40 research outputs found
Graph-Based Classification of Self-Dual Additive Codes over Finite Fields
Quantum stabilizer states over GF(m) can be represented as self-dual additive
codes over GF(m^2). These codes can be represented as weighted graphs, and
orbits of graphs under the generalized local complementation operation
correspond to equivalence classes of codes. We have previously used this fact
to classify self-dual additive codes over GF(4). In this paper we classify
self-dual additive codes over GF(9), GF(16), and GF(25). Assuming that the
classical MDS conjecture holds, we are able to classify all self-dual additive
MDS codes over GF(9) by using an extension technique. We prove that the minimum
distance of a self-dual additive code is related to the minimum vertex degree
in the associated graph orbit. Circulant graph codes are introduced, and a
computer search reveals that this set contains many strong codes. We show that
some of these codes have highly regular graph representations.Comment: 20 pages, 13 figure
On Self-Dual Quantum Codes, Graphs, and Boolean Functions
A short introduction to quantum error correction is given, and it is shown
that zero-dimensional quantum codes can be represented as self-dual additive
codes over GF(4) and also as graphs. We show that graphs representing several
such codes with high minimum distance can be described as nested regular graphs
having minimum regular vertex degree and containing long cycles. Two graphs
correspond to equivalent quantum codes if they are related by a sequence of
local complementations. We use this operation to generate orbits of graphs, and
thus classify all inequivalent self-dual additive codes over GF(4) of length up
to 12, where previously only all codes of length up to 9 were known. We show
that these codes can be interpreted as quadratic Boolean functions, and we
define non-quadratic quantum codes, corresponding to Boolean functions of
higher degree. We look at various cryptographic properties of Boolean
functions, in particular the propagation criteria. The new aperiodic
propagation criterion (APC) and the APC distance are then defined. We show that
the distance of a zero-dimensional quantum code is equal to the APC distance of
the corresponding Boolean function. Orbits of Boolean functions with respect to
the {I,H,N}^n transform set are generated. We also study the peak-to-average
power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove
that PAR_IHN of a quadratic Boolean function is related to the size of the
maximum independent set over the corresponding orbit of graphs. A construction
technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It
is finally shown that both PAR_IHN and APC distance can be interpreted as
partial entanglement measures.Comment: Master's thesis. 105 pages, 33 figure
On the classification of Hermitian self-dual additive codes over GF(9)
Additive codes over GF(9) that are self-dual with respect to the Hermitian
trace inner product have a natural application in quantum information theory,
where they correspond to ternary quantum error-correcting codes. However, these
codes have so far received far less interest from coding theorists than
self-dual additive codes over GF(4), which correspond to binary quantum codes.
Self-dual additive codes over GF(9) have been classified up to length 8, and in
this paper we extend the complete classification to codes of length 9 and 10.
The classification is obtained by using a new algorithm that combines two graph
representations of self-dual additive codes. The search space is first reduced
by the fact that every code can be mapped to a weighted graph, and a different
graph is then introduced that transforms the problem of code equivalence into a
problem of graph isomorphism. By an extension technique, we are able to
classify all optimal codes of length 11 and 12. There are 56,005,876
(11,3^11,5) codes and 6493 (12,3^12,6) codes. We also find the smallest codes
with trivial automorphism group.Comment: 12 pages, 6 figure
Quantum social networks
We introduce a physical approach to social networks (SNs) in which each actor
is characterized by a yes-no test on a physical system. This allows us to
consider SNs beyond those originated by interactions based on pre-existing
properties, as in a classical SN (CSN). As an example of SNs beyond CSNs, we
introduce quantum SNs (QSNs) in which actor is characterized by a test of
whether or not the system is in a quantum state. We show that QSNs outperform
CSNs for a certain task and some graphs. We identify the simplest of these
graphs and show that graphs in which QSNs outperform CSNs are increasingly
frequent as the number of vertices increases. We also discuss more general SNs
and identify the simplest graphs in which QSNs cannot be outperformed.Comment: REVTeX4, 6 pages, 3 figure
Basic exclusivity graphs in quantum correlations
A fundamental problem is to understand why quantum theory only violates some
noncontextuality (NC) inequalities and identify the physical principles that
prevent higher-than-quantum violations. We prove that quantum theory only
violates those NC inequalities whose exclusivity graphs contain, as induced
subgraphs, odd cycles of length five or more, and/or their complements. In
addition, we show that odd cycles are the exclusivity graphs of a well-known
family of NC inequalities and that there is also a family of NC inequalities
whose exclusivity graphs are the complements of odd cycles. We characterize the
maximum noncontextual and quantum values of these inequalities, and provide
evidence supporting the conjecture that the maximum quantum violation of these
inequalities is exactly singled out by the exclusivity principle.Comment: REVTeX4, 7 pages, 2 figure
Iterative Decoding on Multiple Tanner Graphs Using Random Edge Local Complementation
In this paper, we propose to enhance the performance of the sum-product
algorithm (SPA) by interleaving SPA iterations with a random local graph update
rule. This rule is known as edge local complementation (ELC), and has the
effect of modifying the Tanner graph while preserving the code. We have
previously shown how the ELC operation can be used to implement an iterative
permutation group decoder (SPA-PD)--one of the most successful iterative
soft-decision decoding strategies at small blocklengths. In this work, we
exploit the fact that ELC can also give structurally distinct parity-check
matrices for the same code. Our aim is to describe a simple iterative decoder,
running SPA-PD on distinct structures, based entirely on random usage of the
ELC operation. This is called SPA-ELC, and we focus on small blocklength codes
with strong algebraic structure. In particular, we look at the extended Golay
code and two extended quadratic residue codes. Both error rate performance and
average decoding complexity, measured by the average total number of messages
required in the decoding, significantly outperform those of the standard SPA,
and compares well with SPA-PD. However, in contrast to SPA-PD, which requires a
global action on the Tanner graph, we obtain a performance improvement via
local action alone. Such localized algorithms are of mathematical interest in
their own right, but are also suited to parallel/distributed realizations.Comment: 5 pages, to appear in proc. IEEE ISIT, June 200
On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12
We consider additive codes over GF(4) that are self-dual with respect to the
Hermitian trace inner product. Such codes have a well-known interpretation as
quantum codes and correspond to isotropic systems. It has also been shown that
these codes can be represented as graphs, and that two codes are equivalent if
and only if the corresponding graphs are equivalent with respect to local
complementation and graph isomorphism. We use these facts to classify all codes
of length up to 12, where previously only all codes of length up to 9 were
known. We also classify all extremal Type II codes of length 14. Finally, we
find that the smallest Type I and Type II codes with trivial automorphism group
have length 9 and 12, respectively.Comment: 18 pages, 4 figure