Hadamard transform

[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.DFT, normalized to be unitary, if the inputs and outputs are regarded as multidimensional arrays indexed by the nj and kj, respectively.In discrete Fourier transform, when m equal to zeros (mean first row), the result of DFT also is 1.If we calculate zero crossing: The Hadamard transform is in fact equivalent to a multidimensional DFT of size 2 × 2 × ⋯ × 2 × 2., where the multiplication is the boolean dot product on bit strings, so we can identify the input toto take as input the bit string corresponding to the index of an element ofIn quantum computing, the Hadamard gate is a one-qubit rotation, mapping the qubit-basis statesOne 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.This generalizes the preparation of uniform quantum states using Hadamard gates for anyImportantly, neither ancilla qubits nor any quantum gates with multiple controls are needed in this approach for creating the uniform superposition stateThe Hadamard transform can be used to estimate phylogenetic trees from molecular data.[7][8][9] Phylogenetics is the subfield of evolutionary biology focused on understanding the relationships among organisms.A Hadamard transform applied to a vector (or matrix) of site pattern frequencies obtained from a DNA multiple sequence alignment can be used to generate another vector that carries information about the tree topology.The invertible nature of the phylogenetic Hadamard transform also allows the calculation of site likelihoods from a tree topology vector, allowing one to use the Hadamard transform for maximum likelihood estimation of phylogenetic trees.[12][13] The mechanics of the phylogenetic Hadamard transform involve the calculation of a vectorThe invertible nature of this equation allows one to calculate an expected site pattern vector (or matrix) as follows:This makes it possible to encode the multiple sequence alignment as the site pattern vectorThis tree will exhibit long branch attraction if the data are analyzed using the maximum parsimony criterion (assuming the sequence analyzed is long enough for the observed site pattern frequencies to be close to the expected frequencies shown in theThe long branch attraction reflects the fact that the expected number of site patterns with index 6 -- which support the tree ((A,C),(B,D)); -- exceed the expected number of site patterns that support the true tree (index 4).the remaining indices represent site patterns where a single taxon differs from the other three taxa (so they are the equivalent of terminal branch lengths in a standard maximum likelihood phylogenetic tree).If one wishes to use nucleotide data without recoding as R and Y (and ultimately as 0 and 1) it is possible to encode the site patterns as a matrix.If we consider a four-taxon tree there are a total of 256 site patterns (four nucleotides to the 4th power).This can be illustrated using the example from Hendy et al. (1994),[16] which is based on a multiple sequence alignment of four primate hemoglobin pseudogenes: The much larger number of site patterns in column 0 reflects the fact that column 0 corresponds to transition differences, which accumulate more rapidly than transversion differences in virtually all comparisons of genomic regions (and definitely accumulate more rapidly in the hemoglobin pseudogenes used for this worked example[17]).The site patterns GGAA, CCTT, and TTCC would be encoded in the exact same way.The index based on the second Klein group bit pair is multiplied by 8 to yield the column index (in this case it would be column 24) The cell that would include the count of AACT site patterns is indicated with two asterisks; however, the absence of a number in the example indicates that the sequence alignment include no AACT site patterns (likewise, CCAG, GGTC, and TTGA site patterns, which would be encoded in the same way, are absent).In video compression applications, it is usually used in the form of the sum of absolute transformed differences.The Hadamard transform is also applied in experimental techniques such as NMR, mass spectrometry and crystallography.It is additionally used in some versions of locality-sensitive hashing, to obtain pseudo-random matrix rotations.