28 research outputs found
The cohomological reduction method for computing n-dimensional cocyclic matrices
Provided that a cohomological model for is known, we describe a method
for constructing a basis for -cocycles over , from which the whole set of
-dimensional -cocyclic matrices over may be straightforwardly
calculated. Focusing in the case (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 -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 , this method provides an uniform way of looking for higher
dimensional -cocyclic Hadamard matrices for the first time. We illustrate
the method with some examples, for . In particular, we give some
examples of improper 3-dimensional -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-
Equivalence classes of multiplicative central (p n , p n , p n , 1)-relative difference sets
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