Sparse-graph Codes Research Articles

Minimum-Rate Spectrum-Blind Sampling Based on Sparse-Graph Codes

We study the problem of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spectrum-blind</i> sampling, that is, sampling signals with sparse spectra whose frequency support is unknown. The minimum sampling rate for this class of signals has been established as twice the measure of its frequency support; however, constructive sampling schemes that achieve this minimum rate are not known to exist, to the best of our knowledge. We propose a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">constructive</i> sampling framework by leveraging a mix of tools from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">modern coding theory</i> , which has been largely untapped in the field of sampling. We make interesting connections between the fundamental problem of spectrum-blind sampling, and that of designing erasure-correcting codes based on sparse graphs, which have both theoretically and practically revolutionized the design of modern communication systems. Our key idea is to cleverly exploit, rather than avoid, the resulting aliasing artifacts induced by subsampling, which introduce linear mixing of spectral components in the form of parity constraints for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sparse-graph codes</i> . We achieve this by subsampling the input signal after filtering it using a carefully designed ‘sparse-graph coded filter-bank’ structure, where the pass-band on/off patterns of the filters are designed to match the parity-check constraints of a sparse-graph code. We show that the signal reconstruction under this sampling scheme is equivalent to the fast message-passing based “peeling” decoding of sparse-graph codes for reliable transmission over erasure channels. Most importantly, we further show that the achievable sampling rate is determined by the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rate</i> of the sparse-graph codes used in the filter bank. As a result, based on insights derived from the design of capacity-achieving sparse-graph codes (such as Low Density Parity Check codes), we can <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">simultaneously</i> approach the minimum sampling rate for spectrum-blind sampling and low computational complexity based on fast peeling-based decoding with operations per unit of time scaling <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linearly</i> with the sampling rate.

EBP-GEXIT Charts for M-Ary AWGN Channel for Generalized LDPC and Turbo Codes

The maximum a posteriori (MAP) threshold corresponds to the fundamental limit that one can hope to achieve with the given channel code ensemble. Apart from theoretical interests, finding this limit is also desirable since <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spatial-coupled</i> code ensembles approach this MAP threshold due to phenomenon termed as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">threshold saturation</i> . However finding this MAP threshold, in general, is known to be computationally prohibitive. This work proposes a tractable method for estimating the MAP threshold for various families of sparse-graph code ensembles over non-binary complex-input additive white Gaussian noise (AWGN) channel. Towards this, we provide a method to approximate the extended belief propagation generalized extrinsic information transfer (EBP-GEXIT) chart and estimate the MAP threshold by applying the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Maxwell construction</i> to it. To illustrate the validity of our method, we study spatial coupling for serially-concatenated turbo-codes and numerically observe threshold saturation of these codes to the MAP thresholds estimated via our method.

Approach for the construction of non-Calderbank-Steane-Shor low-density-generator-matrix–based quantum codes

Quantum low-density generator matrix (QLDGM) codes based on Calderbank-Steane-Shor (CSS) constructions have shown unprecedented error correction capabilities, displaying much improved performance in comparison to other sparse-graph codes. However, the nature of CSS designs and the manner in which they must be decoded limit the performance that is attainable with codes that are based on this construction. This motivates the search for quantum code design strategies capable of avoiding the drawbacks associated with CSS codes. In this article, we introduce non-CSS quantum code constructions based on classical LDGM codes. The proposed codes are derived from CSS QLDGM designs by performing specific row operations on their quantum parity check matrices to modify the associated decoding graphs. The application of this method results in performance improvements in comparison to CSS QLDGM codes, while also allowing for greater flexibility in the design process. The proposed non-CSS QLDGM scheme outperforms the best quantum low-density parity check codes that have appeared in the literature.

Sub-Linear Time Support Recovery for Compressed Sensing Using Sparse-Graph Codes

We study the support recovery problem for compressed sensing, where the goal is to reconstruct the a high-dimensional $K$-sparse signal $\mathbf{x}\in\mathbb{R}^N$, from low-dimensional linear measurements with and without noise. Our key contribution is a new compressed sensing framework through a new family of carefully designed sparse measurement matrices associated with minimal measurement costs and a low-complexity recovery algorithm. The measurement matrix in our framework is designed based on the well-crafted sparsification through capacity-approaching sparse-graph codes, where the sparse coefficients can be recovered efficiently in a few iterations by performing simple error decoding over the observations. We formally connect this general recovery problem with sparse-graph decoding in packet communication systems, and analyze our framework in terms of the measurement cost, time complexity and recovery performance. In the noiseless setting, our framework can recover any arbitrary $K$-sparse signal in $O(K)$ time using $2K$ measurements asymptotically with high probability. In the noisy setting, when the sparse coefficients take values in a finite and quantized alphabet, our framework can achieve the same goal in time $O(K\log(N/K))$ using $O(K\log(N/K))$ measurements obtained from measurement matrix with elements $\{-1,0,1\}$. When the sparsity $K$ is sub-linear in the signal dimension $K=O(N^\delta)$ for some $0<\delta<1$, our results are order-optimal in terms of measurement costs and run-time, both of which are sub-linear in the signal dimension $N$. The sub-linear measurement cost and run-time can also be achieved with continuous-valued sparse coefficients, with a slight increment in the logarithmic factors. This offers the desired scalability of our framework that can potentially enable real-time or near-real-time processing for massive datasets featuring sparsity.

SAFFRON: A Fast, Efficient, and Robust Framework for Group Testing Based on Sparse-Graph Codes

Group testing is the problem of identifying K defective items among n items by pooling groups of items. In this paper, we design group testing algorithms for approximate recovery with order-optimal sample complexity by leveraging design and analysis tools from modern sparse-graph coding theory. Our algorithm, SAFFRON, recovers at least (1 - e)K defective items w.p.1 - K/nr with m = 2(1 + r)C(e)K log 2 n tests, where e is an arbitrarily small constant, C(e) is a precisely characterizable constant, and r is any positive integer. The decoding complexity is Θ(K log n). We also propose variations of SAFFRON, which are robust to noise and unknown offsets. For example, for n ≃ 4.3 × 10 9 and K = 128, our algorithm is observed to recover all defective items with m ≃ 8.3 × 10 5 tests, even in the presence of noisy test results. Moreover, the decoding time takes less than 4 seconds on a laptop with a 2 GHz Intel Core i7 and 8 GB memory.

Learning Mixtures of Sparse Linear Regressions Using Sparse Graph Codes

In this paper, we consider the mixture of sparse linear regressions model. Let ${\beta}^{(1)},\ldots,{\beta}^{(L)}\in\mathbb{C}^n$ be $ L $ unknown sparse parameter vectors with a total of $ K $ non-zero coefficients. Noisy linear measurements are obtained in the form $y_i={x}_i^H {\beta}^{(\ell_i)} + w_i$, each of which is generated randomly from one of the sparse vectors with the label $ \ell_i $ unknown. The goal is to estimate the parameter vectors efficiently with low sample and computational costs. This problem presents significant challenges as one needs to simultaneously solve the demixing problem of recovering the labels $ \ell_i $ as well as the estimation problem of recovering the sparse vectors $ {\beta}^{(\ell)} $. Our solution to the problem leverages the connection between modern coding theory and statistical inference. We introduce a new algorithm, Mixed-Coloring, which samples the mixture strategically using query vectors $ {x}_i $ constructed based on ideas from sparse graph codes. Our novel code design allows for both efficient demixing and parameter estimation. In the noiseless setting, for a constant number of sparse parameter vectors, our algorithm achieves the order-optimal sample and time complexities of $\Theta(K)$. In the presence of Gaussian noise, for the problem with two parameter vectors (i.e., $L=2$), we show that the Robust Mixed-Coloring algorithm achieves near-optimal $\Theta(K polylog(n))$ sample and time complexities. When $K=O(n^{\alpha})$ for some constant $\alpha\in(0,1)$ (i.e., $K$ is sublinear in $n$), we can achieve sample and time complexities both sublinear in the ambient dimension. In one of our experiments, to recover a mixture of two regressions with dimension $n=500$ and sparsity $K=50$, our algorithm is more than $300$ times faster than EM algorithm, with about one third of its sample cost.

Sparse Graph Codes for Channels with Synchronous Errors

Sparse Graph Codes for Channels with Synchronous Errors

Fast Beam Alignment for Millimeter Wave Communications: A Sparse Encoding and Phaseless Decoding Approach

In this paper, we studied the problem of beam alignment for millimeter wave (mmWave) communications, in which we assume a hybrid analog and digital beamforming structure is employed at the transmitter (i.e. base station), and an omni-directional antenna or an antenna array is used at the receiver (i.e. user). By exploiting the sparse scattering nature of mmWave channels, the beam alignment problem is formulated as a sparse encoding and phaseless decoding problem. More specifically, the problem of interest involves finding a sparse sensing matrix and an efficient recovery algorithm to recover the support and magnitude of the sparse signal from compressive phaseless measurements. A sparse bipartite graph coding (SBG-Coding) algorithm is developed for sparse encoding and phaseless decoding. Our theoretical analysis shows that, in the noiseless case, our proposed algorithm can perfectly recover the support and magnitude of the sparse signal with probability exceeding a pre-specified value from $\mathcal{O}(K^2)$ measurements, where $K$ is the number of nonzero entries of the sparse signal. The proposed algorithm has a simple decoding procedure which is computationally efficient and noise-robust. Simulation results show that our proposed method renders a reliable beam alignment in the low and moderate signal-to-noise ratio (SNR) regimes and presents a clear performance advantage over existing methods.

How to Achieve the Capacity of Asymmetric Channels

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third paradigm is based on an idea of B\"ocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.

FFAST: An Algorithm for Computing an Exactly $ k$ -Sparse DFT in $O( k\log k)$ Time

It is a well-known fact that the Discrete Fourier Transform (DFT) $ \vec {X}$ of an arbitrary $ n$ -length input signal $ \vec {x}$ , can be computed from all the $ n$ time-domain samples in $O( n\log n)$ operations via a Fast Fourier Transform (FFT) algorithm. If the spectrum $ \vec {X}$ is $ k$ -sparse (where $ k\ll n$ ), can we do better? We show that asymptotically in $ k$ and $ n$ , when $ k$ is sub-linear in $ n$ (precisely, $ k= O( n^{ \delta }$ ), where $0 \leq \delta ), and the support of the non-zero DFT coefficients is uniformly random, the fast fourier aliasing-based sparse transform (FFAST) algorithm, proposed in this paper, computes, with asymptotically high probability, the $k$ non-zero DFT coefficients of $\vec {X}$ from $O( k)$ samples of $ \vec {x}$ in $O( k\log k)$ arithmetic operations. Further, the constants in the big Oh notation for both sample and computational cost are small, e.g., when $ \delta , which essentially covers a wide range of sub-linear sparsity cases, the sample cost is less than $4 k$ . Although, in this paper we assume that the samples of the signal $ \vec {x}$ observed by the FFAST are noise-free, a noise-robust extension of the FFAST is provided in a companion (Part II) paper [1] . Our approach is based on filter-less sub-sampling of the input signal using a set of carefully chosen uniform sub-sampling patterns guided by the Chinese Remainder Theorem (CRT) . The idea is to cleverly exploit, rather than avoid, the resulting aliasing artifacts induced by sub-sampling. Specifically, the sub-sampling operation on the time-domain signal $ \vec {x}$ is designed to create aliasing patterns of the non-zero coefficients of the spectrum $ \vec {X}$ to “look like” parity-check constraints of a good erasure-correcting sparse-graph code. Next, we show that computing the sparse DFT $ \vec {X}$ is equivalent to decoding of an appropriate sparse-graph code . The sparse-graph codes further permit a fast peeling-style decoding. Consequently, the resulting DFT computation is low in both the sample and the decoding complexity. We analytically connect our proposed CRT-based aliasing framework to random sparse-graph codes, and analyze the performance of our algorithm using density evolution techniques from coding theory. We provide extensive simulation results, that are in tight agreement with our theoretical findings.

PhaseCode: Fast and Efficient Compressive Phase Retrieval Based on Sparse-Graph Codes

We consider the problem of recovering a complex signal $ {x} \in \mathbb {C}^{n}$ from $m$ intensity measurements of the form $ | {a}_{i} ^{\mathrm{ H}} {x}|, ~1 \leq i \leq m$ , where $ {a}_{i} ^{\mathrm{ H}}$ is the $i$ th row of measurement matrix $ {A} \in \mathbb {C}^{m \times n}$ . Our main focus is on the case where the measurement vectors are unconstrained, and where $ {x}$ is exactly $K$ -sparse, or the so-called general compressive phase retrieval problem. We introduce PhaseCode , a novel family of fast and efficient algorithms that are based on a sparse-graph coding framework. We show that in the noiseless case, the PhaseCode algorithm can recover an arbitrarily-close-to-one fraction of the $K$ nonzero signal components using only slightly more than $4K$ measurements when the support of the signal is uniformly random, with the order-optimal time and memory complexity of $\Theta (K)$ . 1 It is known that the fundamental limit for the number of measurements in compressive phase retrieval problem is $4K - o(K)$ for the more difficult problem of recovering the signal exactly and with no assumptions on its support distribution. This shows that under mild relaxation of the conditions, our algorithm is the first constructive capacity-approaching compressive phase retrieval algorithm: in fact, our algorithm is also order-optimal in complexity and memory. Furthermore, we show that for any signal $ {x}$ , PhaseCode can recover a random $(1 - p)$ -fraction of the nonzero components of $ {x}$ with high probability, where $p$ can be made arbitrarily close to zero, with sample complexity $m = c(p)K$ , where $c(p)$ is a small constant depending on $p$ that can be precisely calculated, with optimal time and memory complexity. As a result, assuming that the nonzero components of $ {x}$ are lower bounded by $\Theta (1)$ and upper bounded by $\Theta (K^{\gamma })$ for some positive constant $\gamma , we are able to provide a strong $\ell _{1}$ guarantee for the estimated signal $\hat { {x}}$ as follows: $\| \hat { {x}} - {x}\|_{1} \leq p \| {x} \|_{1}(1 + o(1))$ , where $p$ can be made arbitrarily close to zero. As one instance, the PhaseCode algorithm can provably recover, with high probability, a random $1 - 10^{-7}$ fraction of the significant signal components, using at most $m = 14K$ measurements. Next, motivated by some important practical classes of optical systems, we consider a “Fourier-friendly” constrained measurement setting, and show that its performance matches that of the unconstrained setting, when the signal is sparse in the Fourier domain with uniform support. In the Fourier-friendly setting that we consider, the measurement matrix is constrained to be a cascade of Fourier matrices (corresponding to optical lenses) and diagonal matrices (corresponding to diffraction mask patterns). Finally, we tackle the compressive phase retrieval problem in the presence of noise, where measurements are in the form of $y_{i}= | {a}_{i} ^{\mathrm{ H}} {x}|^{2}+w_{i}$ , and $w_{i}$ is the additive noise to the $i$ th measurement. We assume that the signal is quantized, and each nonzero component can take $L_{m}$ possible magnitudes and $L_{p}$ possible phases. We consider the regime, where $K=\beta n^\delta $ , $\delta \in (0,1)$ . We use the same architecture of PhaseCode for the noiseless case, and robustify it using two schemes: the almost-linear scheme and the sublinear scheme. We prove that with high probability, the almost-linear scheme recovers $ {x}$ with sample complexity $\Theta (K \log (n))$ and computational complexity $\Theta (L_{m} L_{p} n \log (n))$ , and the sublinear scheme recovers $ {x}$ with sample complexity $\Theta (K\log ^{3}(n))$ and computational complexity $\Theta (L_{m} L_{p} K\log ^{3}(n))$ . Throughout, we provide extensive simulation results that validate the practical power of our proposed algorithms for the sparse unconstrained and Fourier-friendly measurement settings, for noiseless and noisy scenarios. 1 Here, we define the notation $ \mathcal {O}(\cdot )$ , $\Theta (\cdot )$ , and $\Omega (\cdot )$ . We have $f= \mathcal {O}(g)$ if and only if there exists a constant $C_{1}>0$ such that $ \left |{f/g}\right | ; $f=\Theta (g)$ if and only if there exist two constants $C_{1},C_{2}>0$ such that $C_{1} ; and $f=\Omega (g)$ if and only if there exists a constant $C_{1}>0$ such that $ \left |{f/g}\right |>C_{1}$ .

A Fast Hadamard Transform for Signals With Sublinear Sparsity in the Transform Domain

In this paper, we design a new iterative low-complexity algorithm for computing the Walsh-Hadamard transform (WHT) of an N dimensional signal with a K-sparse WHT. We suppose that N is a power of two and K = O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> ), scales sublinearly in N for some α ∈ (0, 1). Assuming a random support model for the nonzero transform-domain components, our algorithm reconstructs the WHT of the signal with a sample complexity O(K log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (N/K)) and a computational complexity O(K log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (K) log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (N/K)). Moreover, the algorithm succeeds with a high probability approaching 1 for large dimension N. Our approach is mainly based on the subsampling (aliasing) property of the WHT, where by a carefully designed subsampling of the time-domain signal, a suitable aliasing pattern is induced in the transform domain. We treat the resulting aliasing patterns as parity-check constraints and represent them by a bipartite graph. We analyze the properties of the resulting bipartite graphs and borrow ideas from codes defined over sparse bipartite graphs to formulate the recovery of the nonzero spectral values as a peeling decoding algorithm for a specific sparse-graph code transmitted over a binary erasure channel. This enables us to use tools from coding theory (belief-propagation analysis) to characterize the asymptotic performance of our algorithm in the very sparse (α ∈ (0, 1/3]) and the less sparse (α ∈ (1/3, 1)) regime. Comprehensive simulation results are provided to assess the empirical performance of the proposed algorithm.

Design of Irregular Repeat Accumulate Codes for Finite Decoder Iterations

This paper deals with the design of non-systematic irregular repeat-accumulate codes that are optimised for a finite number of decoding iterations. In many cases of practical interest, the usual approach for sparse-graph codes, which aims to approach capacity (with arbitrarily many decoder iterations), may not be suitable. This is of particular importance for systems with complexity or delay constraints. In this paper, we provide a design methodology which constrains the number of decoder iterations, as well as other measures of implementation complexity. Our approach uses extrinsic information transfer analysis, and our main contribution is the formulation of code optimisation problems which directly incorporate the number of iterations into the constraints. We focus on the single user binary erasure channel and the two-user binary adder channel, where this transfer analysis is exact. Generalisation to other sparse graph codes and other channels (under usual approximations) is straightforward.

Designing a Good Low-Rate Sparse-Graph Code

This paper deals with the design of low-rate sparse-graph codes, having a linear minimum distance d_{min} in the blocklength n. Its main contributions are: a) a necessary condition on a general family of sparse-graph codes with linear d_{min}; b) a justification of having degree-1 bits in the low-rate code structure; c) a new, efficient ensemble of low-rate sparse-graph codes with bits of degree 1, designed so that the necessary condition (a) is satisfied.

Packet-centric approach to distributed sparse-graph coding in wireless ad hoc networks

In this paper we present a packet-centric approach for distributed coding in decentralized wireless ad hoc networks, for applications in distributed data storage, data persistence and efficient data gathering. We study the setting where each of N network nodes generates an information packet and the goal is to efficiently encode information packets and disseminate produced encoded packets across the network in such fashion that gathering of any subset of slightly more than N encoded packets allows for retrieval of the original information. The process of distributed encoding is performed using packets that randomly walk over the network and sample information packets from network nodes, producing the encoded packets in a simple, elegant, fully decentralized and stateless way. The proposed scheme maintains properties of centralized codes in terms of performance parameters, offering at the same time advantage of robustness to node failures and changes in network topology. We specialize the proposed scheme for several important classes of low-complexity encodable/decodable sparse-graph codes – LDGM, LDPC (IRA), LT, and Raptor codes, evaluating its performance via simulation for various data-gathering scenarios.

Linear Complexity Lossy Compressor for Binary Redundant Memoryless Sources

A lossy compression algorithm for binary redundant memoryless sources is presented. The proposed scheme is based on sparse graph codes. By introducing a nonlinear function, redundant memoryless sequences can be compressed. We propose a linear complexity compressor based on the extended belief propagation, into which an inertia term is heuristically introduced, and show that it has near-optimal performance for moderate block lengths.

Lossy Multicasting Over Binary Symmetric Broadcast Channels

Lossy multicasting of a set of independent, discrete-time, continuous-amplitude source components under the mean square error distortion measure over binary symmetric broadcast channels is investigated. The practically appealing concatenation of successive refinement source coding with broadcast coding and, specifically, time-sharing of linear binary codes, is considered. Three different system optimization criteria are formulated for the lossy multicasting problem. The resulting system optimization is fairly general and applies to a variety of combinations of successive refinement source codes and channel codes. The system optimization is investigated in depth for a class of channel optimized quantization with successive refinement, obtained by using standard embedded scalar quantizers and linear mapping of the (redundant) quantizer bitplanes onto channel codewords by using a systematic Raptor encoder. This scheme is referred to as quantization with linear index coding (QLIC). Unlike existing literature on progressive transmission with unequal error protection or channel optimized quantization, the focus here is on the regime of moderate-to-large code block length and the power of modern sparse-graph codes with iterative belief propagation decoding is leveraged. In this regime, the system optimization takes on the form of simple convex programming that reduces to linear programming for QLIC. The performance of QLIC compares favorably with respect to the state of the art channel optimized quantization in the conventional setting of a single Gaussian source over a binary symmetric channel. For the multicast scenario, the performance gap incurred by the practical QLIC design with respect to ideal source and channel codes is quantified.

Jointly-check iterative decoding algorithm for quantum sparse graph codes

For quantum sparse graph codes with stabilizer formalism, the unavoidable girth-four cycles in their Tanner graphs greatly degrade the iterative decoding performance with a standard belief-propagation (BP) algorithm. In this paper, we present a jointly-check iterative algorithm suitable for decoding quantum sparse graph codes efficiently. Numerical simulations show that this modified method outperforms the standard BP algorithm with an obvious performance improvement.

On the Photonic Implementation of Universal Quantum Gates, Bell States Preparation Circuit, Quantum Relay and Quantum LDPC Encoders and Decoders

We show that any family of universal quantum gates can be implemented based on a single optical hybrid/Mach-Zehnder interferometer (MZI)/directional coupler (DC) and either a highly nonlinear optical fiber or a tap coupler with an avalanche photodiode. We further show how to implement Pauli gates, which are needed in quantum error correction, using the same technology. The use of Bell states in quantum teleportation is essential. We also show how to implement the Bell states preparation circuit. To extend the transmission distance of quantum teleportation systems, the use of quantum relays is necessary. We show how to implement the quantum relay in integrated optics as well. We further study the implementation of encoders/decoders for sparse-graph quantum codes and show that the encoder/decoder for arbitrary quantum sparse-graph code can be implemented in integrated optics as well. We also study the performance of sparse-graph codes and demonstrate that entanglement-assisted sparse-graph codes from balanced incomplete block designs significantly outperform the corresponding dual-containing quantum codes. Finally, we provide several theorems that can be used in the design of entanglement-assisted (EA) quantum codes that require only one qubit to be shared between the source and the destination.

The Generalized Area Theorem and Some of its Consequences

There is a fundamental relationship between belief propagation (BP) and maximum <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> decoding. The case of transmission over the binary erasure channel was investigated in detail in a companion paper (C. MEacuteasson, A. Montanari, and R. Urbanke, "Maxwell's construction: The hidden bridge between iterative and maximum a posteriori decoding," <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IEEE Transactions on Information Theory</i> , submitted for publication). This paper investigates the extension to general memoryless channels (paying special attention to the binary case). An area theorem for transmission over general memoryless channels is introduced and some of its many consequences are discussed. We show that this area theorem gives rise to an upper bound on the maximum <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> threshold for sparse graph codes. In situations where this bound is tight, the extrinsic soft bit estimates delivered by the BP decoder coincide with the correct <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> probabilities above the maximum <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> threshold. More generally, it is conjectured that the fundamental relationship between the maximum <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> probability (MAP) and the BP decoder which was observed for transmission over the binary erasure channel carries over to the general case. We finally demonstrate that in order for the design rate of an ensemble to approach the capacity under BP decoding the component codes have to be perfectly matched, a statement which is well known for the special case of transmission over the binary erasure channel.