Spatially-Coupled QLDPC Codes

Siyi Yang; Robert Calderbank

doi:10.22331/q-2025-04-07-1693

Spatially-Coupled QLDPC Codes

Duke Quantum Center, Duke University, Durham, NC 27708, USA

Published:	2025-04-07, volume 9, page 1693
Editor:	Carlo Beenakker
Eprint:	arXiv:2305.00137v6
Doi:	https://doi.org/10.22331/q-2025-04-07-1693
Citation:	Quantum 9, 1693 (2025).

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Abstract

Spatially-coupled (SC) codes is a class of convolutional LDPC codes that has been well investigated in classical coding theory thanks to their high performance and compatibility with low-latency decoders. We describe toric codes as quantum counterparts of classical two-dimensional spatially-coupled (2D-SC) codes, and introduce spatially-coupled quantum LDPC (SC-QLDPC) codes as a generalization. We use the convolutional structure to represent the parity check matrix of a 2D-SC code as a polynomial in two indeterminates, and derive an algebraic condition that is both necessary and sufficient for a 2D-SC code to be a stabilizer code. This algebraic framework facilitates the construction of new code families. While not the focus of this paper, we note that small memory facilitates physical connectivity of qubits, and it enables local encoding and low-latency windowed decoding. In this paper, we use the algebraic framework to optimize short cycles in the Tanner graph of 2D-SC hypergraph product (HGP) codes that arise from short cycles in either component code. While prior work focuses on QLDPC codes with rate less than 1/10, we construct 2D-SC HGP codes with small memories, higher rates (about 1/3), and superior thresholds.

Popular summary

Surface codes, the leading approach to fault-tolerant quantum computing, offer high thresholds and hardware-friendly implementations. However, their substantial overhead poses a significant challenge for large-scale quantum computing. Quantum low-density parity-check (QLDPC) codes present a promising alternative, significantly reducing qubit overhead while maintaining strong error correction capabilities. Despite their potential, two major challenges remain: (1) designing QLDPC codes that adhere to the connectivity constraints of quantum hardware and (2) optimizing finite-length QLDPC codes, as most high-performance constructions have low rates, while constant-rate codes are typically analyzed asymptotically.
Motivated by the structural similarities between Toric codes and two-dimensional spatially-coupled (SC) LDPC codes, we recognized that the short memory and periodic structure of SC codes offer advantages for quantum hardware implementation, particularly in neutral atom arrays. Building on this insight, we introduced Spatially-Coupled Quantum LDPC (SC-QLDPC) codes, which naturally align with neutral atom arrays, adapt well to modular quantum hardware, and have the potential to enable low-latency windowed decoding. We developed an algebraic framework characterizing SC-QLDPC codes and performed finite-length optimization (cycle reduction) on a special class of SC codes. Our constructions achieve a code rate as high as 0.342 with a decoding threshold of 7%, advancing the feasibility of high-rate quantum error correction.

► BibTeX data

@article{Yang2025spatiallycoupled,
  doi = {10.22331/q-2025-04-07-1693},
  url = {https://doi.org/10.22331/q-2025-04-07-1693},
  title = {Spatially-{C}oupled {QLDPC} {C}odes},
  author = {Yang, Siyi and Calderbank, Robert},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1693},
  month = apr,
  year = {2025}
}

► References

[1] A. Robert Calderbank, Eric M. Rains, Peter W. Shor, and Neil J. A. Sloane. ``Quantum error correction and orthogonal geometry''. Physical Review Letters 78, 405–409 (1997).
https://doi.org/10.1103/PhysRevLett.78.405

[2] A. Robert Calderbank, Eric M. Rains, Peter W. Shor, and Neil J. A. Sloane. ``Quantum error correction via codes over GF(4)''. IEEE Transactions on Information Theory 44, 1369–1387 (1998).
https://doi.org/10.1109/ISIT.1997.613213

[3] Daniel Gottesman. ``Stabilizer codes and quantum error correction''. PhD thesis. California Institute of Technology. (1997).
https://doi.org/10.7907/rzr7-dt72

[4] A Yu Kitaev. ``Quantum computations: algorithms and error correction''. Russian Mathematical Surveys 52, 1191 (1997).
https://doi.org/10.1070/RM1997v052n06ABEH002155

[5] A Yu Kitaev. ``Fault-tolerant quantum computation by anyons''. Annals of physics 303, 2–30 (2003).
https://doi.org/10.1016/S0003-4916(02)00018-0

[6] Sergey B Bravyi and A Yu Kitaev. ``Quantum codes on a lattice with boundary'' (1998). arXiv:9811052.
arXiv:quant-ph/9811052

[7] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. Journal of Mathematical Physics 43, 4452–4505 (2002).
https://doi.org/10.1063/1.1499754

[8] Michael H Freedman, David A Meyer, and Feng Luo. ``Z2-systolic freedom and quantum codes''. Chapter 12, pages 303–338. Chapman and Hall/CRC. (2002).
https://doi.org/10.1201/9781420035377

[9] Sergey Bravyi, David Poulin, and Barbara Terhal. ``Tradeoffs for reliable quantum information storage in 2d systems''. Physical review letters 104, 050503 (2010).
https://doi.org/10.1103/PhysRevLett.104.050503

[10] Austin G Fowler, Matteo Mariantoni, John M Martinis, and Andrew N Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Physical Review A 86, 032324 (2012).
https://doi.org/10.1103/PhysRevA.86.032324

[11] Clare Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter. ``Surface code quantum computing by lattice surgery''. New Journal of Physics 14, 123011 (2012).
https://doi.org/10.22331/q-2019-03-05-128

[12] Nicolas Delfosse and Gilles Zémor. ``Quantum erasure-correcting codes and percolation on regular tilings of the hyperbolic plane''. In 2010 IEEE Information Theory Workshop. Pages 1–5. IEEE (2010).
https://doi.org/10.1109/CIG.2010.5592863

[13] Barbara M Terhal. ``Quantum error correction for quantum memories''. Reviews of Modern Physics 87, 307 (2015).
https://doi.org/10.1103/RevModPhys.87.307

[14] Nicolas Delfosse, Pavithran Iyer, and David Poulin. ``Generalized surface codes and packing of logical qubits'' (2016). arXiv:1606.07116.
arXiv:1606.07116

[15] Nikolas P Breuckmann, Christophe Vuillot, Earl Campbell, Anirudh Krishna, and Barbara M Terhal. ``Hyperbolic and semi-hyperbolic surface codes for quantum storage''. Quantum Science and Technology 2, 035007 (2017).
https://doi.org/10.1088/2058-9565/aa7d3b

[16] Jean-Pierre Tillich and Gilles Zémor. ``Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength''. IEEE Transactions on Information Theory 60, 1193–1202 (2014).
https://doi.org/10.1109/TIT.2013.2292061

[17] Anthony Leverrier, Simon Apers, and Christophe Vuillot. ``Quantum XYZ product codes''. Quantum 6, 766 (2022).
https://doi.org/10.22331/q-2022-07-14-766

[18] Shai Evra, Tali Kaufman, and Gilles Zémor. ``Decodable quantum LDPC codes beyond the n distance barrier using high-dimensional expanders''. SIAM Journal on ComputingPages 276–316 (2022).
https://doi.org/10.1137/20M1383689

[19] M. Sipser and D.A. Spielman. ``Expander codes''. IEEE Transactions on Information Theory 42, 1710–1722 (1996).
https://doi.org/10.1109/18.556667

[20] Gilles Zémor. ``On cayley graphs, surface codes, and the limits of homological coding for quantum error correction''. In Coding and Cryptology: Second International Workshop, IWCC 2009, Zhangjiajie, China, June 1-5, 2009. Proceedings 2. Pages 259–273. Springer (2009).
https://doi.org/10.1007/978-3-642-01877-0_21

[21] Matthew B Hastings, Jeongwan Haah, and Ryan O'Donnell. ``Fiber bundle codes: breaking the n 1/2 polylog (n) barrier for quantum LDPC codes''. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. Pages 1276–1288. (2021).
https://doi.org/10.1145/3406325.3451005

[22] Pavel Panteleev and Gleb Kalachev. ``Quantum LDPC codes with almost linear minimum distance''. IEEE Transactions on Information Theory 68, 213–229 (2022).
https://doi.org/10.1109/TIT.2021.3119384

[23] Nikolas P Breuckmann and Jens N Eberhardt. ``Balanced product quantum codes''. IEEE Transactions on Information Theory 67, 6653–6674 (2021).
https://doi.org/10.1109/TIT.2021.3097347

[24] Pavel Panteleev and Gleb Kalachev. ``Asymptotically good quantum and locally testable classical LDPC codes''. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. Pages 375––388. (2022).
https://doi.org/10.1145/3519935.3520017

[25] A. Leverrier and G. Zemor. ``Quantum tanner codes''. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS). Pages 872–883. (2022).
https://doi.org/10.1109/FOCS54457.2022.00117

[26] Robert Gallager. ``Low-density parity-check codes''. IRE Transactions on information theory 8, 21–28 (1962).
https://doi.org/10.1109/TIT.1962.1057683

[27] Thomas J Richardson, Mohammad Amin Shokrollahi, and Rüdiger L Urbanke. ``Design of capacity-approaching irregular low-density parity-check codes''. IEEE transactions on information theory 47, 619–637 (2001).
https://doi.org/10.1109/18.910578

[28] Thomas J Richardson and Rüdiger L Urbanke. ``The capacity of low-density parity-check codes under message-passing decoding''. IEEE Transactions on information theory 47, 599–618 (2001).
https://doi.org/10.1109/18.910577

[29] David Poulin and Yeojin Chung. ``On the iterative decoding of sparse quantum codes''. Quantum Information & Computation 8, 987–1000 (2008).
https://doi.org/10.26421/QIC8.10-8

[30] Daniel A Lidar and Todd A Brun. ``Quantum error correction''. Cambridge university press. (2013).
https://doi.org/10.1017/CBO9781139034807

[31] D. G. M. Mitchell, M. Lentmaier, and D. J. Costello. ``Spatially coupled LDPC codes constructed from protographs''. IEEE Trans. Information Theory 61, 4866–4889 (2015).
https://doi.org/10.1109/TIT.2015.2453267

[32] David G. M. Mitchell and Eirik Rosnes. ``Edge spreading design of high rate array-based SC-LDPC codes''. In 2017 IEEE International Symposium on Information Theory (ISIT). Pages 2940–2944. (2017).
https://doi.org/10.1109/ISIT.2017.8007068

[33] Allison Beemer, Salman Habib, Christine A. Kelley, and Joerg Kliewer. ``A generalized algebraic approach to optimizing SC-LDPC codes''. In 2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton). Pages 672–679. (2017).
https://doi.org/10.1109/ALLERTON.2017.8262802

[34] Allison Beemer. ``Design and analysis of graph-based codes using algebraic lifts and decoding networks''. PhD thesis. The University of Nebraska-Lincoln. (2018). url: https://digitalcommons.unl.edu/mathstudent/87/.
https://digitalcommons.unl.edu/mathstudent/87/

[35] Ahmed Hareedy, Chinmayi Lanka, and Lara Dolecek. ``A general non-binary LDPC code optimization framework suitable for dense flash memory and magnetic storage''. IEEE Journal on Selected Areas in Communications 34, 2402–2415 (2016).
https://doi.org/10.1109/JSAC.2016.2603719

[36] A. Hareedy, H. Esfahanizadeh, and L. Dolecek. ``High performance non-binary spatially-coupled codes for flash memories''. In 2017 IEEE Information Theory Workshop (ITW). Pages 229–233. (2017).
https://doi.org/10.1109/ITW.2017.8277940

[37] Ahmed Hareedy, Ruiyi Wu, and Lara Dolecek. ``A channel-aware combinatorial approach to design high performance spatially-coupled codes''. IEEE Trans. Information Theory 66, 4834–4852 (2020).
https://doi.org/10.1109/TIT.2020.2979981

[38] Siyi Yang, Ahmed Hareedy, Robert Calderbank, and Lara Dolecek. ``Breaking the computational bottleneck: Probabilistic optimization of high-memory spatially-coupled codes''. IEEE Transactions on Information Theory 69, 886–909 (2023).
https://doi.org/10.1109/TIT.2022.3207321

[39] Manabu Hagiwara, Kenta Kasai, Hideki Imai, and Kohichi Sakaniwa. ``Spatially coupled quasi-cyclic quantum LDPC codes''. In 2011 IEEE International Symposium on Information Theory Proceedings. Pages 638–642. (2011).
https://doi.org/10.1109/ISIT.2011.6034208

[40] M. Lentmaier, A. Sridharan, D. J. Costello, and K. S. Zigangirov. ``Iterative decoding threshold analysis for LDPC convolutional codes''. IEEE Trans. Information Theory 56, 5274–5289 (2010).
https://doi.org/10.1109/TIT.2010.2059490

[41] Aravind R Iyengar, Marco Papaleo, Paul H Siegel, Jack Keil Wolf, Alessandro Vanelli-Coralli, and Giovanni E Corazza. ``Windowed decoding of protograph-based LDPC convolutional codes over erasure channels''. IEEE Transactions on Information Theory 58, 2303–2320 (2011).
https://doi.org/10.1109/TIT.2011.2177439

[42] Aravind R Iyengar, Paul H Siegel, Rüdiger L Urbanke, and Jack Keil Wolf. ``Windowed decoding of spatially coupled codes''. IEEE transactions on Information Theory 59, 2277–2292 (2012).
https://doi.org/10.1109/TIT.2012.2231465

[43] Najeeb Ul Hassan, Ali E Pusane, Michael Lentmaier, Gerhard P Fettweis, and Daniel J Costello. ``Non-uniform window decoding schedules for spatially coupled LDPC codes''. IEEE Transactions on Communications 65, 501–510 (2016).
https://doi.org/10.1049/cmu2.12456

[44] Lai Wei, David GM Mitchell, Thomas E Fuja, and Daniel J Costello. ``Design of spatially coupled LDPC codes over GF$(q) $ for windowed decoding''. IEEE Transactions on Information Theory 62, 4781–4800 (2016).
https://doi.org/10.1109/TIT.2016.2567638

[45] Min Zhu, David GM Mitchell, Michael Lentmaier, Daniel J Costello, and Baoming Bai. ``Braided convolutional codes with sliding window decoding''. IEEE Transactions on Communications 65, 3645–3658 (2017).
https://doi.org/10.1109/TCOMM.2017.2707073

[46] Kevin Klaiber, Sebastian Cammerer, Laurent Schmalen, and Stephan ten Brink. ``Avoiding burst-like error patterns in windowed decoding of spatially coupled LDPC codes''. In 2018 IEEE 10th International Symposium on Turbo Codes & Iterative Information Processing (ISTC). Pages 1–5. IEEE (2018).
https://doi.org/10.1109/ITW46852.2021.9457651

[47] Homa Esfahanizadeh, Lev Tauz, and Lara Dolecek. ``Multi-dimensional spatially-coupled code design: Enhancing the cycle properties''. IEEE Transactions on Communications 68, 2653–2666 (2020).
https://doi.org/10.1109/TCOMM.2020.2971694

[48] Lev Tauz, Homa Esfahanizadeh, and Lara Dolecek. ``Non-uniform windowed decoding for multi-dimensional spatially-coupled LDPC codes''. In 2020 IEEE International Symposium on Information Theory (ISIT). Pages 485–490. IEEE (2020).
https://doi.org/10.1109/ISIT44484.2020.9174511

[49] Eshed Ram and Yuval Cassuto. ``On the decoding performance of spatially coupled LDPC codes with sub-block access''. IEEE Transactions on Information Theory 68, 3700–3718 (2022).
https://doi.org/10.1109/TIT.2022.3152104

[50] Jeongwan Haah. ``Commuting pauli hamiltonians as maps between free modules''. Communications in Mathematical Physics 324, 351–399 (2013).
https://doi.org/10.1007/s00220-013-1810-2

[51] Jeongwan Haah. ``Algebraic methods for quantum codes on lattices''. Revista colombiana de matematicas 50, 299–349 (2016).
https://doi.org/10.15446/recolma.v50n2.62214

[52] Harold Ollivier and Jean-Pierre Tillich. ``Description of a quantum convolutional code''. Physical Review Letters 91, 177902 (2003).
https://doi.org/10.1103/PhysRevLett.91.177902

[53] S. Kudekar, T. J. Richardson, and R. L. Urbanke. ``Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC''. IEEE Trans. Information Theory 57, 803–834 (2011).
https://doi.org/10.1109/TIT.2010.2095072

[54] S. Kumar, A. J. Young, N. Macris, and H. D. Pfister. ``Threshold saturation for spatially coupled LDPC and LDGM codes on BMS channels''. IEEE Trans. Information Theory 60, 7389–7415 (2014).
https://doi.org/10.1109/TIT.2014.2360692

[55] Sergey Bravyi, Andrew W Cross, Jay M Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J Yoder. ``High-threshold and low-overhead fault-tolerant quantum memory''. Nature 627, 778–782 (2024).
https://doi.org/10.1038/s41586-024-07107-7

[56] Qian Xu, J Pablo Bonilla Ataides, Christopher A Pattison, Nithin Raveendran, Dolev Bluvstein, Jonathan Wurtz, Bane Vasić, Mikhail D Lukin, Liang Jiang, and Hengyun Zhou. ``Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays''. Nature PhysicsPages 1–7 (2024).
https://doi.org/10.1038/s41567-024-02479-z

[57] Hanrui Wang, Pengyu Liu, Daniel Bochen Tan, Yilian Liu, Jiaqi Gu, David Z Pan, Jason Cong, Umut A Acar, and Song Han. ``Atomique: A quantum compiler for reconfigurable neutral atom arrays''. In 2024 ACM/IEEE 51st Annual International Symposium on Computer Architecture (ISCA). Pages 293–309. (2024).
https://doi.org/10.1109/ISCA59077.2024.00030

[58] Pavel Panteleev and Gleb Kalachev. ``Degenerate quantum LDPC codes with good finite length performance''. Quantum 5, 585 (2021).
https://doi.org/10.22331/q-2021-11-22-585

[59] M. Battaglioni, A. Tasdighi, G. Cancellieri, F. Chiaraluce, and M. Baldi. ``Design and analysis of time-invariant SC-LDPC convolutional codes with small constraint length''. IEEE Trans. Communications 66, 918–931 (2018).
https://doi.org/10.1109/TCOMM.2017.2774821

[60] M. P. C. Fossorier. ``Quasicyclic low-density parity-check codes from circulant permutation matrices''. IEEE Trans. Information Theory 50, 1788–1793 (2004).
https://doi.org/10.1109/TIT.2004.831841

[61] Joschka Roffe, David R. White, Simon Burton, and Earl Campbell. ``Decoding across the quantum low-density parity-check code landscape''. Physical Review Research 2 (2020).
https://doi.org/10.1103/physrevresearch.2.043423

[62] Julien Du Crest, Mehdi Mhalla, and Valentin Savin. ``Stabilizer inactivation for message-passing decoding of quantum LDPC codes''. In 2022 IEEE Information Theory Workshop (ITW). Pages 488–493. IEEE (2022).
https://doi.org/10.1109/ITW54588.2022.9965902

[63] Hanwen Yao, Waleed Abu Laban, Christian Häger, Alexandre Graell i Amat, and Henry D Pfister. ``Belief propagation decoding of quantum ldpc codes with guided decimation''. In 2024 IEEE International Symposium on Information Theory (ISIT). Pages 2478–2483. IEEE (2024).
https://doi.org/10.1109/ISIT57864.2024.10619083

[64] Joschka Roffe (2022). code: https://pypi.org/project/ldpc/.
https://pypi.org/project/ldpc/

[65] Antoine Grospellier, Lucien Grouès, Anirudh Krishna, and Anthony Leverrier. ``Combining hard and soft decoders for hypergraph product codes''. Quantum 5, 432 (2021).
https://doi.org/10.22331/q-2021-04-15-432

[66] Siyi Yang (2025). code: SiyiPriscaYang/Spatially-Coupled-QLDPC-Codes.
https://github.com/SiyiPriscaYang/Spatially-Coupled-QLDPC-Codes

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