The cohomological reduction method for computing n-dimensional cocyclic matrices

Provided that a cohomological model for $G$ is known, we describe a method for constructing a basis for $n$-cocycles over $G$, from which the whole set of $n$-dimensional $n$-cocyclic matrices over $G$ may be straightforwardly calculated. Focusing in the case $n=2$ (which is of special interest, e.g. for looking for cocyclic Hadamard matrices), this method provides a basis for 2-cocycles in such a way that representative $2$-cocycles are calculated all at once, so that there is no need to distinguish between inflation and transgression 2-cocycles (as it has traditionally been the case until now). When $n>2$, this method provides an uniform way of looking for higher dimensional $n$-cocyclic Hadamard matrices for the first time. We illustrate the method with some examples, for $n=2,3$. In particular, we give some examples of improper 3-dimensional $3$-cocyclic Hadamard matrices.Comment: 17 pages, 0 figure

Quantum Computation with Coherent Spin States and the Close Hadamard Problem

We study a model of quantum computation based on the continuously-parameterized yet finite-dimensional Hilbert space of a spin system. We explore the computational powers of this model by analyzing a pilot problem we refer to as the close Hadamard problem. We prove that the close Hadamard problem can be solved in the spin system model with arbitrarily small error probability in a constant number of oracle queries. We conclude that this model of quantum computation is suitable for solving certain types of problems. The model is effective for problems where symmetries between the structure of the information associated with the problem and the structure of the unitary operators employed in the quantum algorithm can be exploited.Comment: RevTeX4, 13 pages with 8 figures. Accepted for publication in Quantum Information Processing. Article number: s11128-015-1229-

Entropy of the Retina Template

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On twin prime power Hadamard matrices

In this paper, we show that exactly one Hadamard matrix constructed using the twin prime power method is cocyclic. We achieve this by showing that the action of the automorphism group of a Hadamard matrix developed from a difference set induces a 2-transitive action on the rows of the matrix or is intransitive. We then use Ito’s classification of Hadamard matrices with 2-transitive automorphism groups to derive a necessary condition on the order of a cocyclic Hadamard matrix developed from a difference set. This work answers a research problem posed by K.J. Horadam, and exhibits the first known infinite family of Hadamard matrices which are not cocyclic