We introduce a generalized low-density parity-check decoding framework for quantum Tanner codes utilizing soft-output guessing random additive noise decoding (SOGRAND). By soft-output decoding entire component codes, we mitigate trapping sets and cycles, resulting in improved convergence. SOGRAND, combined with ordered statistic decoding (OSD) post-processing, outperforms the standard belief propagation plus OSD baseline by up to three orders of magnitude in logical error rate, providing a way forward for scalable decoding of the emerging class of Tanner-code-based quantum codes.
We introduce a generalized low-density parity-check decoding framework for quantum Tanner codes utilizing soft-output guessing random additive noise decoding (SOGRAND). By soft-output decoding entire component codes, we mitigate trapping sets and cycles, resulting in improved convergence. SOGRAND, combined with ordered statistic decoding (OSD) post-processing, outperforms the standard belief propagation plus OSD baseline by up to three orders of magnitude in logical error rate, providing a way forward for scalable decoding of the emerging class of Tanner-code-based quantum codes.
Long forward error correction codes are typically constructed by concatenating shorter component codes that are then decoded through iterative Soft-Input Soft-Output (SISO) of their components. The recently introduced Soft Output Guessing Random Additive Noise Decoding (SOGRAND) has been shown to enable accurate SISO component decoding for a broad range of component codes. Here we establish that by specializing its SISO computation to Single Parity Check codes, SOGRAND offers an alternative existing Check Node (CN) update for decoding Low Density Parity Check codes. Simulation results demonstrate similar or better decoding performance than Gallager's sum-product algorithm and norm-min-sum, while offering two distinct low complexity, hardware friendly CN update algorithms.
We study the decoding problem for quantum Tanner codes and propose to exploit the underlying local code structure by grouping check nodes into more powerful generalized check nodes for enhanced iterative belief propagation (BP) decoding by decoding the generalized checks using a maximum a posteriori (MAP) decoder as part of the check node processing of each decoding iteration. We mainly study the finite-length setting and show that the proposed enhanced generalized BP decoder for quantum Tanner codes significantly outperforms the standard quaternary BP decoder with memory effects, as well as the recently proposed Relay-BP decoder, even outperforming generalized bicycle (GB) codes with comparable parameters in some cases. For other classes of quantum low-density parity-check (qLDPC) codes, we propose a greedy algorithm to combine checks for generalized BP decoding. However, for GB codes, bivariate bicycle codes, hypergraph product codes, and lifted-product codes, there seems to be limited gain by combining simple checks into more powerful ones. To back up our findings, we also provide a theoretical cycle analysis for the considered qLDPC codes. This leads us to a construction criterion for quantum Tanner codes that produces codes particularly well-suited for BP decoding.
We establish that a large, flexible class of long, high redundancy error correcting codes can be efficiently and accurately decoded with guessing random additive noise decoding (GRAND). Performance evaluation demonstrates that it is possible to construct simple product codes with lengths of approximately 200 to 4000 bits and rates between 0.2 and 0.8 that outperform low-density parity-check (LDPC) codes from the 5G New Radio standard in both AWGN and fading channels. The concatenated structure enables many desirable features, including: low-complexity hardware-friendly encoding and decoding; significant flexibility in length and rate through modularity; and high levels of parallelism in encoding and decoding that enable low latency. Central is the development of a method through which any soft-input (SI) GRAND algorithm can provide soft-output (SO) in the form of an accurate a-posteriori estimate of the likelihood that a decoding is correct or, in the case of list decoding, the likelihood that each element of the list is correct. The distinguishing feature of soft-output GRAND (SOGRAND) is the provision of an estimate that the correct decoding has not been found, even when providing a single decoding. Per-block SO can be converted into accurate per-bit SO by a weighted sum that includes a term for the SI. Implementing SOGRAND adds negligible computation and memory to the existing decoding process, and using it results in a practical, low-latency alternative to LDPC codes.
The recent success in constructing asymptotically good quantum low-density parity-check (QLDPC) codes makes this family of codes a promising candidate for error-correcting schemes in quantum computing. However, conventional belief propagation (BP) decoding of QLDPC codes does not yield satisfying performance due to the presence of unavoidable short cycles in their Tanner graph and the special degeneracy phenomenon. In this work, we propose to decode QLDPC codes based on a check matrix with redundant rows, generated from linear combinations of the rows in the original check matrix. This approach yields a significant improvement in decoding performance with the additional advantage of very low decoding latency. Furthermore, we propose a novel neural belief propagation decoder based on the quaternary BP decoder of QLDPC codes which leads to further decoding performance improvements.
Quantum low-density parity-check (qLDPC) codes are promising for realizing scalable fault-tolerant quantum computation due to their potential for low-overhead protocols. A common approach to decoding qLDPC codes is to use the belief propagation (BP) decoder, followed by a post-processing step to enhance decoding accuracy. For fast decoding, the post-processing algorithm is desirable to have a small computational cost and rely only on local operations on the Tanner graph to facilitate parallel implementation. To address this requirement, we propose degeneracy cutting (DC), an efficient post-processing technique for the BP decoder that operates on information restricted to the support of each stabilizer generator. DC selectively removes one variable node with the lowest error probability for each stabilizer generator, significantly improving decoding performance while retaining the favorable computational scaling and structure amenable to parallelization inherent to BP. We further extend our method to realistic noise models, including phenomenological and circuit-level noise models, by introducing the detector degeneracy matrix, which generalizes the notion of stabilizer-induced degeneracy to these settings. Numerical simulations demonstrate that BP+DC achieves decoding performance approaching that of BP followed by ordered statistics decoding (BP+OSD) in several settings, while requiring significantly less computational cost. Our results present BP+DC as a promising decoder for fault-tolerant quantum computing, offering a valuable balance of accuracy, efficiency, and suitability for parallel implementation.
Quantum low-density parity-check (QLDPC) codes represent a promising approach for error correction in quantum computing. The recently proposed Tesseract decoder uses the A* search algorithm that guarantees finding the most likely error pattern. However, practical implementations of Tesseract often involve an extremely large graph, and the inherently sequential nature of the search results in high computational overhead and long runtime. To improve decoding efficiency, we propose a two-stage decoding framework. First, a belief propagation (BP) decoder efficiently processes the syndrome. This step generates hard decisions (a binary error vector) and soft information (per-qubit confidence levels). In non-convergent BP cases, a gating mechanism examines the BP decoder's output to identify and filter out qubits with oscillating confidence values. In these cases, the refined output serves as input to Tesseract, which then attempts to identify and correct any residual errors. This hybrid approach leverages the high speed of BP decoding and delegates the more challenging decoding instances to Tesseract. Numerical results demonstrate a substantial reduction in overall decoding complexity while maintaining the logical error rate (LER) of the stand-alone Tesseract. Across all tested physical error rates, the proposed method achieves at least 5x reduction in the number of expanded nodes in the Tesseract, with a peak reduction of nearly 15x at a physical error rate of p = 0.05 for the [[126, 12, d < 11]] T1 code, and approximately 8.8x at p = 0.05 for the [[72, 12, 6]] bicycle bivariate (BB) code.