Equiangular Lines and Antipodal Covers
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Date
2010-09-22T14:02:27Z
Authors
Mirjalalieh Shirazi, Mirhamed
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
It is not hard to see that the number of equiangular lines in a complex space of dimension $d$ is at most $d^{2}$. A set of $d^{2}$ equiangular lines in a $d$-dimensional complex space is of significant importance in Quantum Computing as it corresponds to a measurement for which its statistics determine completely the quantum state on which the measurement is carried out. The existence of $d^{2}$ equiangular lines in a $d$-dimensional complex space is only known for a few values of $d$, although physicists conjecture that they do exist for any value of $d$.
The main results in this thesis are:
\begin{enumerate}
\item Abelian covers of complete graphs that have certain parameters can be used to construct sets of $d^2$ equiangular lines in $d$-dimen\-sion\-al space;
\item we exhibit infinitely many parameter sets that satisfy all the known necessary conditions for the existence of such a cover; and
\item we find the decompose of the space into irreducible modules over the Terwilliger algebra of covers of complete graphs.
\end{enumerate}
A few techniques are known for constructing covers of complete graphs, none of which can be used to construct covers that lead to sets of $d^{2}$ equiangular lines in $d$-dimensional complex spaces. The third main result is developed in the hope of assisting such construction.
Description
Keywords
Algebraic Combinatorics, Quantum Computing, Graph Theory