Hadamard transform

About: Hadamard transform is a research topic. Over the lifetime, 7262 publications have been published within this topic receiving 94328 citations.

Note on Hadamard's determinant theorem

Abstract: Introduction. We shall call a square matrix A of order n an Hadamard matrix or for brevity an iî-matrix, if each element of A has the value ± 1 and if the determinant of A has the maximum possible value w. It is known that such a matrix A is an iï-matrix [ l] if, and only if, AA'~nEn where A f is the transpose of A and En is the unit matrix of order n. It is also known that, if an iï-matrix of order n > 1 exists, n must have the value 2 or be divisible by 4. The existence of an iî-matrix of order n has been proved [2,3] only for the following values of n>\\\\ (a) w = 2, (b) w = £*+ls~0 mod 4, p a prime, (c) n — m(p-\\-l) where m ^ 2 is the order of an ü-matrix and p is a prime, (d) n = q(q — l) where q is a product of factors of types (a) and (b), (e) n = 172 and for n a product of any number of factors of types (a), (b), (c), (d) and (e). In this note we shall show that an iï-matrix of order n also exists when (f) n — q(q+3) where q and g+4 are both products of factors of types (a) and (b), (g) n = nin2(p+l)p, where Wi>l and w2>l are orders of i7-matrices and p is an odd prime, and (h) n — nin2in(tn+3) where Wi>l and W2>1 are orders of jff-matrices and m and ra+4 are both of the form p + l, p an odd prime. It is interesting to note the presence of the factors tii and w2 in the types (g) and (h) and their absence in the types (d) and (f).Thus, if p is a prime and £*+ls=0 mod 4, an iJ-matrix of order p(p+l) exists but, if p + l =2 mod 4, we can only be sure of the existence of an iJ-matrix of order nitt2p(p+l) where tii>l and ti2>l are orders of iï-matrices. This is analogous to the simpler result that, iî p + 1^0 mod 4 an ü-matrix of order p + l exists but, if p+lz=2 mod 4, we can only be sure of the existence of an i?-matrix of order n(p+l) where n > 1 is the order of an iï-matrix. We shall denote the direct product of two matrices A and B by A B and the unit matrix of order n by Ew.

Hadamard codewords as orthogonal spreading sequences in synchronous DS CDMA systems for mobile radio channels

Abstract: This paper addresses the question if orthogonal spreading sequences, especially Hadamard codewords, can improve the performance of synchronous direct-sequence code-division multiple-access systems applied in a mobile radio environment. For this purpose, the correlation properties of Hadamard codewords are determined and performance simulations are carried out. The results show, that Hadamard codewords have rather poor autocorrelation properties and very inhomogeous crosscorrelation properties ranging from excellent up to very poor depending on which codeword is considered. The performance simulations reveal a behaviour similar to the crosscorrelation properties. Using additional scrambling the autocorrelation properties are improved and the crosscorrelation properties are made more homogenous yielding performance results similar to those obtained for conventional spreading using Gold-codes or preferentially phased Gold-codes. >

Hadamard-Free Circuits Expose the Structure of the Clifford Group

Abstract: The Clifford group plays a central role in quantum randomized benchmarking, quantum tomography, and error correction protocols. Here we study the structural properties of this group. We show that any Clifford operator can be uniquely written in the canonical form $F_{1}HSF_{2}$ , where $H$ is a layer of Hadamard gates, $S$ is a permutation of qubits, and $F_{i}$ are parameterized Hadamard-free circuits chosen from suitable subgroups of the Clifford group. Our canonical form provides a one-to-one correspondence between Clifford operators and layered quantum circuits. We report a polynomial-time algorithm for computing the canonical form. We employ this canonical form to generate a random uniformly distributed $n$ -qubit Clifford operator in runtime $O(n^{2})$ . The number of random bits consumed by the algorithm matches the information-theoretic lower bound. A surprising connection is highlighted between random uniform Clifford operators and the Mallows distribution on the symmetric group. The variants of the canonical form, one with a short Hadamard-free part and one allowing a circuit depth $9n$ implementation of arbitrary Clifford unitaries in the Linear Nearest Neighbor architecture are also discussed. Finally, we study computational quantum advantage where a classical reversible linear circuit can be implemented more efficiently using Clifford gates, and show an explicit example where such an advantage takes place.