5,086 research outputs found
On set systems with restricted intersections modulo p and p-ary t-designs
We consider bounds on the size of families ℱ of subsets of a v-set subject to restrictions modulo a prime p on the cardinalities of the pairwise intersections. We improve the known bound when ℱ is allowed to contain sets of different sizes, but only in a special case. We show that if the bound for uniform families ℱ holds with equality, then ℱ is the set of blocks of what we call a p-ary t-design for certain values of t. This motivates us to make a few observations about p-ary t-designs for their own sake
On structures in hypergraphs of models of a theory
We define and study structural properties of hypergraphs of models of a
theory including lattice ones. Characterizations for the lattice properties of
hypergraphs of models of a theory, as well as for structures on sets of
isomorphism types of models of a theory, are given
Order 3 Symmetry in the Clifford Hierarchy
We investigate the action of the first three levels of the Clifford hierarchy
on sets of mutually unbiased bases comprising the Ivanovic MUB and the Alltop
MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the
dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop
vectors splits into three Clifford orbits. These vectors form configurations
with so-called Zauner subspaces, eigenspaces of order 3 elements of the
Clifford group highly relevant to the SIC problem. We identify Alltop vectors
as the magic states that appear in the context of fault-tolerant universal
quantum computing, wherein the appearance of distinct Clifford orbits implies a
surprising inequivalence between some magic states.Comment: 20 pages, 2 figures. Published versio
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