55,912 research outputs found
Quantum error-correcting codes associated with graphs
We present a construction scheme for quantum error correcting codes. The
basic ingredients are a graph and a finite abelian group, from which the code
can explicitly be obtained. We prove necessary and sufficient conditions for
the graph such that the resulting code corrects a certain number of errors.
This allows a simple verification of the 1-error correcting property of
fivefold codes in any dimension. As new examples we construct a large class of
codes saturating the singleton bound, as well as a tenfold code detecting 3
errors.Comment: 8 pages revtex, 5 figure
New and Updated Semidefinite Programming Bounds for Subspace Codes
We show that and, more generally, by semidefinite programming for . Furthermore, we extend results by Bachoc et al. on SDP bounds for
, where is odd and is small, to for small and
small
Construction of Almost Disjunct Matrices for Group Testing
In a \emph{group testing} scheme, a set of tests is designed to identify a
small number of defective items among a large set (of size ) of items.
In the non-adaptive scenario the set of tests has to be designed in one-shot.
In this setting, designing a testing scheme is equivalent to the construction
of a \emph{disjunct matrix}, an matrix where the union of supports
of any columns does not contain the support of any other column. In
principle, one wants to have such a matrix with minimum possible number of
rows (tests). One of the main ways of constructing disjunct matrices relies on
\emph{constant weight error-correcting codes} and their \emph{minimum
distance}. In this paper, we consider a relaxed definition of a disjunct matrix
known as \emph{almost disjunct matrix}. This concept is also studied under the
name of \emph{weakly separated design} in the literature. The relaxed
definition allows one to come up with group testing schemes where a
close-to-one fraction of all possible sets of defective items are identifiable.
Our main contribution is twofold. First, we go beyond the minimum distance
analysis and connect the \emph{average distance} of a constant weight code to
the parameters of an almost disjunct matrix constructed from it. Our second
contribution is to explicitly construct almost disjunct matrices based on our
average distance analysis, that have much smaller number of rows than any
previous explicit construction of disjunct matrices. The parameters of our
construction can be varied to cover a large range of relations for and .Comment: 15 Page
Spin glass reflection of the decoding transition for quantum error correcting codes
We study the decoding transition for quantum error correcting codes with the
help of a mapping to random-bond Wegner spin models.
Families of quantum low density parity-check (LDPC) codes with a finite
decoding threshold lead to both known models (e.g., random bond Ising and
random plaquette gauge models) as well as unexplored earlier generally
non-local disordered spin models with non-trivial phase diagrams. The decoding
transition corresponds to a transition from the ordered phase by proliferation
of extended defects which generalize the notion of domain walls to non-local
spin models. In recently discovered quantum LDPC code families with finite
rates the number of distinct classes of such extended defects is exponentially
large, corresponding to extensive ground state entropy of these codes.
Here, the transition can be driven by the entropy of the extended defects, a
mechanism distinct from that in the local spin models where the number of
defect types (domain walls) is always finite.Comment: 15 pages, 2 figure
- …