Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE Access

Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies

Zunaira Babar, Panagiotis Botsinis, Dimitrios Alanis, Soon Xin Ng, Lajos Hanzo

Published Jan 1, 2015• 111 citations
Publisher

Abstract

The near-capacity performance of classical low-density parity check (LDPC) codes and their efficient iterative decoding makes quantum LDPC (QLPDC) codes a promising candidate for quantum error correction. In this paper, we present a comprehensive survey of QLDPC codes from the perspective of code design as well as in terms of their decoding algorithms. We also conceive a modified non-binary decoding algorithm for homogeneous Calderbank-Shor-Steane-type QLDPC codes, which is capable of alleviating the problems imposed by the unavoidable length-four cycles. Our modified decoder outperforms the state-of-the-art decoders in terms of their word error rate performance, despite imposing a reduced decoding complexity. Finally, we intricately amalgamate our modified decoder with the classic uniformly reweighted belief propagation for the sake of achieving an improved performance.

DOI
10.1109/access.2015.2503267

Related papers

Quantum Low-Density Parity-Check Codes

Bane V. Vasic, Valentin Savin, Michele Pacenti +2 more

Quantum error correction (QEC) is a cornerstone of quantum computing, enabling reliable information processing in the presence of noise. Sparse stabilizer codes -- referred to generally as quantum low-density parity-check (QLDPC) codes -- have risen to the forefront of QEC research in recent years. This can be attributed to several key factors. First, classical LDPC codes admit low-complexity belief propagation iterative decoding and near-capacity performance, which contributed to the early interest in QLDPC codes. Then, the result promising constant overhead fault tolerance using QLDPC codes led to the search for code families that go beyond the long-holding $\sqrt{n}$ scaling barrier of minimum distance for codelength $n$. This resulted in recent breakthroughs in the construction of QLDPC codes, which, combined with efficient decoding algorithms and the development of fault-tolerant protocols operating on QLDPC-encoded quantum information, provide a promising pathway to low-overhead, fault-tolerant quantum computation. However, despite their potential, challenges remain, particularly in constructing and decoding finite-length codes that account for, or efficiently leverage, specific characteristics of quantum hardware, such as connectivity, topology, native gate sets, and noise models. This article provides an in-depth examination of QLDPC codes and their iterative decoders, catering to an information theory audience with no or limited background in quantum mechanics. We discuss the theoretical underpinnings, explore unique characteristics of quantum channels, and delineate key code constructions and decoding algorithms, ultimately highlighting the impact and future prospects of QLDPC codes in quantum information science.

arXiv:2510.14090•IEEE BITS the Information Theory Magazine•Dec 1, 2025•0 citations

A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes

Sisi Miao, Jonathan Mandelbaum, Holger Jäkel +1 more

Quantum low-density parity-check (QLDPC) codes are among the most promising candidates for future quantum error correction schemes. However, a limited number of short to moderate-length QLDPC codes have been designed and their decoding performance is sub-optimal with a quaternary belief propagation (BP) decoder due to unavoidable short cycles in their Tanner graphs. In this paper, we propose a novel joint code and decoder design for QLDPC codes. The constructed codes have a minimum distance of about the square root of the block length. In addition, it is, to the best of our knowledge, the first QLDPC code family where BP decoding is not impaired by short cycles of length 4. This is achieved by using an ensemble BP decoder mitigating the influence of assembled short cycles. We outline two code construction methods based on classical quasi-cyclic codes and finite geometry codes. Numerical results demonstrate outstanding decoding performance over depolarizing channels.

arXiv:2401.06874•IEEE International Symposium on Information Theory (ISIT)•Jul 7, 2024•3 citations

Quaternary Neural Belief Propagation Decoding of Quantum LDPC Codes With Overcomplete Check Matrices

Sisi Miao, Alexander Schnerring, Haizheng Li +1 more

Quantum low-density parity-check (QLDPC) codes are promising candidates for error correction in quantum computers. One of the major challenges in implementing QLDPC codes in quantum computers is the lack of a universal decoder. In this work, we first propose to decode QLDPC codes with a belief propagation (BP) decoder operating on overcomplete check matrices. Then, we extend the neural BP (NBP) decoder, which was originally studied for suboptimal binary BP decoding of QLPDC codes, to quaternary BP decoders. Numerical simulation results demonstrate that both approaches as well as their combination yield a low-latency, high-performance decoder for several short to moderate length QLDPC codes.

arXiv:2308.08208•IEEE Access•Jan 1, 2025•7 citations

Quaternary-Binary Message-Passing Decoder for Quantum LDPC Codes

Dimitris Chytas, Nithin Raveendran, Asit Kumar Pradhan +1 more

We introduce a low-complexity message-passing quantum error correction algorithm for decoding Quantum Low-Density Parity-Check (QLDPC) stabilizer codes. The proposed decoder operates on the quaternary stabilizer graph but only exchanges binary messages. This leads to a significantly reduced complexity compared to other quaternary belief propagation (BP) algorithms that pass floating-point messages. The efficacy of the proposed decoder is evaluated by providing decoding examples, performance metrics using Monte-Carlo simulations, and complexity analysis. Despite its reduced complexity, the performance loss of the proposed decoder is modest compared to floating-point parallel quaternary decoders for a Calderbank-Shor-Steane (CSS) code family. In particular, experiments obtained over the [[1054, 140, 20]] lifted product (LP) Tanner code demonstrated that for low error rates (< 0.01), the proposed quaternary-binary message-passing decoder approaches the performance of quaternary BP by converging in almost the same number of iterations while requiring less complex operations. Additionally, for non-CSS codes, our decoder performs similarly as quaternary floating-point decoders despite its lower complexity.

Globecom•Dec 4, 2023•0 citations

Belief Propagation Decoding of Quantum LDPC Codes with Guided Decimation

Hanwen Yao, Waleed Abu Laban, Christian Häger +2 more

Quantum low-density parity-check (QLDPC) codes have emerged as a promising technique for quantum error correction. A variety of decoders have been proposed for QLDPC codes and many utilize belief propagation (BP) decoding in some fashion. However, the use of BP decoding for degenerate QLDPC codes is known to have issues with convergence. These issues are typically attributed to short cycles in the Tanner graph and error patterns with the same syndrome due to code degeneracy. In this work, we propose a decoder for QLDPC codes based on BP guided decimation (BPGD), which has been previously studied for constraint satisfaction and lossy compression problems. This decimation process is applicable to both binary and quaternary BP and it involves sequentially freezing the value of the most reliable qubits to encourage BP convergence. We find that BPGD significantly reduces the BP failure rate due to non-convergence, achieving performance on par with BP with ordered statistics decoding and BP with stabilizer inactivation, without the need to solve systems of linear equations. To explore how and why BPGD improves performance, we discuss several interpretations of BPGD and their connection to BP syndrome decoding.

arXiv:2312.10950•IEEE International Symposium on Information Theory (ISIT)•Jul 7, 2024•29 citations

A comprehensive Study of Decoding Algorithms for Low Density Parity-Check(LDPC)

C S Anil Kumar, M Vedhashri, L Vanditha +2 more

The near-capacity performance and effective decoding algorithms of Low-Density Parity-Check (LDPC) codes have made them a mainstay of contemporary error correction. With an emphasis on iterative techniques like the Belief Propagation (BP) algorithm, the Min-Sum algorithm, and their variations, this paper offers a thorough analysis of decoding algorithms for LDPC codes. We investigate the trade-offs between various algorithmic techniques’ computational complexity, decoding latency, and error correction performance. Particular focus is placed on new methods and hardware-friendly algorithms that are tailored for next-generation communication systems, such 5 G and beyond.

International Conference on Knowledge Engineering and Communication Systems (ICKECS)•Apr 28, 2025•0 citations
The near-capacity performance of classical low-density parity check (LDPC) codes and their efficient iterative decoding makes quantum LDPC (QLPDC) codes a promising candidate for quantum error correction. In this paper, we present a comprehensive survey of QLDPC codes from the perspective of code design as well as in terms of their decoding algorithms. We also conceive a modified non-binary decoding algorithm for homogeneous Calderbank-Shor-Steane-type QLDPC codes, which is capable of alleviating the problems imposed by the unavoidable length-four cycles. Our modified decoder outperforms the state-of-the-art decoders in terms of their word error rate performance, despite imposing a reduced decoding complexity. Finally, we intricately amalgamate our modified decoder with the classic uniformly reweighted belief propagation for the sake of achieving an improved performance.
Quantum error correction (QEC) is a cornerstone of quantum computing, enabling reliable information processing in the presence of noise. Sparse stabilizer codes -- referred to generally as quantum low-density parity-check (QLDPC) codes -- have risen to the forefront of QEC research in recent years. This can be attributed to several key factors. First, classical LDPC codes admit low-complexity belief propagation iterative decoding and near-capacity performance, which contributed to the early interest in QLDPC codes. Then, the result promising constant overhead fault tolerance using QLDPC codes led to the search for code families that go beyond the long-holding n\sqrt{n}n​ scaling barrier of minimum distance for codelength nnn. This resulted in recent breakthroughs in the construction of QLDPC codes, which, combined with efficient decoding algorithms and the development of fault-tolerant protocols operating on QLDPC-encoded quantum information, provide a promising pathway to low-overhead, fault-tolerant quantum computation. However, despite their potential, challenges remain, particularly in constructing and decoding finite-length codes that account for, or efficiently leverage, specific characteristics of quantum hardware, such as connectivity, topology, native gate sets, and noise models. This article provides an in-depth examination of QLDPC codes and their iterative decoders, catering to an information theory audience with no or limited background in quantum mechanics. We discuss the theoretical underpinnings, explore unique characteristics of quantum channels, and delineate key code constructions and decoding algorithms, ultimately highlighting the impact and future prospects of QLDPC codes in quantum information science.
Quantum low-density parity-check (QLDPC) codes are among the most promising candidates for future quantum error correction schemes. However, a limited number of short to moderate-length QLDPC codes have been designed and their decoding performance is sub-optimal with a quaternary belief propagation (BP) decoder due to unavoidable short cycles in their Tanner graphs. In this paper, we propose a novel joint code and decoder design for QLDPC codes. The constructed codes have a minimum distance of about the square root of the block length. In addition, it is, to the best of our knowledge, the first QLDPC code family where BP decoding is not impaired by short cycles of length 4. This is achieved by using an ensemble BP decoder mitigating the influence of assembled short cycles. We outline two code construction methods based on classical quasi-cyclic codes and finite geometry codes. Numerical results demonstrate outstanding decoding performance over depolarizing channels.
Quantum low-density parity-check (QLDPC) codes are promising candidates for error correction in quantum computers. One of the major challenges in implementing QLDPC codes in quantum computers is the lack of a universal decoder. In this work, we first propose to decode QLDPC codes with a belief propagation (BP) decoder operating on overcomplete check matrices. Then, we extend the neural BP (NBP) decoder, which was originally studied for suboptimal binary BP decoding of QLPDC codes, to quaternary BP decoders. Numerical simulation results demonstrate that both approaches as well as their combination yield a low-latency, high-performance decoder for several short to moderate length QLDPC codes.
We introduce a low-complexity message-passing quantum error correction algorithm for decoding Quantum Low-Density Parity-Check (QLDPC) stabilizer codes. The proposed decoder operates on the quaternary stabilizer graph but only exchanges binary messages. This leads to a significantly reduced complexity compared to other quaternary belief propagation (BP) algorithms that pass floating-point messages. The efficacy of the proposed decoder is evaluated by providing decoding examples, performance metrics using Monte-Carlo simulations, and complexity analysis. Despite its reduced complexity, the performance loss of the proposed decoder is modest compared to floating-point parallel quaternary decoders for a Calderbank-Shor-Steane (CSS) code family. In particular, experiments obtained over the [[1054, 140, 20]] lifted product (LP) Tanner code demonstrated that for low error rates (< 0.01), the proposed quaternary-binary message-passing decoder approaches the performance of quaternary BP by converging in almost the same number of iterations while requiring less complex operations. Additionally, for non-CSS codes, our decoder performs similarly as quaternary floating-point decoders despite its lower complexity.
Quantum low-density parity-check (QLDPC) codes have emerged as a promising technique for quantum error correction. A variety of decoders have been proposed for QLDPC codes and many utilize belief propagation (BP) decoding in some fashion. However, the use of BP decoding for degenerate QLDPC codes is known to have issues with convergence. These issues are typically attributed to short cycles in the Tanner graph and error patterns with the same syndrome due to code degeneracy. In this work, we propose a decoder for QLDPC codes based on BP guided decimation (BPGD), which has been previously studied for constraint satisfaction and lossy compression problems. This decimation process is applicable to both binary and quaternary BP and it involves sequentially freezing the value of the most reliable qubits to encourage BP convergence. We find that BPGD significantly reduces the BP failure rate due to non-convergence, achieving performance on par with BP with ordered statistics decoding and BP with stabilizer inactivation, without the need to solve systems of linear equations. To explore how and why BPGD improves performance, we discuss several interpretations of BPGD and their connection to BP syndrome decoding.
The near-capacity performance and effective decoding algorithms of Low-Density Parity-Check (LDPC) codes have made them a mainstay of contemporary error correction. With an emphasis on iterative techniques like the Belief Propagation (BP) algorithm, the Min-Sum algorithm, and their variations, this paper offers a thorough analysis of decoding algorithms for LDPC codes. We investigate the trade-offs between various algorithmic techniques’ computational complexity, decoding latency, and error correction performance. Particular focus is placed on new methods and hardware-friendly algorithms that are tailored for next-generation communication systems, such 5 G and beyond.