The Hadamard transform is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on real numbers. The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms, and is in fact equivalent to a multidimensional DFT of size. It decomposes an arbitrary input vector into a superposition of Walsh functions. The transform is named for the FrenchmathematicianJacques Hadamard, the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh.
Definition
The Hadamard transform Hm is a 2m × 2mmatrix, theHadamard matrix, 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 representation of the indices n and k. Recursively, we define the 1 × 1 Hadamard transform H0 by the identityH0 = 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 -th entry by writing where the kj and nj are the binary digits 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. where is the bitwise dot product of the binary representations of the numbers i and j. For example, if, then, agreeing with the above. Note that the first row, first column element of the matrix is denoted by. H1 is precisely the size-2 DFT. It can also be regarded as the Fourier transform on the two-element additive group of Z/. The rows of the Hadamard matrices are the Walsh functions.
Computational complexity
In the classical domain, the Hadamard transform can be computed in operations, using the fast Hadamard transform algorithm. In the quantum domain, the Hadamard transform can be computed in time, as it is a quantum logic gate that can be parallelized.
In quantum information processing the Hadamard transformation, more often called Hadamard gate in this context, 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, also known as the computational basis. The states and are known as and respectively, and together constitute the polar basis in quantum computing. Many quantum algorithms use the Hadamard transform as an initial step, since it maps m qubits initialized with to a superposition of all 2m 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 log 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. 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, then the final state is always the same as the initial state.
