Hadamard transform

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

Quasi-convex Functions And Hadamard's Inequality

Abstract: In this paper we establish some new inequalities of Hadamard's type for quasi-convex functions The results obtained include earlier known results in existing literature as special cases of our inequalities.

On Composition of Four-Symbol δ-Codes and Hadamard Matrices

Abstract: It is shown that key instruments for composition of four-symbol 6-codes are the Lagrange identity for polynomials, a certain type of quasisymmetric sequences (i.e., a set of normal or near normal sequences) and base sequences. The following is proved: If a set of base sequences for length t and a set of normal (or near normal) sequences for length n exist then four-symbol 3-codes of length (2n + 1 )t (or nt) can be composed by application of the Lagrange identity. Consequently a new infinite family of Hadamard matrices of order 4uw can be constructed, where w is the order of Williamson matrices and u = (2n + 1 )t (or nt) . Other related topics are also discussed.

Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes

Abstract: In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes of finite fields, and they generalize the constructions given by Feng and Xiang [10,12]. Three infinite families of strongly regular graphs with new parameters are obtained. The main tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.

ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity

Abstract: We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called 'spider' associated with the computational basis, as well as a new arity-N generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of non-linear classical functions. For example, the AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in the ZX-calculus. Consequently, N-controlled gates, hypergraph states, Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the Clifford hierarchy also enjoy encodings with low constant overhead. This suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical non-linear behaviour (e.g. in an oracle) and purely quantum operations. After presenting the calculus, we will prove it is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form.