to the transformation matrix


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

Hadamard gate operations: