Factor graph optimization of error-correcting codes for belief propagation decoding

Abstract

A computer-implemented system and method for optimizing error-correcting codes is disclosed. In one embodiment, the system comprises a learned factor graph representing a binary parity-check matrix that is iteratively refined via tensor-based belief propagation and gradient descent, thereby improving decoding performance under defined noise conditions. In another embodiment, the system comprises a learned factor graph representing a quantum stabilizer matrix that is iteratively refined via tensor-based belief propagation and gradient descent, thereby enhancing error correction on qubits.

Description

[0001] FACTOR GRAPH OPTIMIZATION OF ERROR-CORRECTING CODES FOR BELIEF PROPAGATION DECODING

[0002] RELATED APPLICATION / S

[0003] This application claims the benefit of priority of U.S. Provisional Patent Application No. 63 / 634,566 filed on April 16, 2024, the contents of which are incorporated herein by reference in their entirety.

[0004] FIELD AND BACKGROUND OF THE INVENTION

[0005] The present invention, optionally, relates to error-correcting codes and, more particularly, but not exclusively, to optimizing error-correction performance via a learned factor graph approach employing tensor-based belief propagation and gradient descent for both classical binary codes and quantum stabilizer codes.

[0006] The present invention provides concrete technological solutions to specific problems in data transmission and communication hardware systems. Specifically, the invention addresses the inefficiencies and high error rates in existing physical signal processing systems that implement error correction codes, particularly in resource-constrained environments such as loT devices and wireless communication systems where processing power, memory, and energy consumption are critical limitations.

[0007] Reliable digital communication is of major importance in the modern information age and involves the design of codes that can be robustly and efficiently decoded despite noisy transmission channels. Significant research has been dedicated to the study of capacity-approaching Error Correcting Codes (ECC). Despite the initial focus on short and medium-length linear block codes, the use of long channel codes became the prevailing approach to approaching channel capacity. However, the assumptions underlying long-block-length code design are increasingly misaligned with the constraints of modem communication systems. Today’s emerging applications — particularly in wireless domains — require the transmission of short data units, such as those seen in remote command links, Internet of Things (loT) devices, and messaging services, motivating a renewed focus on short and medium-length codes optimized for realistic and diverse channel models.

[0008] The design of optimal linear block codes capable of being efficiently decoded is a key concern, especially for short block lengths. Low-Density Parity-Check (LDPC) codes have emerged as a leading class of near-capacity-approaching codes, due in large part to their sparse structure and the availability of efficient iterative decoding algorithms such as belief propagation (BP). A wide range of LDPC code design techniques have been developed to optimize various criteria such as decoding threshold, code structure, and error-floor. These methods include constructions based on concatenation of randomly permuted sub-matrices, density evolution and differential evolution optimization, progressive edge growth, finite geometry, repeat- accumulate approaches, combinatorial design constraints, array codes, structure learning for Bayesian networks, and code discovery via genetic algorithms. Despite the extensive body of work, developing efficient sparse codes well-suited to modem demands remains an open challenge. Existing classical methods are largely not data-driven and are difficult to flexibly adapt to short code lengths, wireless channels or structural constraints.

[0009] These challenges motivate the development of data-driven solutions capable of adapting to a variety of practical constraints and operating conditions. The majority of machine learning efforts in the ECC context focus on the design of neural decoders. These include parametrized versions of the BP decoder with a neural model, wherein the Tanner graph is unfolded into an neural network in which scalar weights are assigned to each variable edge, transformer-based decoders, denoising diffusion error correction codes, foundation models capable of unified decoding of codes from different families, lengths and rates, and unified frameworks that co-learn both binary linear block codes and their corresponding neural decoders. However, despite their flexibility, neural decoding approaches typically incur significant computational and memory overhead compared to classical decoders. This increased complexity, combined with the challenges of hardware acceleration and deployment, has limited the practical adoption of neural decoders in real-world communication systems. Notably, these methods have primarily focused on short and moderate-length codes, where traditional decoding techniques are less effective and there is a clear need for further advancement — especially in light of today’s emerging applications.

[0010] SUMMARY OF THE INVENTION

[0011] The present invention relates, in various embodiments, to a gradient-based data-driven approach for the design of sparse error correction codes.

[0012] According to an aspect of some embodiments of the present invention there is provided a hardware-implemented communication system for error correction in noisy communication channels, comprising a decoding circuitry, comprising a non-transitory memory storing computer- readable instructions, and at least one processor configured to execute the computer-readable instructions to cause the computer system to receive a message encoded with an error correction code (ECC), retrieve a learned factor graph representing a binary parity check matrix of the ECC, the binary parity check matrix having been computed based on a plurality of codewords and a plurality of respective channel outputs, and perform belief propagation decoding using the retrieved learned factor graph, thereby decoding the message.

[0013] Optionally, computing the binary parity check matrix based on a plurality of codewords and a plurality of respective channel outputs comprises performing a plurality of iterations based on an initial binary parity check matrix via tensor belief backpropagation, and each iteration in the plurality of iterations comprises determining, based on the plurality of codewords and the plurality of respective channel outputs, a gradient of a parameter matrix corresponding to a current binary parity check matrix, and performing a line search to determine an optimal step size applied to the gradient, the optimal step size inducing a sign flip in at least one element of the current binary parity check matrix.

[0014] Optionally, the binary parity check matrix comprises a constrained binary parity check matrix.

[0015] Optionally, the tensor belief backpropagation comprises a Tanner graph learning approach.

[0016] Optionally, the ECC comprises a binary linear block code.

[0017] Optionally, the computer-readable instructions further cause the computer system to unroll the learned factor graph into a Trellis graph comprising an initial layer of variable nodes and successive interleaved layers of check nodes and variable nodes.

[0018] According to an aspect of some embodiments of the present invention there is provided a processor-implemented method for decoding messages encoded with an ECC, comprising retrieving a learned factor graph representing a binary parity check matrix of the ECC, the binary parity check matrix having been computed based on a plurality of codewords and a plurality of respective channel outputs, and performing belief propagation decoding using the retrieved learned factor graph, thereby decoding a message.

[0019] According to an aspect of some embodiments of the present invention there is provided a computer quantum error-correction system for decoding quantum codewords encoded with a quantum stabilizer code (QSC), comprising a non-transitory memory storing computer-readable instructions, and at least one processor configured to execute the computer-readable instructions to cause the computer system to receive a quantum codeword encoded with a QSC, retrieve a learned factor graph representing a quantum stabilizer matrix, the quantum stabilizer matrix having been computed based on a plurality of quantum codewords and a plurality of respective channel outputs, perform stabilizer measurements on the quantum codeword according to the learned factor graph, thereby obtaining syndrome data, perform belief propagation decoding using the learned factor graph and the syndrome data, thereby estimating one or more error operators acting on the quantum codeword, and apply inverses of the one or more error operators to the quantum codeword, thereby correcting the quantum codeword.

[0020] Optionally, computing the quantum stabilizer matrix based on a plurality of quantum codewords and a plurality of respective channel outputs comprises performing a plurality of iterations based on an initial quantum stabilizer check matrix via tensor belief backpropagation, and each iteration in the plurality of iterations comprises determining, based on the plurality of quantum codewords and the plurality of respective channel outputs, a gradient of a parameter matrix corresponding to a current quantum stabilizer matrix, and performing a line search to determine an optimal step size applied to the gradient, the optimal step size inducing a sign flip in at least one element of the current quantum stabilizer matrix.

[0021] Optionally, the quantum stabilizer matrix comprises a plurality of pairwise commuting stabilizer generators.

[0022] Optionally, the tensor belief backpropagation comprises a Tanner graph learning approach.

[0023] Optionally, the QSC comprises a Calderbank-Shor-Steane (CSS) code.

[0024] Optionally, the computer-readable instructions further cause the computer system to unroll the learned factor graph into a Trellis graph comprising an initial layer of variable nodes and successive interleaved layers of check nodes and variable nodes.

[0025] According to an aspect of some embodiments of the present invention there is provided a processor-implemented method for decoding quantum codewords encoded with a QSC, comprising retrieving a learned factor graph representing a quantum stabilizer matrix, the quantum stabilizer matrix having been computed based on a plurality of quantum codewords and a plurality of respective channel outputs, performing stabilizer measurements on a quantum codeword according to the learned factor graph, thereby obtaining syndrome data, performing belief propagation decoding using the learned factor graph and the syndrome data, thereby estimating one or more error operators acting on the quantum codeword, and applying inverses of the one or more error operators to the quantum codeword, thereby correcting the quantum codeword.

[0026] Unless otherwise defined, all technical and / or scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the invention pertains. Although methods and materials similar or equivalent to those described herein can be used in the practice or testing of embodiments of the invention, exemplary methods and / or materials are described below. In case of conflict, the patent specification, including definitions, will control. In addition, the materials, methods, and examples are illustrative only and are not intended to be necessarily limiting. Implementation of the method and / or system of embodiments of the invention can involve performing or completing selected tasks manually, automatically, or a combination thereof. Moreover, according to actual instrumentation and equipment of embodiments of the method and / or system of the invention, several selected tasks could be implemented by hardware, by software or by firmware or by a combination thereof using an operating system.

[0027] For example, hardware for performing selected tasks according to embodiments of the invention could be implemented as a chip or a circuit. As software, selected tasks according to embodiments of the invention could be implemented as a plurality of software instructions being executed by a computer using any suitable operating system. In an exemplary embodiment of the invention, one or more tasks according to exemplary embodiments of method and / or system as described herein are performed by a data processor, such as a computing platform for executing a plurality of instructions. Optionally, the data processor includes a volatile memory for storing instructions and / or data and / or a non-volatile storage, for example, a magnetic hard-disk and / or removable media, for storing instructions and / or data. Optionally, a network connection is provided as well. A display and / or a user input device such as a keyboard or mouse are optionally provided as well.

[0028] BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0029] Some embodiments of the invention are herein described, by way of example only, with reference to the accompanying drawings. With specific reference now to the drawings in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of embodiments of the invention. In this regard, the description taken with the drawings makes apparent to those skilled in the art how embodiments of the invention may be practiced.

[0030] In the drawings:

[0031] FIG. 1 is a visual representation of select Hamming(7,4) code properties, according to the background of the invention;

[0032] FIG. 2 is an illustration of an end-to-end communication system based on an ECC, according to some embodiments of the invention;

[0033] FIG. 3 is a pseudocode and a flowchart of a tensor belief propagation algorithm, according to some embodiments of the invention;

[0034] FIG. 4 is a pseudocode and a flowchart of a belief propagation codes optimization algorithm, according to some embodiments of the invention;

[0035] FIG. 5 is a table of bit error rate based comparison of a learned locally-optimal code with classical codes, according to some embodiments of the invention; FIG. 6 is a visualization of original and learned parity check matrices, according to some embodiments of the invention; and

[0036] FIG. 7 is an illustration of sparsity reduction of learned codes, according to some embodiments of the invention.

[0037] DESCRIPTION OF SPECIFIC EMBODIMENTS OF THE INVENTION

[0038] The present invention, optionally, relates to error-correcting codes and, more particularly, but not exclusively, to a computer-implemented system and method for optimizing error-correction performance via a learned factor graph approach employing tensor-based belief propagation and gradient descent for both classical binary codes and quantum stabilizer codes.

[0039] As used herein, the term “ECC” means error-correcting code.

[0040] According to certain embodiments of the present invention, a data-driven, gradient-based method is provided for the design of sparse ECCs. The method described herein enables automatic optimization of parity-check or stabilizer matrices tailored to specific channel noise characteristics. The method is particularly useful for modem communication scenarios involving short and medium-length codes, where traditional, asymptotically-motivated design techniques may be suboptimal.

[0041] As used herein, the term “BP” means belief propagation.

[0042] The method described herein is adapted to optimize ECCs with respect to BP decoding performance via the learning of a factor graph structure — such as a Tanner graph or its quantum analog — through simulations over a noisy communication channel. This method yields considerable practical improvements given the ubiquity and advantages of using the BP algorithm for decoding sparse ECCs.

[0043] As used herein, the term “LDPC” means low-density parity-check.

[0044] The invention is not limited to classical decoding of LDPC codes; in certain embodiments, the structure learning method is extended to quantum stabilizer codes, where the learned generators improve decoding success under quantum noise models, or may be applied to other classical code families.

[0045] The invention offers several notable advantages. While maximum likelihood decoding remains a theoretical benchmark, it is computationally infeasible for most practical codes. Existing design strategies often rely on fixed algebraic structures or code ensembles optimized via asymptotic metrics, such as decoding thresholds. The invention disclosed herein departs from such fixed-rule design by enabling a learned optimization pipeline that operates directly in the finite- block-length regime and is adaptable to application- specific channels. The proposed approach considerably outperforms the decoding performance of existing popular codes, while inducing a very low overhead for integration into the existing decoding solutions.

[0046] For purposes of better understanding some embodiments of the present invention, as illustrated in Figures 2-7 of the drawings, reference is first made to a visual representation of select Hamming(7,4) code properties as illustrated in Fig. 1.

[0047] Fig. 1(a) illustrates a parity check matrix H of a Hamming(7,4) code, wherein each row corresponds to a parity-check constraint and each column corresponds to a code symbol. The parity check matrix H serves as a fundamental representation of a code’s error-checking properties.

[0048] Fig. 1(b) illustrates a induced Tanner graph corresponding to the parity check matrix illustrated in Fig. 1(a). In this graph, variable nodes represent code symbols and check nodes represent parity-check constraints, with edges connecting nodes where a corresponding entry in the parity check matrix is nonzero.

[0049] Fig. 1(c) illustrates an unrolled Trellis graph with two iterations, corresponding to the Tanner graph illustrated in Fig. 1(b), with odd layers in black and even layers in white.

[0050] Fig. 1(d) illustrates the approach of the invention for structure learning via learned binary weighting of the edges of a complete bipartite factor graph in contrast with conventional sparse representation.

[0051] Optionally, the BP algorithm enables iterative propagation of a current codeword estimate (belief) via a Trellis graph determined according to a factor graph defined by a code (Tanner graph). The factor graph is unrolled into a Trellis graph, initiated with n variable nodes, and is composed of two types of interleaved layers defined by the check / factor nodes and variable nodes.

[0052] Before explaining at least one embodiment of the invention in detail, it is to be understood that the invention is not necessarily limited in its application to the details of construction and the arrangement of the components and / or methods set forth in the following description and / or illustrated in the drawings. The invention is capable of other embodiments or of being practiced or carried out in various ways.

[0053] According to some embodiments of the invention, the BP algorithm is reformulated into a differentiable tensor-based representation that permits the learning of graph connectivity through backpropagation, and an efficient line- search routine is adapted to relaxed binary programming problems, enabling discrete edge-weight updates in the factor graph in a continuous optimization setting.

[0054] Reference is now made to Fig. 2, illustrating an end-to-end communication system based on an ECC. According to some embodiments of the invention, a communication system utilizes a linear block code C defined by a generator matrix G G {0,l}fcxnand a parity check matrix H G , wherein H satisfies the constraint GHT= 0 over the Galois field GF(2). The parity check matrix H defines a Tanner graph representation, comprising n variable nodes and (n — k) check nodes, wherein variable nodes correspond to encoded symbols and check nodes correspond to parity constraints.

[0055] In an embodiment, an encoder 200 receives 201 an input message m ∈ {0, 1}kand applies 202 the generator matrix G to encode the input message to a codeword c ∈ ⊂ {0, 1}n, wherein the codeword c satisfies the constraint He = 0. The encoder 200 further modulates 203 the codeword via a Binary Phase Shift Keying (BPSK) scheme, thereby producing a transmitted signal . The encoder 200 transmits the transmitted signal csvia a Binary-Input Symmetric- Output (BISO) channel 204, thereby producing a channel output y, to a BP decoder 205, wherein the BP decoder 205 comprises a decoder function , aiming to predict a soft approximation c' = fH(y) of the codeword c.

[0056] Optionally, the BISO channel 204 comprises an additive white Gaussian noise (AWGN) channel, wherein the transmitted signal csexperiences a random noise y = cs+ E, wherein is a random noise independent of the transmitted signal cs.

[0057] Optionally, the BISO channel 204 comprises a Rayleigh fading channel, wherein the transmitted signal csexperiences multipath fading , wherein h is the independent and identically distributed (i.i.d.) Rayleigh distributed fading vector with coefficient 1, and E is the regular AWGN noise.

[0058] Optionally, the BISO channel 204 comprises a Gaussian mixture channel, or a bursty noise channel, wherein the transmitted signal csexperiences channel interference y = cs+ E + wherein E is the regular AWGN noise and with a probability p = 0.1 and ζ = 0 with a probability 1 — p.

[0059] Optionally, the BP algorithm operates on a Trellis graph by propagating messages from variable nodes to check nodes and from check nodes to variable nodes, in an alternative and iterative fashion, wherein the input layer corresponds to a vector of log-likelihood ratios (LLR) L E of the channel output y: wherein yvis an element of the channel output y, and cvis a respective codeword bit.

[0060] Optionally, xlis a vector of messages propagated from a layer i — 1 to a layer i, e = (c, v) is an edge on a Tanner graph, and each message is indexed by an edge e.

[0061] Optionally, at an odd (variable) layer the BP algorithm comprises a computation such that wherein N (v) = {(c, v) \H (c, v) = 1} is a set of all edges in which v participates.

[0062] Optionally, at input layer k = 0 the vector of messages comprises x0= 0, and thereby a vector of messages x1is determined solely by the vector L.

[0063] Optionally, at an even (check) layer the BP algorithm comprises a computation such that wherein N(c) = {(c, v) |H(c, v) = 1} is a set of all edges in which row c of a parity matrix H participates.

[0064] Optionally, at output layer the BP algorithm comprises a soft-decision output of a codeword such that

[0065] Optionally, given an original parity check matrix H, the method aims to compute an optimized parity check matrix H* such that wherein is a modulation function such that cs= 0(c), Z is a channel noise distribution, fH,T; ) is a BP decoder function built upon H with T discrete iterations (sampled uniformly from a set), D is a distance metric, and R is a potential hard / soft regularization (i.e. a plurality of sparsity or code structure constraints).

[0066] Several challenges arise from this optimization problem: (i) the optimization is highly non- differentiable and results in an NP-hard binary non-linear integer programming problem, (ii) computation of the codewords c = Gm is both highly non-differentiable (matrix-vector multiplication over GF(2) in case symmetry is not maintained during the optimization) and computationally expensive (inverse via Gaussian elimination of H), (iii) the modulation function can be non-differentiable, and (iv) the BP decoder assumes a fixed set of factor graph edges e corresponding to the upon which the BP decoder is conditioned.

[0067] Reference is now made to Fig. 3, illustrating a pseudocode and a flowchart of a tensor belief propagation algorithm 300. According to some embodiments of the invention, the optimization method comprises a structure graph (Tanner graph) learning approach, wherein the bipartite graph is assumed as complete with learnable binary-weighted edges, thereby enabling a tensor reformulation of the BP algorithm weighted by H and a differentiable optimization of the Tanner graph. In the tensor formulation the BP algorithm comprises a composition of differentiable functions and is therefore differentiable with respect to H . The tensor formulation further solves the challenge (iv) of graph learning. As the conditional independence of error probability under symmetry is satisfied for message passing algorithms for any H, it is sufficient to perform optimization solely for the zero codeword c = Gm = 0, thereby removing dependency on G in the optimization objective and solving challenge (ii). The reformulated optimization problem is invariant to the choice of the modulation function , thereby solving challenge (iii).

[0068] Optionally, at a variable layer the BP algorithm comprises a computation such that and at a check layer the BP algorithm comprises a computation such that wherein Ciand Vjcomprise a plurality of non-zero elements in column i and row j, respectively, and the ones elements in satisfy the identity element of multiplication.

[0069] The process proceeds through the following steps.

[0070] At step 301, the process receives an LLR vector L, a parity check matrix H and a desired number of iterations, wherein each iteration comprises a variable layer and a check layer. Next, the process proceeds to a decision step 302.

[0071] At the decision step 302, the process makes a determination of whether the desired number of iterations has been completed. If yes, the process proceeds to a step 307. If not, the process proceeds to a decision step 303.

[0072] At the decision step 303, the process makes a determination of whether the current iteration is the first iteration. If yes, the process proceeds to a step 304. If not, the process proceeds to a step 305.

[0073] At a step 304, the process computes a result of a variable layer Q = L, wherein L is the LLR vector, and proceeds to a step 306.

[0074] At the step 305, the process computes a result of a variable layer Q comprising a function of the LLR vector L, the parity matrix H and a result of a check layer R as disclosed hereinabove, and proceeds to a step 306.

[0075] At the step 306, the process computes a result of a check layer R comprising a function of the parity matrix H and a result of a variable layer Q as disclosed hereinabove, and returns to the step 302. At the step 307, the process computes the result of an output layer 0 comprising a function of the LLR vector L, the parity matrix H and a result of a check layer R as disclosed hereinabove, outputs the result and terminates.

[0076] Reference is now made to Fig. 4, illustrating a pseudocode and a flowchart of a belief propagation codes optimization algorithm 400. According to some embodiments of the invention, optimization of H comprises relaxing the NP-hard binary programming problems to an unconstrained objective, wherein is a parameter matrix, and bin(-) is an element-wise binarization operator.

[0077] Optionally, the bin(-) operator comprises a shifted straight-through-estimator (STE) such that

[0078] Optionally, the empirical risk objective comprises a binary cross-entropy loss (BCE) as the binary error rate (BER) discrepancy measure D : wherein cs= 0(c) comprises a modulated zero codeword and εicomprises an i-th noise sample drawn from a channel noise distribution, thereby aiming to enable optimal decoding on a varying number of decoding iterations t. The empirical risk objective is (sub)differentiable and thus optimizable via classical first-order methods, given the STE definition of the gradient.

[0079] In a conventional small learning rate regime, solely gradient steps modifying a sign in fl induce a modification of the loss function, since the parity check matrix is binary, thereby reducing the efficiency of a gradient descent. In a conventional small learning rate regime, convergence to a local minimum is further complicated due to oscillating behaviors around zero. Efficiency improvements require determining an optimal step size given a gradient computed on a sufficiently representative batch.

[0080] Optionally, a line search problem to determine an optimal step size inducing a sign flip in Ω and thereby provably limiting the maximum number of relevant grid samples to n(n — k), comprises a parallelizable objective evaluation on a grid such that

[0081] Optionally, the line search objective comprises a non-differentiable BER instead of a Bayesian BCE loss. Optionally, the line search objective comprises a Frame Error Rate (FER) instead of a Bayesian BCE loss.

[0082] Optionally, the optimization method parameters comprise an original parity check matrix H subject to optimization (i.e. an initial Ω ), a maximum number of optimization steps in case a convergence is not reached, count and quality of data samples, grid search length and BP iteration number.

[0083] Optionally, the number of optimization steps comprises 20 iterations.

[0084] Optionally, training noise is sampled randomly per batch in the {3, ... ,7} normalized SNR range.

[0085] Optionally, the number of data samples per optimization iteration comprises 4.9M. Optionally, and the data samples are required to have a non-zero syndrome.

[0086] Optionally, the number of BP iterations comprises 5 iterations.

[0087] Optionally, a grid search is heuristically restricted to a plurality of smallest step sizes.

[0088] The process proceeds through the following steps.

[0089] At a step 401, the process receives an original parity check matrix H subject to optimization and proceeds to a step 402.

[0090] At the step 402, the process initializes a parameter matrix Ω based on the received parity check matrix H . Next, the process proceeds to a decision step 403.

[0091] At the decision step 403, the process makes a determination as of whether the number of completed iterations exceeds the configured maximum number of iterations. If yes, the process proceeds to a step 410. If not, the process proceeds to a step 404.

[0092] At the step 404, the process receives a batch of training data, comprising a plurality of messages x and a respective plurality of channel outputs y. Next, the process proceeds to a step 405.

[0093] At the step 405, the process binarizes the current parameter matrix Ω , thereby producing a current binary parity check matrix H ', and proceeds to a step 406.

[0094] At the step 406, the process determines a BCE loss based on the current binary parity check matrix H ' and the batch of training data. Next, the process proceeds to a step 407.

[0095] At the step 407, the process applies the line search method disclosed hereinabove to determine a plurality of optimal step size candidates A, wherein each optimal size candidate flips a sign of at least one value within Ω . Next, the process proceeds to a step 408.

[0096] At the step 408, the process computes, for each optimal step size candidate in the plurality of optimal step size candidates λ, a BCE loss based on the current parameter matrix Ω modified according to the optimal step size candidate and the batch of training data, and updates the current parameter matrix (1 based on the best step candidate. Next, the process proceeds to a decision step

[0097] 409.

[0098] At the decision step 409, the process makes a determination as of whether the optimization has converged. If not, the process returns to the step 403. If yes, the process proceeds to the step

[0099] 410.

[0100] At the step 410, the process the process binarizes the current parameter matrix Ω , thereby producing an optimized binary parity check matrix H*, outputs the result and terminates.

[0101] Optionally, the system is implemented through specialized hardware components specifically designed for the tensor-based belief propagation approach described herein. The implementation may comprise one or more of:

[0102] (i) signal processing circuits with dedicated tensor operation units that efficiently implement the reformulated BP algorithm as described in Fig. 3;

[0103] (ii) custom memory architecture optimized for the specific access patterns of the learned factor graph structure;

[0104] (iii) line search co-processors that efficiently implement the optimization algorithm as described in Fig. 4;

[0105] (iv) physical layer interfaces that directly convert analog channel signals to log-likelihood ratios for processing; and

[0106] (v) low-power FPGA implementations enabling deployment in resource-constrained loT devices.

[0107] These hardware components are referred to as "decoding circuitry" and "signal processor", and may provide the physical infrastructure that transforms encoded signals into reliably decoded messages.

[0108] Reference is now made to Fig. 5, illustrating a comparison of the negative natural logarithm of Bit Error Rate (BER) for several normalized SNR values of the present invention with classical codes. The disclosed method is applied to a respective classical code to produce an optimized code (“Our” columns), and compared against the original classical code (“BP” columns) for AWGN, fading and bursting noise channels. PEGX denotes a Progressive Edge Growth construction code, wherein the degree of each node is X%.

[0109] The results are reported as negative natural logarithm bit error rates (BER) for three , following a conventional testing benchmark. BP-based results are obtained after iterations in the first row and iterations in the second row of the results table. During testing, at least 105random codewords are decoded, to obtain at least 50 frames with errors at each SNR value. Higher is better. The best results are marked in bold. Reference is now made to Fig. 6, visualizing original and learned parity check matrices. Visualization of the original (first row) and the respective learned parity check matrices (second row) for (a) LDPC(32,16), (b) LDPC(128,64), (c) BCH(63,45) and (d) Random(64,32,p = 0.5). “PCM iter X” denotes the number of optimization iterations applied to produce final optimized parity check matrix H* . For low-density codes modifications induced by the optimization process remain small, since the code is already near local optimum, while for denser codes the modifications are substantial. The inductive bias of stochastic gradient descent does not impact BP inductive bias under the condition of sparse connections.

[0110] Reference is now made to Fig. 7, illustrating sparsity reduction of learned codes. Sparsity ratio is computed as , wherein Sband Socomprise sparsity of the baseline (original) code and sparsity of optimized (learned) code, respectively. The optimization process always generates sparser codes but does not affect code girth.

[0111] As used herein, the term “QSC” means quantum stabilizer code.

[0112] Optionally, the present invention is not limited to classical ECCs but is equally applicable to quantum error correction. A classical binary parity-check matrix may be analogously replaced by a quantum stabilizer matrix that defines a QSC. Optionally, the optimized quantum stabilizer matrix is generated using the data-driven, gradient-based optimization framework disclosed hereinabove, based on an original quantum stabilizer matrix.

[0113] Optionally, a learned factor graph represents a quantum stabilizer matrix of a QSC, wherein each node corresponds to a qubit or to a stabilizer measurement, and the training data comprises quantum codewords and the syndrome data obtained from stabilizer measurements performed on a quantum channel subject to noise. Optionally, the tensor-based BP algorithm is accordingly modified to process the syndrome data, thereby facilitating BP decoding to estimate one or more error operators acting on a quantum codeword.

[0114] Optionally, the iterative optimization process is performed on an initial quantum stabilizer matrix using tensor-based belief backpropagation. At each iteration, the method computes a gradient with respect to a parameter matrix that serves as a relaxed representation of the quantum stabilizer matrix. A line- search routine is employed to determine an optimal step size that induces critical modifications — such as sign flips or the equivalent changes in a binary or symplectic representation — to optimize the matrix. Optionally, the learned quantum stabilizer matrix comprises a plurality of pairwise commuting stabilizer generators, thereby satisfying the fundamental commutation constraints necessary for quantum error correction.

[0115] Optionally, the quantum stabilizer matrix is derived from or configured as a Calderbank- Shor-Steane (CSS) code, which utilizes two classical binary linear block codes to respectively address bit-flip and phase-flip errors. Optionally, the present invention is to other families of quantum codes. The fundamental steps — receiving a quantum codeword, obtaining syndrome data via stabilizer measurements, performing belief propagation decoding, and applying corrective operators — remain analogous to the classical method, thus ensuring that the data-driven framework is inherently adaptable.

[0116] HARDWARE IMPLEMENTATION DETAILS

[0117] Optionally, the system disclosed herein is implemented through specialized hardware components configured for efficient error correction in noisy communication channels. The hardware implementation comprises:

[0118] (i) dedicated signal processing circuits optimized for tensor-based belief propagation operations;

[0119] (ii) specialized memory architecture for efficient storage and retrieval of learned factor graphs;

[0120] (iii) hardware accelerators designed to implement the line search algorithm for optimal step size determination;

[0121] (iv) physical layer integration components that directly interface with communication channel hardware; and

[0122] (v) energy-efficient circuit design that enables implementation in resource-constrained devices.

[0123] These hardware components work together to transform physical signals received through noisy channels into reliably decoded messages with significantly improved error correction performance compared to conventional systems, as quantified in Figure 5.

[0124] Optionally, a communication system comprising one or more such hardware components comprises a training and optimization system that processes actual channel data to capture real- world noise characteristics, implements the tensor-based belief propagation algorithm disclosed hereinabove in specialized hardware; produces optimized binary parity check matrices that are then stored in the communication system's memory; and creates factor graphs specifically optimized for the physical characteristics of a target communication channel.

[0125] The hardware system disclosed herein enables a number of practical applications, for example: loT device networks operating in noisy environments, enabling reliable communication despite strict power and computational constraints; 5G and future wireless communications, resulting in higher data rates and extended coverage; ultra-low-latency vehicle-to-vehicle communication systems, enabling safe autonomous driving applications; medical device communication with guaranteed message integrity for implantable devices, meeting FDA safety requirements; satellite communications, where a specialized hardware implementation may provide error correction performance that would be unattainable with conventional decoders given strict power limitations; quantum communication systems, enabling practical quantum key distribution and quantum networking.

[0126] The present invention provides several measurable technical improvements over prior art systems:

[0127] 1. Error rate reduction: As demonstrated in Fig. 5, the system achieves significant improvements in bit error rate compared to conventional systems, with up to 2-3 orders of magnitude improvement at higher SNR values.

[0128] 2. Computational efficiency: The tensor-based approach reduces computational complexity by approximately 30% compared to traditional BP decoders.

[0129] 3. Sparsity improvement: As shown in Fig. 7, the optimized factor graphs are consistently sparser than the original codes while maintaining superior performance.

[0130] 4. Adaptability: The system may be configured to dynamically optimize the code based on current physical channel conditions.

[0131] Optionally, the system disclosed herein comprises specialized quantum hardware components that interface with quantum communication channels and quantum processors. The implementation comprises one or more of:

[0132] (i) quantum stabilizer measurement circuits that physically interact with qubits;

[0133] (ii) custom quantum error syndrome detection hardware;

[0134] (iii) specialized classical-quantum interface components; and

[0135] (iv) hardware-optimized quantum error correction circuits.

[0136] The system thereby achieves measurable improvement in quantum state preservation and fidelity compared to conventional quantum error correction systems, enabling more reliable quantum computation and communication.

[0137] The terms "comprises", "comprising", "includes", "including", “having” and their conjugates mean "including but not limited to".

[0138] The term “consisting of’ means “including and limited to”.

[0139] The term "consisting essentially of" means that the composition, method or structure may include additional ingredients, steps and / or parts, but only if the additional ingredients, steps and / or parts do not materially alter the basic and novel characteristics of the claimed composition, method or structure. As used herein, the singular form "a", "an" and "the" include plural references unless the context clearly dictates otherwise. For example, the term "a compound" or "at least one compound" may include a plurality of compounds, including mixtures thereof.

[0140] It is appreciated that certain features of the invention, which are, for clarity, described in the context of separate embodiments, may also be provided in combination in a single embodiment. Conversely, various features of the invention, which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable subcombination or as suitable in any other described embodiment of the invention. Certain features described in the context of various embodiments are not to be considered essential features of those embodiments, unless the embodiment is inoperative without those elements.

[0141] Although the invention has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications and variations that fall within the spirit and broad scope of the appended claims.

[0142] It is the intent of the Applicant(s) that all publications, patents and patent applications referred to in this specification are to be incorporated in their entirety by reference into the specification, as if each individual publication, patent or patent application was specifically and individually noted when referenced that it is to be incorporated herein by reference. In addition, citation or identification of any reference in this application shall not be construed as an admission that such reference is available as prior art to the present invention. To the extent that section headings are used, they should not be construed as necessarily limiting. In addition, any priority document(s) of this application is / are hereby incorporated herein by reference in its / their entirety.

Claims

WHAT IS CLAIMED IS:

1. A hardware-implemented communication system for error correction in noisy communication channels, comprising: a decoding circuitry comprising: a non-transitory memory storing computer-readable instructions; and at least one processor configured to execute the computer-readable instructions to cause the computer system to: receive a message encoded with an error correction code (ECC); retrieve a learned factor graph representing a binary parity check matrix of the ECC, the binary parity check matrix having been computed based on a plurality of codewords and a plurality of respective channel outputs; and perform belief propagation decoding using the retrieved learned factor graph, thereby decoding the message.

2. The system according to claim 1, wherein computing the binary parity check matrix based on a plurality of codewords and a plurality of respective channel outputs comprises performing a plurality of iterations based on an initial binary parity check matrix via tensor belief backpropagation, and wherein each iteration in the plurality of iterations comprises: determining, based on the plurality of codewords and the plurality of respective channel outputs, a gradient of a parameter matrix corresponding to a current binary parity check matrix; and performing a line search to determine an optimal step size applied to the gradient, the optimal step size inducing a sign flip in at least one element of the current binary parity check matrix.

3. The system according to claim 1, wherein the binary parity check matrix comprises a constrained binary parity check matrix.

4. The system according to claim 2, wherein the tensor belief backpropagation comprises a Tanner graph learning approach.

5. The system according to claim 1, wherein the ECC comprises a binary linear block code.

6. The system according to claim 1, wherein the computer-readable instructions further cause the computer system to unroll the learned factor graph into a Trellis graphcomprising an initial layer of variable nodes and successive interleaved layers of check nodes and variable nodes.

7. A processor-implemented method for decoding messages encoded with an ECC, comprising: retrieving a learned factor graph representing a binary parity check matrix of the ECC, the binary parity check matrix having been computed based on a plurality of codewords and a plurality of respective channel outputs; and performing belief propagation decoding using the retrieved learned factor graph, thereby decoding a message.

8. A computer quantum error-correction system for decoding quantum codewords encoded with a quantum stabilizer code (QSC), comprising: a non-transitory memory storing computer-readable instructions; and at least one processor configured to execute the computer-readable instructions to cause the computer system to: receive a quantum codeword encoded with a QSC; retrieve a learned factor graph representing a quantum stabilizer matrix, the quantum stabilizer matrix having been computed based on a plurality of quantum codewords and a plurality of respective channel outputs; perform stabilizer measurements on the quantum codeword according to the learned factor graph, thereby obtaining syndrome data; perform belief propagation decoding using the learned factor graph and the syndrome data, thereby estimating one or more error operators acting on the quantum codeword; and apply inverses of the one or more error operators to the quantum codeword, thereby correcting the quantum codeword.

9. The system according to claim 8, wherein computing the quantum stabilizer matrix based on a plurality of quantum codewords and a plurality of respective channel outputs comprises performing a plurality of iterations based on an initial quantum stabilizer check matrix via tensor belief backpropagation, and wherein each iteration in the plurality of iterations comprises: determining, based on the plurality of quantum codewords and the plurality of respective channel outputs, a gradient of a parameter matrix corresponding to a current quantum stabilizer matrix; andperforming a line search to determine an optimal step size applied to the gradient, the optimal step size inducing a sign flip in at least one element of the current quantum stabilizer matrix.

10. The system according to claim 8, wherein the quantum stabilizer matrix comprises a plurality of pairwise commuting stabilizer generators.

11. The system according to claim 9, wherein the tensor belief backpropagation comprises a Tanner graph learning approach.

12. The system according to claim 8, wherein the QSC comprises a Calderbank-Shor- Steane (CSS) code.

13. The system according to claim 8, wherein the computer-readable instructions further cause the computer system to unroll the learned factor graph into a Trellis graph comprising an initial layer of variable nodes and successive interleaved layers of check nodes and variable nodes.

14. A processor-implemented method for decoding quantum codewords encoded with a QSC, comprising: retrieving a learned factor graph representing a quantum stabilizer matrix, the quantum stabilizer matrix having been computed based on a plurality of quantum codewords and a plurality of respective channel outputs; performing stabilizer measurements on a quantum codeword according to the learned factor graph, thereby obtaining syndrome data; performing belief propagation decoding using the learned factor graph and the syndrome data, thereby estimating one or more error operators acting on the quantum codeword; and applying inverses of the one or more error operators to the quantum codeword, thereby correcting the quantum codeword.

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