42 research outputs found
Examples of minimal-memory, non-catastrophic quantum convolutional encoders
One of the most important open questions in the theory of quantum
convolutional coding is to determine a minimal-memory, non-catastrophic,
polynomial-depth convolutional encoder for an arbitrary quantum convolutional
code. Here, we present a technique that finds quantum convolutional encoders
with such desirable properties for several example quantum convolutional codes
(an exposition of our technique in full generality will appear elsewhere). We
first show how to encode the well-studied Forney-Grassl-Guha (FGG) code with an
encoder that exploits just one memory qubit (the former Grassl-Roetteler
encoder requires 15 memory qubits). We then show how our technique can find an
online decoder corresponding to this encoder, and we also detail the operation
of our technique on a different example of a quantum convolutional code.
Finally, the reduction in memory for the FGG encoder makes it feasible to
simulate the performance of a quantum turbo code employing it, and we present
the results of such simulations.Comment: 5 pages, 2 figures, Accepted for the International Symposium on
Information Theory 2011 (ISIT 2011), St. Petersburg, Russia; v2 has minor
change
Analog network coding in general SNR regime: Performance of a greedy scheme
The problem of maximum rate achievable with analog network coding for a
unicast communication over a layered relay network with directed links is
considered. A relay node performing analog network coding scales and forwards
the signals received at its input. Recently this problem has been considered
under certain assumptions on per node scaling factor and received SNR.
Previously, we established a result that allows us to characterize the optimal
performance of analog network coding in network scenarios beyond those that can
be analyzed using the approaches based on such assumptions.
The key contribution of this work is a scheme to greedily compute a lower
bound to the optimal rate achievable with analog network coding in the general
layered networks. This scheme allows for exact computation of the optimal
achievable rates in a wider class of layered networks than those that can be
addressed using existing approaches. For the specific case of Gaussian N-relay
diamond network, to the best of our knowledge, the proposed scheme provides the
first exact characterization of the optimal rate achievable with analog network
coding. Further, for general layered networks, our scheme allows us to compute
optimal rates within a constant gap from the cut-set upper bound asymptotically
in the source power.Comment: 11 pages, 5 figures. Fixed an issue with the notation in the
statement and proof of Lemma 1. arXiv admin note: substantial text overlap
with arXiv:1204.2150 and arXiv:1202.037
Merging Belief Propagation and the Mean Field Approximation: A Free Energy Approach
We present a joint message passing approach that combines belief propagation
and the mean field approximation. Our analysis is based on the region-based
free energy approximation method proposed by Yedidia et al. We show that the
message passing fixed-point equations obtained with this combination correspond
to stationary points of a constrained region-based free energy approximation.
Moreover, we present a convergent implementation of these message passing
fixedpoint equations provided that the underlying factor graph fulfills certain
technical conditions. In addition, we show how to include hard constraints in
the part of the factor graph corresponding to belief propagation. Finally, we
demonstrate an application of our method to iterative channel estimation and
decoding in an orthogonal frequency division multiplexing (OFDM) system
Speeding up Glauber Dynamics for Random Generation of Independent Sets
The maximum independent set (MIS) problem is a well-studied combinatorial
optimization problem that naturally arises in many applications, such as
wireless communication, information theory and statistical mechanics.
MIS problem is NP-hard, thus many results in the literature focus on fast
generation of maximal independent sets of high cardinality. One possibility is
to combine Gibbs sampling with coupling from the past arguments to detect
convergence to the stationary regime. This results in a sampling procedure with
time complexity that depends on the mixing time of the Glauber dynamics Markov
chain.
We propose an adaptive method for random event generation in the Glauber
dynamics that considers only the events that are effective in the coupling from
the past scheme, accelerating the convergence time of the Gibbs sampling
algorithm
Stabilizer codes from modified symplectic form
Stabilizer codes form an important class of quantum error correcting codes
which have an elegant theory, efficient error detection, and many known
examples. Constructing stabilizer codes of length is equivalent to
constructing subspaces of which are
"isotropic" under the symplectic bilinear form defined by . As a
result, many, but not all, ideas from the theory of classical error correction
can be translated to quantum error correction. One of the main theoretical
contribution of this article is to study stabilizer codes starting with a
different symplectic form.
In this paper, we concentrate on cyclic codes. Modifying the symplectic form
allows us to generalize the previous known construction for linear cyclic
stabilizer codes, and in the process, circumvent some of the Galois theoretic
no-go results proved there. More importantly, this tweak in the symplectic form
allows us to make use of well known error correcting algorithms for cyclic
codes to give efficient quantum error correcting algorithms. Cyclicity of error
correcting codes is a "basis dependent" property. Our codes are no more
"cyclic" when they are derived using the standard symplectic forms (if we
ignore the error correcting properties like distance, all such symplectic forms
can be converted to each other via a basis transformation). Hence this change
of perspective is crucial from the point of view of designing efficient
decoding algorithm for these family of codes. In this context, recall that for
general codes, efficient decoding algorithms do not exist if some widely
believed complexity theoretic assumptions are true
Haplotype Assembly: An Information Theoretic View
This paper studies the haplotype assembly problem from an information
theoretic perspective. A haplotype is a sequence of nucleotide bases on a
chromosome, often conveniently represented by a binary string, that differ from
the bases in the corresponding positions on the other chromosome in a
homologous pair. Information about the order of bases in a genome is readily
inferred using short reads provided by high-throughput DNA sequencing
technologies. In this paper, the recovery of the target pair of haplotype
sequences using short reads is rephrased as a joint source-channel coding
problem. Two messages, representing haplotypes and chromosome memberships of
reads, are encoded and transmitted over a channel with erasures and errors,
where the channel model reflects salient features of high-throughput
sequencing. The focus of this paper is on the required number of reads for
reliable haplotype reconstruction, and both the necessary and sufficient
conditions are presented with order-wise optimal bounds.Comment: 30 pages, 5 figures, 1 tabel, journa