Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs

Belief propagation -- a powerful heuristic method to solve inference problems involving a large number of random variables -- was recently generalized to quantum theory. Like its classical counterpart, this algorithm is exact on trees when the appropriate independence conditions are met and is expected to provide reliable approximations when operated on loopy graphs. In this paper, we benchmark the performances of loopy quantum belief propagation (QBP) in the context of finite-tempereture quantum many-body physics. Our results indicate that QBP provides reliable estimates of the high-temperature correlation function when the typical loop size in the graph is large. As such, it is suitable e.g. for the study of quantum spin glasses on Bethe lattices and the decoding of sparse quantum error correction codes.Comment: 5 pages, 4 figure

An efficient CDMA decoder for correlated information sources

We consider the detection of correlated information sources in the ubiquitous Code-Division Multiple-Access (CDMA) scheme. We propose a message-passing based scheme for detecting correlated sources directly, with no need for source coding. The detection is done simultaneously over a block of transmitted binary symbols (word). Simulation results are provided demonstrating a substantial improvement in bit-error-rate in comparison with the unmodified detector and the alternative of source compression. The robustness of the error-performance improvement is shown under practical model settings, including wrong estimation of the generating Markov transition matrix and finite-length spreading codes.Comment: 11 page

Finite size effects and error-free communication in Gaussian channels

The efficacy of a specially constructed Gallager-type error-correcting code to communication in a Gaussian channel is being examined. The construction is based on the introduction of complex matrices, used in both encoding and decoding, which comprise sub-matrices of cascading connection values. The finite size effects are estimated for comparing the results to the bounds set by Shannon. The critical noise level achieved for certain code-rates and infinitely large systems nearly saturates the bounds set by Shannon even when the connectivity used is low

Statistical Physics of Irregular Low-Density Parity-Check Codes

Low-density parity-check codes with irregular constructions have been recently shown to outperform the most advanced error-correcting codes to date. In this paper we apply methods of statistical physics to study the typical properties of simple irregular codes. We use the replica method to find a phase transition which coincides with Shannon's coding bound when appropriate parameters are chosen. The decoding by belief propagation is also studied using statistical physics arguments; the theoretical solutions obtained are in good agreement with simulations. We compare the performance of irregular with that of regular codes and discuss the factors that contribute to the improvement in performance.Comment: 20 pages, 9 figures, revised version submitted to JP

Statistical mechanics of error exponents for error-correcting codes

Error exponents characterize the exponential decay, when increasing message length, of the probability of error of many error-correcting codes. To tackle the long standing problem of computing them exactly, we introduce a general, thermodynamic, formalism that we illustrate with maximum-likelihood decoding of low-density parity-check (LDPC) codes on the binary erasure channel (BEC) and the binary symmetric channel (BSC). In this formalism, we apply the cavity method for large deviations to derive expressions for both the average and typical error exponents, which differ by the procedure used to select the codes from specified ensembles. When decreasing the noise intensity, we find that two phase transitions take place, at two different levels: a glass to ferromagnetic transition in the space of codewords, and a paramagnetic to glass transition in the space of codes.Comment: 32 pages, 13 figure

Survey Propagation as local equilibrium equations

It has been shown experimentally that a decimation algorithm based on Survey Propagation (SP) equations allows to solve efficiently some combinatorial problems over random graphs. We show that these equations can be derived as sum-product equations for the computation of marginals in an extended space where the variables are allowed to take an additional value -- $*$ -- when they are not forced by the combinatorial constraints. An appropriate ``local equilibrium condition'' cost/energy function is introduced and its entropy is shown to coincide with the expected logarithm of the number of clusters of solutions as computed by SP. These results may help to clarify the geometrical notion of clusters assumed by SP for the random K-SAT or random graph coloring (where it is conjectured to be exact) and helps to explain which kind of clustering operation or approximation is enforced in general/small sized models in which it is known to be inexact.Comment: 13 pages, 3 figure

Error-correcting codes that nearly saturate Shannon's bound

Gallager-type error-correcting codes that nearly saturate Shannon's bound are constructed using insight gained from mapping the problem onto that of an Ising spin system. The performance of the suggested codes is evaluated for different code rates in both finite and infinite message length

Determinants of the voltage dependence of G protein modulation within calcium channel β subunits

CaVβ subunits of voltage-gated calcium channels contain two conserved domains, a src-homology-3 (SH3) domain and a guanylate kinase-like (GK) domain with an intervening HOOK domain. We have shown in a previous study that, although Gβγ-mediated inhibitory modulation of CaV2.2 channels did not require the interaction of a CaVβ subunit with the CaVα1 subunit, when such interaction was prevented by a mutation in the α1 subunit, G protein modulation could not be removed by a large depolarization and showed voltage-independent properties (Leroy et al., J Neurosci 25:6984–6996, 2005). In this study, we have investigated the ability of mutant and truncated CaVβ subunits to support voltage-dependent G protein modulation in order to determine the minimal domain of the CaVβ subunit that is required for this process. We have coexpressed the CaVβ subunit constructs with CaV2.2 and α2δ-2, studied modulation by the activation of the dopamine D2 receptor, and also examined basal tonic modulation. Our main finding is that the CaVβ subunit GK domains, from either β1b or β2, are sufficient to restore voltage dependence to G protein modulation. We also found that the removal of the variable HOOK region from β2a promotes tonic voltage-dependent G protein modulation. We propose that the absence of the HOOK region enhances Gβγ binding affinity, leading to greater tonic modulation by basal levels of Gβγ. This tonic modulation requires the presence of an SH3 domain, as tonic modulation is not supported by any of the CaVβ subunit GK domains alone

Statistical physics of low density parity check error correcting codes

We study the performance of Low Density Parity Check (LDPC) error-correcting codes using the methods of statistical physics. LDPC codes are based on the generation of codewords using Boolean sums of the original message bits by employing two randomly-constructed sparse matrices. These codes can be mapped onto Ising spin models and studied using common methods of statistical physics. We examine various regular constructions and obtain insight into their theoretical and practical limitations. We also briefly report on results obtained for irregular code constructions, for codes with non-binary alphabet, and on how a finite system size effects the error probability

A survey of three-dimensional turbo codes and recent performance enhancements

This paper presents a survey of two techniques intended for improving the performance of conventional turbo codes (TCs). The first part of this work is dedicated to explore a hybrid concatenation structure combining both parallel and serial concatenation based on a three-dimensional (3D) code. The 3D structure, recently introduced by Berrou et al., is able to ensure large asymptotic gains at very low error rates at the expense of an increase in complexity and a loss in the convergence threshold. In order to reduce the loss in the convergence threshold, the authors consider first a time-varying construction of the post-encoded parity. Then, they investigate the association of the 3D TC with high-order modulations according to the bit-interleaved coded modulation approach. The second part of this study deals with irregular TCs. In contrast to 3D TCs, although irregular TCs can achieve performance closer to capacity, their asymptotic performance is very poor. Therefore, the authors propose irregular turbo coding schemes with suitable interleavers in order to improve their distance properties. Finally, a modified encoding procedure, inspired from the 3D TC, makes it possible to obtain irregular TCs which perform better than the corresponding regular codes in both the waterfall and the error floor regions