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 performs an orthogonal, symmetric, involutive, linear operation on 2m 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 2 × 2 × ⋯ × 2 × 2. It decomposes an arbitrary input vector into a superposition of Walsh functions.
The Hadamard transform Hm is a 2m × 2m matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2m real numbers xn into 2m real numbers Xk. 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 Hm for m > 0 by:
where the 1/√ is a normalization that is sometimes omitted.
For m > 1, we can also define Hm by:
where represents the Kronecker product. 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
where the kj and nj are the binary digits (0 or 1) of k and n, respectively. Note that for the element in the top left corner, we define: . 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 nj and kj, 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, if , then , agreeing with the above (ignoring the overall constant). Note that the first row, first column element of the matrix is denoted by .
The rows of the Hadamard matrices are the Walsh functions.
The Hadamard transform can be computed in n log n operations (n = 2m), using the fast Hadamard transform algorithm.
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
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.
Notably, computing the quantum Hadamard transform is simply the application of a Hadamard gate to each qubit individually because of the tensor product structure of the Hadamard transform. This simple result means the quantum Hadamard transform requires n operations, compared to the classical case of n log n operations.
Hadamard gate operations
One application of the Hadamard gate to either a 0 or 1 qubit will produce a quantum state that, if observed, will be a 0 or 1 with equal probability (as seen in the first two operations). This is exactly like flipping a fair coin in the standard probabilistic model of computation. However, if the Hadamard gate is applied twice in succession (as is effectively being done in the last two operations), then the final state is always the same as the initial state. This would be like taking a fair coin that is showing heads, flipping it twice, and it always landing on heads after the second flip.
The Hadamard transform is also used in data encryption, as well as many signal processing and data compression algorithms, such as JPEG XR and MPEG-4 AVC. In video compression applications, it is usually used in the form of the sum of absolute transformed differences. It is also a crucial part of Grover's algorithm and Shor's algorithm in quantum computing. The Hadamard transform is also applied in experimental techniques such as NMR, mass spectrometry and crystallography.
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