0% example non-article
% consider using an archive.org link in case the original resource is not reliably hosted

@book{eisenbud2013commutative,
  title={Commutative algebra: with a view toward algebraic geometry},
  author={Eisenbud, David},
  volume={150},
  year={2013},
  publisher={Springer Science \& Business Media}
}

@book{weibel1994introduction,
  title={An introduction to homological algebra},
  author={Weibel, Charles A},
  number={38},
  year={1994},
  publisher={Cambridge university press}
}

@misc{jacob2025singleshotdecodingfaulttolerantgates,
      title={Single-Shot Decoding and Fault-tolerant Gates with Trivariate Tricycle Codes}, 
      author={Abraham Jacob and Campbell McLauchlan and Dan E. Browne},
      year={2025},
      eprint={2508.08191},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2508.08191}, 
}

@misc{breuckmann2024cupsgatesicohomology,
      title={Cups and Gates I: Cohomology invariants and logical quantum operations}, 
      author={Nikolas P. Breuckmann and Margarita Davydova and Jens N. Eberhardt and Nathanan Tantivasadakarn},
      year={2024},
      eprint={2410.16250},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2410.16250}, 
}

@article{Bravyi_2024,
   title={High-threshold and low-overhead fault-tolerant quantum memory},
   volume={627},
   ISSN={1476-4687},
   url={http://dx.doi.org/10.1038/s41586-024-07107-7},
   DOI={10.1038/s41586-024-07107-7},
   number={8005},
   journal={Nature},
   publisher={Springer Science and Business Media LLC},
   author={Bravyi, Sergey and Cross, Andrew W. and Gambetta, Jay M. and Maslov, Dmitri and Rall, Patrick and Yoder, Theodore J.},
   year={2024},
   month=mar, pages={778–782} }

@misc{lin2023quantumtwoblockgroupalgebra,
      title={Quantum two-block group algebra codes}, 
      author={Hsiang-Ku Lin and Leonid P. Pryadko},
      year={2023},
      eprint={2306.16400},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2306.16400}, 
}

@misc{eberhardt2024logicaloperatorsfoldtransversalgates,
      title={Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes}, 
      author={Jens Niklas Eberhardt and Vincent Steffan},
      year={2024},
      eprint={2407.03973},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2407.03973}, 
}

@article{Campbell_2019,
   title={A theory of single-shot error correction for adversarial noise},
   volume={4},
   ISSN={2058-9565},
   url={http://dx.doi.org/10.1088/2058-9565/aafc8f},
   DOI={10.1088/2058-9565/aafc8f},
   number={2},
   journal={Quantum Science and Technology},
   publisher={IOP Publishing},
   author={Campbell, Earl T},
   year={2019},
   month=feb, pages={025006} }

@article{breuckmann2021balanced,
  title={Balanced product quantum codes},
  author={Breuckmann, Nikolas P and Eberhardt, Jens N},
  journal={IEEE Transactions on Information Theory},
  volume={67},
  number={10},
  pages={6653--6674},
  year={2021},
  publisher={IEEE}
}

@article{Breuckmann_2021,
   title={Quantum Low-Density Parity-Check Codes},
   volume={2},
   ISSN={2691-3399},
   url={http://dx.doi.org/10.1103/PRXQuantum.2.040101},
   DOI={10.1103/prxquantum.2.040101},
   number={4},
   journal={PRX Quantum},
   publisher={American Physical Society (APS)},
   author={Breuckmann, Nikolas P. and Eberhardt, Jens Niklas},
   year={2021},
   month=oct }


@article{Zeng_2019,
   title={Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates},
   volume={122},
   ISSN={1079-7114},
   url={http://dx.doi.org/10.1103/PhysRevLett.122.230501},
   DOI={10.1103/physrevlett.122.230501},
   number={23},
   journal={Physical Review Letters},
   publisher={American Physical Society (APS)},
   author={Zeng, Weilei and Pryadko, Leonid P.},
   year={2019},
   month=jun }

@misc{xu2024fastparallelizablelogicalcomputation,
      title={Fast and Parallelizable Logical Computation with Homological Product Codes}, 
      author={Qian Xu and Hengyun Zhou and Guo Zheng and Dolev Bluvstein and J. Pablo Bonilla Ataides and Mikhail D. Lukin and Liang Jiang},
      year={2024},
      eprint={2407.18490},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2407.18490}, 
}


@article{Quintavalle_2021,
   title={Single-Shot Error Correction of Three-Dimensional Homological Product Codes},
   volume={2},
   ISSN={2691-3399},
   url={http://dx.doi.org/10.1103/PRXQuantum.2.020340},
   DOI={10.1103/prxquantum.2.020340},
   number={2},
   journal={PRX Quantum},
   publisher={American Physical Society (APS)},
   author={Quintavalle, Armanda O. and Vasmer, Michael and Roffe, Joschka and Campbell, Earl T.},
   year={2021},
   month=jun }


@article{Kubica_2022,
   title={Single-shot quantum error correction with the three-dimensional subsystem toric code},
   volume={13},
   ISSN={2041-1723},
   url={http://dx.doi.org/10.1038/s41467-022-33923-4},
   DOI={10.1038/s41467-022-33923-4},
   number={1},
   journal={Nature Communications},
   publisher={Springer Science and Business Media LLC},
   author={Kubica, Aleksander and Vasmer, Michael},
   year={2022},
   month=oct }

@article{Berthusen_2024,
   title={Experiments with the four-dimensional surface code on a quantum charge-coupled device quantum computer},
   volume={110},
   ISSN={2469-9934},
   url={http://dx.doi.org/10.1103/PhysRevA.110.062413},
   DOI={10.1103/physreva.110.062413},
   number={6},
   journal={Physical Review A},
   publisher={American Physical Society (APS)},
   author={Berthusen, Noah and Dreiling, Joan and Foltz, Cameron and Gaebler, John P. and Gatterman, Thomas M. and Gresh, Dan and Hewitt, Nathan and Mills, Michael and Moses, Steven A. and Neyenhuis, Brian and Siegfried, Peter and Hayes, David},
   year={2024},
   month=dec }

@article{Stahl_2024,
   title={Single-shot quantum error correction in intertwined toric codes},
   volume={110},
   ISSN={2469-9969},
   url={http://dx.doi.org/10.1103/PhysRevB.110.075143},
   DOI={10.1103/physrevb.110.075143},
   number={7},
   journal={Physical Review B},
   publisher={American Physical Society (APS)},
   author={Stahl, Charles},
   year={2024},
   month=aug }


@article{steane1996simple,
  title={Simple quantum error-correcting codes},
  author={Steane, Andrew M},
  journal={Physical Review A},
  volume={54},
  number={6},
  pages={4741},
  year={1996},
  publisher={APS}
}

@article{calderbank1996good,
  title={Good quantum error-correcting codes exist},
  author={Calderbank, A Robert and Shor, Peter W},
  journal={Physical Review A},
  volume={54},
  number={2},
  pages={1098},
  year={1996},
  publisher={APS}
}

@article{sather2004commutative,
  title={Commutative Algebra Mini-Course},
  author={Sather-Wagstaff, Sean},
  year={2004}
}

@incollection{beck2014somewhat,
  title={A somewhat gentle introduction to differential graded commutative algebra},
  author={Beck, Kristen A and Sather-Wagstaff, Sean},
  booktitle={Connections between algebra, combinatorics, and geometry},
  pages={3--99},
  year={2014},
  publisher={Springer}
}

@article{Chen_2025,
   title={Anyon Theory and Topological Frustration of High-Efficiency Quantum Low-Density Parity-Check Codes},
   volume={135},
   ISSN={1079-7114},
   url={http://dx.doi.org/10.1103/86j7-cmsw},
   DOI={10.1103/86j7-cmsw},
   number={7},
   journal={Physical Review Letters},
   publisher={American Physical Society (APS)},
   author={Chen, Keyang and Liu, Yuanting and Zhang, Yiming and Liang, Zijian and Chen, Yu-An and Liu, Ke and Song, Hao},
   year={2025},
   month=aug }

@misc{murfet2024linearlogicquantumerror,
      title={Linear Logic and Quantum Error Correcting Codes}, 
      author={Daniel Murfet and William Troiani},
      year={2024},
      eprint={2405.19051},
      archivePrefix={arXiv},
      primaryClass={math.LO},
      url={https://arxiv.org/abs/2405.19051}, 
}

@article{august2015differential,
  title={Differential graded algebras and applications},
  author={August, Jenny and Booth, Matt and Cooke, Juliet and Weelinck, Tim},
  journal={URL: https://www. maths. ed. ac. uk/\~{} mbooth/hodgeproject. pdf},
  year={2015}
}

@article{green1984koszul,
  title={Koszul cohomology and the geometry of projective varieties},
  author={Green, Mark L},
  journal={Journal of Differential Geometry},
  volume={19},
  number={1},
  pages={125--171},
  year={1984},
  publisher={Lehigh University}
}

@book{aprodu2010koszul,
  title={Koszul cohomology and algebraic geometry},
  author={Aprodu, Marian and Nagel, Jan},
  volume={52},
  year={2010},
  publisher={American Mathematical Soc.}
}

@book{negri2008structural,
  title={Structural proof theory},
  author={Negri, Sara and Von Plato, Jan},
  year={2008},
  publisher={Cambridge university press}
}

@article{voss2025multivariate,
  title={Multivariate bicycle codes},
  author={Voss, Lukas and Xian, Sim Jian and Haug, Tobias and Bharti, Kishor},
  journal={Physical Review A},
  volume={111},
  number={6},
  pages={L060401},
  year={2025},
  publisher={APS}
}

@misc{OSCAR,
  key          = {OSCAR},
  organization = {The OSCAR Team},
  title        = {O{SCAR} -- {O}pen {S}ource {C}omputer {A}lgebra {R}esearch system, {V}ersion 1.5.0},
  year         = {2025},
  url          = {https://www.oscar-system.org},
}

@book{OSCAR-book,
  editor = {Decker, Wolfram and Eder, Christian and Fieker, Claus and Horn, Max and Joswig, Michael},
  title = {The {C}omputer {A}lgebra {S}ystem {OSCAR}: {A}lgorithms and {E}xamples},
  year = {2025},
  publisher = {Springer},
  series = {Algorithms and {C}omputation in {M}athematics},
  volume = {32},
  edition = {1},
  url = {https://link.springer.com/book/9783031621260},
  issn = {1431-1550},
  doi = {10.1007/978-3-031-62127-7},
}

@software{quantumclifford,
  author       = {{QuantumClifford Contributors}},
  title        = {{QuantumSavory/QuantumClifford.jl: v0.10.1}},
  month        = sep,
  year         = 2025,
  publisher    = {Zenodo},
  version      = {v0.10.1},
  doi          = {10.5281/zenodo.13950271},
  url          = {https://doi.org/10.5281/zenodo.13950271}
}

@article{Dennis_2002,
   title={Topological quantum memory},
   volume={43},
   ISSN={1089-7658},
   url={http://dx.doi.org/10.1063/1.1499754},
   DOI={10.1063/1.1499754},
   number={9},
   journal={Journal of Mathematical Physics},
   publisher={AIP Publishing},
   author={Dennis, Eric and Kitaev, Alexei and Landahl, Andrew and Preskill, John},
   year={2002},
   month=sep, pages={4452–4505} }


@book{atiyah2018introduction,
  title={Introduction to commutative algebra},
  author={Atiyah, Michael F and Macdonald, Ian Grant},
  year={2018},
  publisher={CRC Press}
}
@book{bruns1998cohen,
  title={Cohen-macaulay rings},
  author={Bruns, Winfried and Herzog, H J{\"u}rgen},
  number={39},
  year={1998},
  publisher={Cambridge university press}
}

@book{matsumura1989commutative,
  title={Commutative ring theory},
  author={Matsumura, Hideyuki},
  number={8},
  year={1989},
  publisher={Cambridge university press}
}

@misc{yoder2025tourgrossmodularquantum,
      title={Tour de gross: A modular quantum computer based on bivariate bicycle codes}, 
      author={Theodore J. Yoder and Eddie Schoute and Patrick Rall and Emily Pritchett and Jay M. Gambetta and Andrew W. Cross and Malcolm Carroll and Michael E. Beverland},
      year={2025},
      eprint={2506.03094},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2506.03094}, 
}


@article{Breuckmann_2024,
   title={Fold-Transversal Clifford Gates for Quantum Codes},
   volume={8},
   ISSN={2521-327X},
   url={http://dx.doi.org/10.22331/q-2024-06-13-1372},
   DOI={10.22331/q-2024-06-13-1372},
   journal={Quantum},
   publisher={Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
   author={Breuckmann, Nikolas P. and Burton, Simon},
   year={2024},
   month=jun, pages={1372} }

@misc{sayginel2025faulttolerantlogicalcliffordgates,
      title={Fault-Tolerant Logical Clifford Gates from Code Automorphisms}, 
      author={Hasan Sayginel and Stergios Koutsioumpas and Mark Webster and Abhishek Rajput and Dan E Browne},
      year={2025},
      eprint={2409.18175},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2409.18175}, 
}

@misc{zheng2025highratesurgeryconstantoverheadlogical,
      title={High-Rate Surgery: towards constant-overhead logical operations}, 
      author={Guo Zheng and Liang Jiang and Qian Xu},
      year={2025},
      eprint={2510.08523},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2510.08523}, 
}

@article{Ide_2025,
   title={Fault-Tolerant Logical Measurements via Homological Measurement},
   volume={15},
   ISSN={2160-3308},
   url={http://dx.doi.org/10.1103/PhysRevX.15.021088},
   DOI={10.1103/physrevx.15.021088},
   number={2},
   journal={Physical Review X},
   publisher={American Physical Society (APS)},
   author={Ide, Benjamin and Gowda, Manoj G. and Nadkarni, Priya J. and Dauphinais, Guillaume},
   year={2025},
   month=jun }

@article{cowtan2024css,
  title={CSS code surgery as a universal construction},
  author={Cowtan, Alexander and Burton, Simon},
  journal={Quantum},
  volume={8},
  pages={1344},
  year={2024},
  publisher={Verein zur F{\"o}rderung des Open Access Publizierens in den Quantenwissenschaften}
}

@article{cowtan2025parallel,
  title={Parallel logical measurements via quantum code surgery},
  author={Cowtan, Alexander and He, Zhiyang and Williamson, Dominic J and Yoder, Theodore J},
  journal={arXiv preprint arXiv:2503.05003},
  year={2025}
}

@article{Cohen_2022,
   title={Low-overhead fault-tolerant quantum computing using long-range connectivity},
   volume={8},
   ISSN={2375-2548},
   url={http://dx.doi.org/10.1126/sciadv.abn1717},
   DOI={10.1126/sciadv.abn1717},
   number={20},
   journal={Science Advances},
   publisher={American Association for the Advancement of Science (AAAS)},
   author={Cohen, Lawrence Z. and Kim, Isaac H. and Bartlett, Stephen D. and Brown, Benjamin J.},
   year={2022},
   month=may }

@article{bombin2015single,
  title={Single-shot fault-tolerant quantum error correction},
  author={Bomb{\'\i}n, H{\'e}ctor},
  journal={Physical Review X},
  volume={5},
  number={3},
  pages={031043},
  year={2015},
  publisher={APS}
}

@article{tan2025single,
  title={Single-Shot Universality in Quantum LDPC Codes via Code-Switching},
  author={Tan, Shi Jie Samuel and Hong, Yifan and Lin, Ting-Chun and Gullans, Michael J and Hsieh, Min-Hsiu},
  journal={arXiv preprint arXiv:2510.08552},
  year={2025}
}

@article{PhysRevA.88.012311,
  title = {Quantum Kronecker sum-product low-density parity-check codes with finite rate},
  author = {Kovalev, Alexey A. and Pryadko, Leonid P.},
  journal = {Phys. Rev. A},
  volume = {88},
  issue = {1},
  pages = {012311},
  numpages = {13},
  year = {2013},
  month = {Jul},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevA.88.012311},
  url = {https://link.aps.org/doi/10.1103/PhysRevA.88.012311}
}

@article{liang2025generalized,
  title={Generalized toric codes on twisted tori for quantum error correction},
  author={Liang, Zijian and Liu, Ke and Song, Hao and Chen, Yu-An},
  journal={PRX Quantum},
  volume={6},
  number={2},
  pages={020357},
  year={2025},
  publisher={APS}
}

@misc{rabeti2025listdecodingnewbicycle,
      title={List Decoding and New Bicycle Code Constructions for Quantum LDPC Codes}, 
      author={Sheida Rabeti and Hessam Mahdavifar},
      year={2025},
      eprint={2511.02951},
      archivePrefix={arXiv},
      primaryClass={cs.IT},
      url={https://arxiv.org/abs/2511.02951}, 
}

@misc{panteleev2022asymptoticallygoodquantumlocally,
      title={Asymptotically Good Quantum and Locally Testable Classical LDPC Codes}, 
      author={Pavel Panteleev and Gleb Kalachev},
      year={2022},
      eprint={2111.03654},
      archivePrefix={arXiv},
      primaryClass={cs.IT},
      url={https://arxiv.org/abs/2111.03654}, 
}

@article{kobayashi2025clifford,
  title={Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic},
  author={Kobayashi, Ryohei and Zhu, Guanyu and Hsin, Po-Shen},
  journal={arXiv preprint arXiv:2511.02900},
  year={2025}
}

@misc{lin2025singleshottwoshotdecodinggeneralized,
      title={Single-shot and two-shot decoding with generalized bicycle codes}, 
      author={Hsiang-Ku Lin and Xingrui Liu and Pak Kau Lim and Leonid P. Pryadko},
      year={2025},
      eprint={2502.19406},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2502.19406}, 
}

@article{Gu_2024,
   title={Single-Shot Decoding of Good Quantum LDPC Codes},
   volume={405},
   ISSN={1432-0916},
   url={http://dx.doi.org/10.1007/s00220-024-04951-6},
   DOI={10.1007/s00220-024-04951-6},
   number={3},
   journal={Communications in Mathematical Physics},
   publisher={Springer Science and Business Media LLC},
   author={Gu, Shouzhen and Tang, Eugene and Caha, Libor and Choe, Shin Ho and He, Zhiyang and Kubica, Aleksander},
   year={2024},
   month=mar }

@misc{leverrier2022quantumtannercodes,
      title={Quantum Tanner codes}, 
      author={Anthony Leverrier and Gilles Zémor},
      year={2022},
      eprint={2202.13641},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2202.13641}, 
}

@inproceedings{Leverrier_2015,
   title={Quantum Expander Codes},
   url={http://dx.doi.org/10.1109/FOCS.2015.55},
   DOI={10.1109/focs.2015.55},
   booktitle={2015 IEEE 56th Annual Symposium on Foundations of Computer Science},
   publisher={IEEE},
   author={Leverrier, Anthony and Tillich, Jean-Pierre and Zemor, Gilles},
   year={2015},
   month=oct, pages={810–824} }

@misc{breuckmann2025logicaloperatorsderivedautomorphisms,
      title={Logical Operators and Derived Automorphisms of Tile Codes}, 
      author={Nikolas P. Breuckmann and Shin Ho Choe and Jens Niklas Eberhardt and Francisco Revson Fernandes Pereira and Vincent Steffan},
      year={2025},
      eprint={2511.14589},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2511.14589}, 
}

@article{steffan2025tile,
  title={Tile codes: High-efficiency quantum codes on a lattice with boundary},
  author={Steffan, Vincent and Choe, Shin Ho and Breuckmann, Nikolas P and Pereira, Francisco Revson Fernandes and Eberhardt, Jens Niklas},
  journal={Physical Review Letters},
  volume={135},
  number={17},
  pages={170601},
  year={2025},
  publisher={APS}
}

@book{harris2013algebraic,
  title={Algebraic geometry: a first course},
  author={Harris, Joe},
  volume={133},
  year={2013},
  publisher={Springer Science \& Business Media}
}

@article{sather2009homological,
  title={Homological algebra notes},
  author={Sather-Wagstaff, Sean},
  journal={Unpublished lecture notes. Available at http://ssather. people. clemson. edu/notes. html},
  year={2009}
}

@article{pecorari2025high,
  title={High-rate quantum LDPC codes for long-range-connected neutral atom registers},
  author={Pecorari, Laura and Jandura, Sven and Brennen, Gavin K and Pupillo, Guido},
  journal={Nature Communications},
  volume={16},
  number={1},
  pages={1111},
  year={2025},
  publisher={Nature Publishing Group UK London}
}

@misc{albuquerque2011hopfalgebraicpropertiescirculant,
      title={Some (Hopf) algebraic properties of circulant matrices}, 
      author={Helena Albuquerque and Florin Panaite},
      year={2011},
      eprint={1110.1546},
      archivePrefix={arXiv},
      primaryClass={math.RA},
      url={https://arxiv.org/abs/1110.1546}, 
}

@book{davis1979circulant,
  title={Circulant Matrices},
  author={Davis, P.J.},
  isbn={9780471057710},
  lccn={79010551},
  series={Monographs and textbooks in pure and applied mathematics},
  url={https://books.google.com/books?id=2wrvAAAAMAAJ},
  year={1979},
  publisher={Wiley}
}

@misc{lin2024transversalnoncliffordgatesquantum,
      title={Transversal non-Clifford gates for quantum LDPC codes on sheaves}, 
      author={Ting-Chun Lin},
      year={2024},
      eprint={2410.14631},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2410.14631}, 
}

@book{MR1867354,
  added-at = {2011-11-11T03:25:59.000+0100},
  address = {Cambridge},
  author = {Hatcher, Allen},
  biburl = {https://www.bibsonomy.org/bibtex/2a001f8991d1935a94831c18d9e85756d/mwpb479},
  file = {:C\:\\Users\\mat\\Documents\\Pdfs\\AT.pdf:PDF},
  groups = {public},
  interhash = {bf666f5059c799f3b1a723fd79c616d6},
  intrahash = {a001f8991d1935a94831c18d9e85756d},
  isbn = {0-521-79160-X; 0-521-79540-0},
  keywords = {},
  mrclass = {55-01 (55-00)},
  mrnumber = {1867354 (2002k:55001)},
  mrreviewer = {Donald W. Kahn},
  pages = {xii+544},
  publisher = {Cambridge University Press},
  timestamp = {2011-11-11T03:25:59.000+0100},
  title = {Algebraic topology},
  username = {mwpb479},
  year = 2002
}

@article{hutchings2011cup,
  title={Cup product and intersections},
  author={Hutchings, Michael},
  journal={Course notes. math. berkeley. edu/„hutching/teach/215b-2011/cup. pdf(Cited on page 51.)},
  year={2011}
}

@misc{lin2025abelianmulticyclecodessingleshot,
      title={Abelian multi-cycle codes for single-shot error correction}, 
      author={Hsiang-Ku Lin and Pak Kau Lim and Alexey A. Kovalev and Leonid P. Pryadko},
      year={2025},
      eprint={2506.16910},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2506.16910}, 
}

@misc{mullerImprovedBeliefPropagation2025,
  title = {Improved Belief Propagation Is Sufficient for Real-Time Decoding of Quantum Memory},
  author = {M{\"u}ller, Tristan and Alexander, Thomas and Beverland, Michael E. and B{\"u}hler, Markus and Johnson, Blake R. and Maurer, Thilo and Vandeth, Drew},
  year = {2025},
  month = jun,
  number = {arXiv:2506.01779},
  eprint = {2506.01779},
  primaryclass = {quant-ph},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2506.01779},
  urldate = {2025-06-06},
  archiveprefix = {arXiv},
  keywords = {Quantum Physics},
}

@misc{zhu2025transversalnoncliffordgatesqldpc,
      title={Transversal non-Clifford gates on qLDPC codes breaking the $\sqrt{N}$ distance barrier and quantum-inspired geometry with $\mathbb{Z}_2$ systolic freedom}, 
      author={Guanyu Zhu},
      year={2025},
      eprint={2507.15056},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2507.15056}, 
}

@misc{hsin2025classifyinglogicalgatesquantum,
      title={Classifying Logical Gates in Quantum Codes via Cohomology Operations and Symmetry}, 
      author={Po-Shen Hsin and Ryohei Kobayashi and Guanyu Zhu},
      year={2025},
      eprint={2411.15848},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2411.15848}, 
}

@article{Hsin_2025,
   title={Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate Beyond the
                    n
                    1/3
                    Distance Barrier},
   volume={6},
   ISSN={2691-3399},
   url={http://dx.doi.org/10.1103/hkgn-hhqx},
   DOI={10.1103/hkgn-hhqx},
   number={4},
   journal={PRX Quantum},
   publisher={American Physical Society (APS)},
   author={Hsin, Po-Shen and Kobayashi, Ryohei and Zhu, Guanyu},
   year={2025},
   month=dec }


@misc{zhu2025noncliffordparallelizablefaulttolerantlogical,
      title={Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries}, 
      author={Guanyu Zhu and Shehryar Sikander and Elia Portnoy and Andrew W. Cross and Benjamin J. Brown},
      year={2025},
      eprint={2310.16982},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2310.16982}, 
}

@misc{zhu2025topologicaltheoryqldpcnonclifford,
      title={A topological theory for qLDPC: non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the $N^{1/3}$ distance barrier}, 
      author={Guanyu Zhu},
      year={2025},
      eprint={2501.19375},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2501.19375}, 
}

@misc{golowich2024quantumldpccodestransversal,
      title={Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes}, 
      author={Louis Golowich and Ting-Chun Lin},
      year={2024},
      eprint={2410.14662},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2410.14662}, 
}

@article{Panteleev_2021,
   title={Degenerate Quantum LDPC Codes With Good Finite Length Performance},
   volume={5},
   ISSN={2521-327X},
   url={http://dx.doi.org/10.22331/q-2021-11-22-585},
   DOI={10.22331/q-2021-11-22-585},
   journal={Quantum},
   publisher={Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
   author={Panteleev, Pavel and Kalachev, Gleb},
   year={2021},
   month=nov, pages={585} }

@article{PhysRevX.5.031043,
  title = {Single-Shot Fault-Tolerant Quantum Error Correction},
  author = {Bomb\'{\i}n, H\'ector},
  journal = {Phys. Rev. X},
  volume = {5},
  issue = {3},
  pages = {031043},
  numpages = {26},
  year = {2015},
  month = {Sep},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevX.5.031043},
  url = {https://link.aps.org/doi/10.1103/PhysRevX.5.031043}
}

@article{Bombín_2015,
doi = {10.1088/1367-2630/17/8/083002},
url = {https://doi.org/10.1088/1367-2630/17/8/083002},
year = {2015},
month = {aug},
publisher = {IOP Publishing},
volume = {17},
number = {8},
pages = {083002},
author = {Bombín, Héctor},
title = {Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes},
journal = {New Journal of Physics},
abstract = {Color codes are topological stabilizer codes with unusual transversality properties. Here I show that their group of transversal gates is optimal and only depends on the spatial dimension, not the local geometry. I also introduce a generalized, subsystem version of color codes. In 3D they allow the transversal implementation of a universal set of gates by gauge fixing, while error-dectecting measurements involve only four or six qubits.}
}

@misc{müller2025improvedbeliefpropagationsufficient,
      title={Improved belief propagation is sufficient for real-time decoding of quantum memory}, 
      author={Tristan Müller and Thomas Alexander and Michael E. Beverland and Markus Bühler and Blake R. Johnson and Thilo Maurer and Drew Vandeth},
      year={2025},
      eprint={2506.01779},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2506.01779}, 
}

@article{Kuo_2020,
   title={Refined Belief Propagation Decoding of Sparse-Graph Quantum Codes},
   volume={1},
   ISSN={2641-8770},
   url={http://dx.doi.org/10.1109/JSAIT.2020.3011758},
   DOI={10.1109/jsait.2020.3011758},
   number={2},
   journal={IEEE Journal on Selected Areas in Information Theory},
   publisher={Institute of Electrical and Electronics Engineers (IEEE)},
   author={Kuo, Kao-Yueh and Lai, Ching-Yi},
   year={2020},
   month=aug, pages={487–498} }

@misc{landahl2011color,
      title={Fault-tolerant quantum computing with color codes}, 
      author={Andrew J. Landahl and Jonas T. Anderson and Patrick R. Rice},
      year={2011},
      eprint={1108.5738},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/1108.5738}, 
}

@article{roffe_decoding_2020,
   title={Decoding across the quantum low-density parity-check code landscape},
   volume={2},
   ISSN={2643-1564},
   url={http://dx.doi.org/10.1103/PhysRevResearch.2.043423},
   DOI={10.1103/physrevresearch.2.043423},
   number={4},
   journal={Physical Review Research},
   publisher={American Physical Society (APS)},
   author={Roffe, Joschka and White, David R. and Burton, Simon and Campbell, Earl},
   year={2020},
   month={Dec}
}

@software{Roffe_LDPC_Python_tools_2022,
author = {Roffe, Joschka},
title = {{LDPC: Python tools for low density parity check codes}},
url = {https://pypi.org/project/ldpc/},
year = {2022}
}

@article{huangfu2018parallelizing,
  title={Parallelizing the dual revised simplex method},
  author={Huangfu, Qi and Hall, JA Julian},
  journal={Mathematical Programming Computation},
  volume={10},
  number={1},
  pages={119--142},
  year={2018},
  publisher={Springer}
}

@article{Liang_2025,
   title={Generalized Toric Codes on Twisted Tori for Quantum Error Correction},
   volume={6},
   ISSN={2691-3399},
   url={http://dx.doi.org/10.1103/rmy6-9n89},
   DOI={10.1103/rmy6-9n89},
   number={2},
   journal={PRX Quantum},
   publisher={American Physical Society (APS)},
   author={Liang, Zijian and Liu, Ke and Song, Hao and Chen, Yu-An},
   year={2025},
   month=jun }

@misc{beni2025tesseractsearchbaseddecoderquantum,
      title={Tesseract: A Search-Based Decoder for Quantum Error Correction}, 
      author={Laleh Aghababaie Beni and Oscar Higgott and Noah Shutty},
      year={2025},
      eprint={2503.10988},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2503.10988}, 
}

@article{gidney2021stim,
  doi = {10.22331/q-2021-07-06-497},
  url = {https://doi.org/10.22331/q-2021-07-06-497},
  title = {Stim: a fast stabilizer circuit simulator},
  author = {Gidney, Craig},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens
                in den Quantenwissenschaften}},
  volume = 5,
  pages = 497,
  month = jul,
  year = 2021
}

@misc{kimlee2025quantumsavorywritesymbolicallyrun,
      title={QuantumSavory: Write Symbolically, Run on Any Backend -- A Unified Simulation Toolkit for Quantum Computing and Networking}, 
      author={Hana KimLee and Leonardo Bacciottini and Abhishek Bhatt and Andrew Kille and Stefan Krastanov},
      year={2025},
      eprint={2512.16752},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2512.16752}, 
}

@misc{bezanson2015juliafreshapproachnumerical,
      title={Julia: A Fresh Approach to Numerical Computing}, 
      author={Jeff Bezanson and Alan Edelman and Stefan Karpinski and Viral B. Shah},
      year={2015},
      eprint={1411.1607},
      archivePrefix={arXiv},
      primaryClass={cs.MS},
      url={https://arxiv.org/abs/1411.1607}, 
}

@book{arndt2010matters,
  title={Matters Computational: ideas, algorithms, source code},
  author={Arndt, J{\"o}rg},
  year={2010},
  publisher={Springer Science \& Business Media}
}

@article{Higgott_2023,
   title={Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes},
   volume={4},
   ISSN={2691-3399},
   url={http://dx.doi.org/10.1103/PRXQuantum.4.020332},
   DOI={10.1103/prxquantum.4.020332},
   number={2},
   journal={PRX Quantum},
   publisher={American Physical Society (APS)},
   author={Higgott, Oscar and Breuckmann, Nikolas P.},
   year={2023},
   month=may }

@misc{stacks-project,
  author       = {The {Stacks project authors}},
  title        = {The Stacks project},
  howpublished = {\url{https://stacks.math.columbia.edu}},
  year         = {2025},
}

@article{Hong_2025,
   title={Single-shot preparation of hypergraph product codes via dimension jump},
   volume={9},
   ISSN={2521-327X},
   url={http://dx.doi.org/10.22331/q-2025-10-07-1879},
   DOI={10.22331/q-2025-10-07-1879},
   journal={Quantum},
   publisher={Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
   author={Hong, Yifan},
   year={2025},
   month=oct, pages={1879} }

@article{Bombin_2013,
doi = {10.1088/1367-2630/15/5/055023},
url = {https://doi.org/10.1088/1367-2630/15/5/055023},
year = {2013},
month = {may},
publisher = {IOP Publishing},
volume = {15},
number = {5},
pages = {055023},
author = {Bombin, H and Chhajlany, R W and Horodecki, M and Martin-Delgado, M A},
title = {Self-correcting quantum computers},
journal = {New Journal of Physics},
abstract = {Is the notion of a quantum computer (QC) resilient to thermal noise unphysical? We address this question from a constructive perspective and show that local quantum Hamiltonian models provide self-correcting QCs. To this end, we first give a sufficient condition on the connectedness of excitations for a stabilizer code model to be a self-correcting quantum memory. We then study the two main examples of topological stabilizer codes in arbitrary dimensions and establish their self-correcting capabilities. Also, we address the transversality properties of topological color codes, showing that six-dimensional color codes provide a self-correcting model that allows the transversal and local implementation of a universal set of operations in seven spatial dimensions. Finally, we give a procedure for initializing such quantum memories at finite temperature.}
}


@software{stefan_krastanov_2025_17932783,
  author       = {Stefan Krastanov and
                  Feroz Ahmed Mian and
                  Zhongyi Ni and
                  Kanyang Ying and
                  Praveen Viswanathan and
                  Shayan Pardis and
                  Sagnik Pal and
                  brohan-byte and
                  John Lapeyre and
                  H.A. and
                  Anthony Micciche and
                  Yuxuan Yan and
                  Stefan Krastanov's bot and
                  Tommy Hofmann and
                  Quinten Preiß and
                  gsommers and
                  Lars Göttgens and
                  IsaacP1234 and
                  Abhishek Bhatt and
                  Adriano Meligrana and
                  Benzillaist and
                  Chen Zhao and
                  ShuGe-MIT and
                  Tim Holy and
                  Trung Dang and
                  adrianariton and
                  ismoldayev and
                  riteshbhirud},
  title        = {QuantumSavory/QuantumClifford.jl: v0.10.1},
  month        = dec,
  year         = 2025,
  publisher    = {Zenodo},
  version      = {v0.10.1},
  doi          = {10.5281/zenodo.17932783},
  url          = {https://doi.org/10.5281/zenodo.17932783},
}

@article{Horsman_2012,
   title={Surface code quantum computing by lattice surgery},
   volume={14},
   ISSN={1367-2630},
   url={http://dx.doi.org/10.1088/1367-2630/14/12/123011},
   DOI={10.1088/1367-2630/14/12/123011},
   number={12},
   journal={New Journal of Physics},
   publisher={IOP Publishing},
   author={Horsman, Dominic and Fowler, Austin G and Devitt, Simon and Meter, Rodney Van},
   year={2012},
   month=dec, pages={123011} }


@article{Litinski_2019,
   title={A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery},
   volume={3},
   ISSN={2521-327X},
   url={http://dx.doi.org/10.22331/q-2019-03-05-128},
   DOI={10.22331/q-2019-03-05-128},
   journal={Quantum},
   publisher={Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
   author={Litinski, Daniel},
   year={2019},
   month=mar, pages={128} }


@misc{menon2025magictricyclesefficientmagic,
      title={Magic tricycles: Efficient magic state generation with finite block-length quantum LDPC codes}, 
      author={Varun Menon and J. Pablo Bonilla-Ataides and Rohan Mehta and Andi Gu and Daniel Bochen Tan and Mikhail D. Lukin},
      year={2025},
      eprint={2508.10714},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2508.10714}, 
}

@INPROCEEDINGS{11249819,
  author={Campbell, Devon},
  booktitle={2025 IEEE International Conference on Quantum Computing and Engineering (QCE)}, 
  title={Single-Shot Decoding of Biased-Tailored Quantum LDPC Codes}, 
  year={2025},
  volume={02},
  number={},
  pages={284-287},
  keywords={Quantum computing;Program processors;Noise;Measurement uncertainty;Parity check codes;Reproducibility of results;Loss measurement;Decoding;Noise measurement;Resilience;fault-tolerant design;hybrid classical-quantum systems;quantum error correction;biased noise;single-shot decoding;belief propagation;ordered statistics decoding},
  doi={10.1109/QCE65121.2025.10336}}


@article{gottesman2013fault,
  title={Fault-tolerant quantum computation with constant overhead},
  author={Gottesman, Daniel},
  journal={arXiv preprint arXiv:1310.2984},
  year={2013}
}

@misc{dinur2021locallytestablecodesconstant,
      title={Locally Testable Codes with constant rate, distance, and locality}, 
      author={Irit Dinur and Shai Evra and Ron Livne and Alexander Lubotzky and Shahar Mozes},
      year={2021},
      eprint={2111.04808},
      archivePrefix={arXiv},
      primaryClass={cs.IT},
      url={https://arxiv.org/abs/2111.04808}, 
}

@article{Leverrier_2022,
   title={Towards local testability for quantum coding},
   volume={6},
   ISSN={2521-327X},
   url={http://dx.doi.org/10.22331/q-2022-02-24-661},
   DOI={10.22331/q-2022-02-24-661},
   journal={Quantum},
   publisher={Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
   author={Leverrier, Anthony and Londe, Vivien and Zémor, Gilles},
   year={2022},
   month=feb, pages={661} }


@misc{leverrier2022efficientdecodingconstantfraction,
      title={Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes}, 
      author={Anthony Leverrier and Gilles Zémor},
      year={2022},
      eprint={2206.07571},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2206.07571}, 
}

@misc{dinur2022goodquantumldpccodes,
      title={Good Quantum LDPC Codes with Linear Time Decoders}, 
      author={Irit Dinur and Min-Hsiu Hsieh and Ting-Chun Lin and Thomas Vidick},
      year={2022},
      eprint={2206.07750},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2206.07750}, 
}

@misc{gu2022efficientdecoderlineardistance,
      title={An efficient decoder for a linear distance quantum LDPC code}, 
      author={Shouzhen Gu and Christopher A. Pattison and Eugene Tang},
      year={2022},
      eprint={2206.06557},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2206.06557}, 
}

@article{Tan_2025,
   title={Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes},
   volume={9},
   ISSN={2521-327X},
   url={http://dx.doi.org/10.22331/q-2025-10-28-1897},
   DOI={10.22331/q-2025-10-28-1897},
   journal={Quantum},
   publisher={Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
   author={Tan, Shi Jie Samuel and Stambler, Lev},
   year={2025},
   month=oct, pages={1897} }


@article{PRXQuantum.5.040302,
  title = {Weight-Reduced Stabilizer Codes with Lower Overhead},
  author = {Sabo, Eric and Gunderman, Lane G. and Ide, Benjamin and Vasmer, Michael and Dauphinais, Guillaume},
  journal = {PRX Quantum},
  volume = {5},
  issue = {4},
  pages = {040302},
  numpages = {43},
  year = {2024},
  month = {Oct},
  publisher = {American Physical Society},
  doi = {10.1103/PRXQuantum.5.040302},
  url = {https://link.aps.org/doi/10.1103/PRXQuantum.5.040302}
}

@article{hastings2016weight,
  title={Weight reduction for quantum codes},
  author={Hastings, Matthew B},
  journal={arXiv preprint arXiv:1611.03790},
  year={2016}
}
@article{hastings2021quantum,
  title={On quantum weight reduction},
  author={Hastings, Matthew B},
  journal={arXiv preprint arXiv:2102.10030},
  year={2021}
}

@article{Gidney_2023,
   title={A Pair Measurement Surface Code on Pentagons},
   volume={7},
   ISSN={2521-327X},
   url={http://dx.doi.org/10.22331/q-2023-10-25-1156},
   DOI={10.22331/q-2023-10-25-1156},
   journal={Quantum},
   publisher={Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
   author={Gidney, Craig},
   year={2023},
   month=oct, pages={1156} }

@article{hurley2006group,
  title={Group rings and rings of matrices},
  author={Hurley, Ted},
  journal={Int. J. Pure Appl. Math},
  volume={31},
  number={3},
  pages={319--335},
  year={2006}
}

@article{Derks_2025,
   title={Designing fault-tolerant circuits using detector error models},
   volume={9},
   ISSN={2521-327X},
   url={http://dx.doi.org/10.22331/q-2025-11-06-1905},
   DOI={10.22331/q-2025-11-06-1905},
   journal={Quantum},
   publisher={Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
   author={Derks, Peter-Jan H.S. and Townsend-Teague, Alex and Burchards, Ansgar G. and Eisert, Jens},
   year={2025},
   month=nov, pages={1905} }

@article{Brown_2016,
   title={Fault-tolerant error correction with the gauge color code},
   volume={7},
   ISSN={2041-1723},
   url={http://dx.doi.org/10.1038/ncomms12302},
   DOI={10.1038/ncomms12302},
   number={1},
   journal={Nature Communications},
   publisher={Springer Science and Business Media LLC},
   author={Brown, Benjamin J. and Nickerson, Naomi H. and Browne, Dan E.},
   year={2016},
   month=jul }

@article{breuckmann2016local,
  title={Local decoders for the 2D and 4D toric code},
  author={Breuckmann, Nikolas P and Duivenvoorden, Kasper and Michels, Dominik and Terhal, Barbara M},
  journal={arXiv preprint arXiv:1609.00510},
  year={2016}
}


@misc{li2025poincaredualitymultiplicativestructures,
      title={Poincar\'e Duality and Multiplicative Structures on Quantum Codes}, 
      author={Yiming Li and Zimu Li and Zi-Wen Liu and Quynh T. Nguyen},
      year={2025},
      eprint={2512.21922},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2512.21922}, 
}

@article{barkeshli2023codimension,
  title={Codimension-2 defects and higher symmetries in (3+ 1) D topological phases},
  author={Barkeshli, Maissam and Chen, Yu-An and Huang, Sheng-Jie and Kobayashi, Ryohei and Tantivasadakarn, Nathanan and Zhu, Guanyu},
  journal={SciPost Physics},
  volume={14},
  number={4},
  pages={065},
  year={2023}
}

@misc{jacobson2009basic,
  title={Basic Algebra II, 2a Edi{\c{c}}ao},
  author={Jacobson, N},
  year={2009},
  publisher={Dover, New York}
}

@book{darling1994differential,
  title={Differential forms and connections},
  author={Darling, Richard WR},
  year={1994},
  publisher={Cambridge University Press}
}

@book{abraham2012manifolds,
  title={Manifolds, tensor analysis, and applications},
  author={Abraham, Ralph and Marsden, Jerrold E and Ratiu, Tudor},
  volume={75},
  year={2012},
  publisher={Springer Science \& Business Media}
}

@misc{leverrier2025smallquantumtannercodes,
      title={Small quantum Tanner codes from left--right Cayley complexes}, 
      author={Anthony Leverrier and Wouter Rozendaal and Gilles Zémor},
      year={2025},
      eprint={2512.20532},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2512.20532}, 
}

@misc{radebold2025explicitinstancesquantumtanner,
      title={Explicit Instances of Quantum Tanner Codes}, 
      author={Rebecca Katharina Radebold and Stephen D. Bartlett and Andrew C. Doherty},
      year={2025},
      eprint={2508.05095},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2508.05095}, 
}

@misc{guémard2025moderatelengthliftedquantumtanner,
      title={Moderate-length lifted quantum Tanner codes}, 
      author={Virgile Guémard and Gilles Zémor},
      year={2025},
      eprint={2502.20297},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2502.20297}, 
}

@misc{evra2020decodablequantumldpccodes,
      title={Decodable quantum LDPC codes beyond the $\sqrt{n}$ distance barrier using high dimensional expanders}, 
      author={Shai Evra and Tali Kaufman and Gilles Zémor},
      year={2020},
      eprint={2004.07935},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2004.07935}, 
}

@article{Tillich_2014,
   title={Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength},
   volume={60},
   ISSN={1557-9654},
   url={http://dx.doi.org/10.1109/TIT.2013.2292061},
   DOI={10.1109/tit.2013.2292061},
   number={2},
   journal={IEEE Transactions on Information Theory},
   publisher={Institute of Electrical and Electronics Engineers (IEEE)},
   author={Tillich, Jean-Pierre and Zemor, Gilles},
   year={2014},
   month=feb, pages={1193–1202} }

@misc{postema2025existencecharacterisationbivariatebicycle,
      title={Existence and Characterisation of Bivariate Bicycle Codes}, 
      author={Jasper Johannes Postema and Servaas J. J. M. F. Kokkelmans},
      year={2025},
      eprint={2502.17052},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2502.17052}, 
}

@misc{wang2025coprimebivariatebicyclecodes,
      title={Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms}, 
      author={Ming Wang and Frank Mueller},
      year={2025},
      eprint={2408.10001},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2408.10001}, 
}

@article{symons2025sequences,
  title={Sequences of Bivariate Bicycle Codes from Covering Graphs},
  author={Symons, Benjamin CB and Rajput, Abhishek and Browne, Dan E},
  journal={arXiv preprint arXiv:2511.13560},
  year={2025}
}

@misc{liang2025selfdualbivariatebicyclecodes,
      title={Self-dual bivariate bicycle codes with transversal Clifford gates}, 
      author={Zijian Liang and Yu-An Chen},
      year={2025},
      eprint={2510.05211},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2510.05211}, 
}

@misc{cross2025improvedqldpcsurgerylogical,
      title={Improved QLDPC Surgery: Logical Measurements and Bridging Codes}, 
      author={Andrew W. Cross and Zhiyang He and Patrick J. Rall and Theodore J. Yoder},
      year={2025},
      eprint={2407.18393},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2407.18393}, 
}

@misc{williamson2024lowoverheadfaulttolerantquantumcomputation,
      title={Low-overhead fault-tolerant quantum computation by gauging logical operators}, 
      author={Dominic J. Williamson and Theodore J. Yoder},
      year={2024},
      eprint={2410.02213},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2410.02213}, 
}

@article{eastin2009restrictions,
  title={Restrictions on transversal encoded quantum gate sets},
  author={Eastin, Bryan and Knill, Emanuel},
  journal={Physical review letters},
  volume={102},
  number={11},
  pages={110502},
  year={2009},
  publisher={APS}
}

@misc{mostad2024generalizingquantumtannercodes,
      title={Generalizing Quantum Tanner Codes}, 
      author={Olai Å. Mostad and Eirik Rosnes and Hsuan-Yin Lin},
      year={2024},
      eprint={2405.07980},
      archivePrefix={arXiv},
      primaryClass={cs.IT},
      url={https://arxiv.org/abs/2405.07980}, 
}


@article{huang2023improved,
  title={Improved noisy syndrome decoding of quantum ldpc codes with sliding window},
  author={Huang, Shilin and Puri, Shruti},
  journal={arXiv preprint arXiv:2311.03307},
  year={2023}
}

@article{lin2024single,
  title={Single-shot error correction on toric codes with high-weight stabilizers},
  author={Lin, Yingjia and Huang, Shilin and Brown, Kenneth R},
  journal={Physical Review A},
  volume={109},
  number={5},
  pages={052438},
  year={2024},
  publisher={APS}
}


@misc{lin2025dynamiclocalsingleshotchecks,
      title={Dynamic local single-shot checks for toric codes}, 
      author={Yingjia Lin and Abhinav Anand and Kenneth R. Brown},
      year={2025},
      eprint={2511.20576},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2511.20576}, 
}

@misc{guemard2024liftsquantumcsscodes,
      title={Lifts of quantum CSS codes}, 
      author={Virgile Guemard},
      year={2024},
      eprint={2404.16736},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2404.16736}, 
}

@inproceedings{Hastings_2021, series={STOC ’21},
   title={Fiber bundle codes: breaking the
            n
            1/2
            polylog(
            n
            ) barrier for Quantum LDPC codes},
   url={http://dx.doi.org/10.1145/3406325.3451005},
   DOI={10.1145/3406325.3451005},
   booktitle={Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing},
   publisher={ACM},
   author={Hastings, Matthew B. and Haah, Jeongwan and O’Donnell, Ryan},
   year={2021},
   month=jun, pages={1276–1288},
   collection={STOC ’21} }

@misc{gulshen2025quantumtannercolorcodes,
      title={Quantum Tanner Color Codes on Qubits with Transversal Gates}, 
      author={Kyle Gulshen and Tali Kaufman},
      year={2025},
      eprint={2510.07864},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2510.07864}, 
}

@article{haah2011local,
  title={Local stabilizer codes in three dimensions without string logical operators},
  author={Haah, Jeongwan},
  journal={Physical Review A—Atomic, Molecular, and Optical Physics},
  volume={83},
  number={4},
  pages={042330},
  year={2011},
  publisher={APS}
}


@book{dummit2004abstract,
  title={Abstract algebra},
  author={Dummit, David Steven and Foote, Richard M and others},
  volume={3},
  year={2004},
  publisher={Wiley Hoboken}
}

@misc{salescabrera2025codesalgebrasdirectproducts,
      title={Codes in algebras of direct products of groups}, 
      author={Miguel Sales-Cabrera and Xaro Soler-Escrivà and Víctor Sotomayor},
      year={2025},
      eprint={2412.09695},
      archivePrefix={arXiv},
      primaryClass={cs.IT},
      url={https://arxiv.org/abs/2412.09695}, 
}

@misc{salescabrera2025dualitiesdihedralgeneralisedquaternion,
      title={Dualities of dihedral and generalised quaternion codes and applications to quantum codes}, 
      author={Miguel Sales-Cabrera and Xaro Soler-Escrivà and Víctor Sotomayor},
      year={2025},
      eprint={2512.07354},
      archivePrefix={arXiv},
      primaryClass={cs.IT},
      url={https://arxiv.org/abs/2512.07354}, 
}

@misc{willenborg2025dihedralquantumcodes,
      title={Dihedral Quantum Codes}, 
      author={Nadja Willenborg and Martino Borello and Anna-Lena Horlemann and Habibul Islam},
      year={2025},
      eprint={2310.15092},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2310.15092}, 
}

@article{Vasmer_2021,
   title={Cellular automaton decoders for topological quantum codes with noisy measurements and beyond},
   volume={11},
   ISSN={2045-2322},
   url={http://dx.doi.org/10.1038/s41598-021-81138-2},
   DOI={10.1038/s41598-021-81138-2},
   number={1},
   journal={Scientific Reports},
   publisher={Springer Science and Business Media LLC},
   author={Vasmer, Michael and Browne, Dan E. and Kubica, Aleksander},
   year={2021},
   month=jan }

@article{Herold_2017,
doi = {10.1088/1367-2630/aa7099},
url = {https://doi.org/10.1088/1367-2630/aa7099},
year = {2017},
month = {jun},
publisher = {IOP Publishing},
volume = {19},
number = {6},
pages = {063012},
author = {Herold, Michael and Kastoryano, Michael J and Campbell, Earl T and Eisert, Jens},
title = {Cellular automaton decoders of topological quantum memories in the fault tolerant setting},
journal = {New Journal of Physics},
abstract = {Active error decoding and correction of topological quantum codes—in particular the toric code—remains one of the most viable routes to large scale quantum information processing. In contrast, passive error correction relies on the natural physical dynamics of a system to protect encoded quantum information. However, the search is ongoing for a completely satisfactory passive scheme applicable to locally interacting two-dimensional systems. Here, we investigate dynamical decoders that provide passive error correction by embedding the decoding process into local dynamics. We propose a specific discrete time cellular-automaton decoder in the fault tolerant setting and provide numerical evidence showing that the logical qubit has a survival time extended by several orders of magnitude over that of a bare unencoded qubit. We stress that (asynchronous) dynamical decoding gives rise to a Markovian dissipative process. We hence equate cellular-automaton decoding to a fully dissipative topological quantum memory, which removes errors continuously. In this sense, uncontrolled and unwanted local noise can be corrected for by a controlled local dissipative process. We analyze the required resources, commenting on additional polylogarithmic factors beyond those incurred by an ideal constant resource dynamical decoder.}
}

@article{brown2016fault,
  title={Fault-tolerant error correction with the gauge color code},
  author={Brown, Benjamin J and Nickerson, Naomi H and Browne, Dan E},
  journal={Nature communications},
  volume={7},
  number={1},
  pages={12302},
  year={2016},
  publisher={Nature Publishing Group UK London}
}

@ARTICLE{9585444,
  author={Breuckmann, Nikolas P. and Londe, Vivien},
  journal={IEEE Transactions on Information Theory}, 
  title={Single-Shot Decoding of Linear Rate LDPC Quantum Codes With High Performance}, 
  year={2022},
  volume={68},
  number={1},
  pages={272-286},
  keywords={Codes;Manifolds;Decoding;Encoding;Parity check codes;Qubit;Measurement uncertainty;Quantum codes;quantum error-correction;single-shot decoding;hyperbolic;quantum fault-tolerance;Coxeter groups;cellular automata;belief-propagation},
  doi={10.1109/TIT.2021.3122352}}

@book{gallian2021contemporary,
  title={Contemporary abstract algebra},
  author={Gallian, Joseph},
  year={2021},
  publisher={Chapman and Hall/CRC}
}

@misc{zhu2026nonabelianqldpctqftformalism,
      title={Non-Abelian qLDPC: TQFT Formalism, Addressable Gauging Measurement and Application to Magic State Fountain on 2D Product Codes}, 
      author={Guanyu Zhu and Ryohei Kobayashi and Po-Shen Hsin},
      year={2026},
      eprint={2601.06736},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2601.06736}, 
}

@INPROCEEDINGS{10756140,
  author={Dinur, Irit and Lin, Ting-Chun and Vidick, Thomas},
  booktitle={2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)}, 
  title={Expansion of High-Dimensional Cubical Complexes: with Application to Quantum Locally Testable Codes}, 
  year={2024},
  volume={},
  number={},
  pages={379-385},
  keywords={Computer science;Parity check codes;high-dimensional expansion;quantum locally testable codes},
  doi={10.1109/FOCS61266.2024.00031}}

@misc{Pryadko-2025-distm4ri,
  author = {Pryadko, L. P. and Zeng, W.},
  title = {dist-m4ri--- distance of a classical or quantum {CSS} code},
  howpublished = {\url{https://github.com/QEC-pages/dist-m4ri}},
  year = {2024},
  note = {Accessed: 2026}
}

@misc{sriram2025diffusioncodesselfcorrectionsmallerset,
      title={Diffusion Codes: Self-Correction from Small(er)-Set Expansion with Tunable Non-locality}, 
      author={Adithya Sriram and Vedika Khemani and Benedikt Placke},
      year={2025},
      eprint={2510.07179},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2510.07179}, 
}

@article{ostrev2024classical,
  title={Classical product code constructions for quantum Calderbank-Shor-Steane codes},
  author={Ostrev, Dimiter and Orsucci, Davide and L{\'a}zaro, Francisco and Matuz, Balazs},
  journal={Quantum},
  volume={8},
  pages={1420},
  year={2024},
  publisher={Verein zur F{\"o}rderung des Open Access Publizierens in den Quantenwissenschaften}
}

@article{PRXQuantum.5.040328,
  title = {Towards Early Fault Tolerance on a $2\ifmmode\times\else\texttimes\fi{}N$ Array of Qubits Equipped with Shuttling},
  author = {Siegel, Adam and Strikis, Armands and Fogarty, Michael},
  journal = {PRX Quantum},
  volume = {5},
  issue = {4},
  pages = {040328},
  numpages = {22},
  year = {2024},
  month = {Nov},
  publisher = {American Physical Society},
  doi = {10.1103/PRXQuantum.5.040328},
  url = {https://link.aps.org/doi/10.1103/PRXQuantum.5.040328}
}

@misc{tan2025singleshotuniversalityquantumldpc,
      title={Single-Shot Universality in Quantum LDPC Codes via Code-Switching}, 
      author={Shi Jie Samuel Tan and Yifan Hong and Ting-Chun Lin and Michael J. Gullans and Min-Hsiu Hsieh},
      year={2025},
      eprint={2510.08552},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2510.08552}, 
}

@misc{wang2026checkweightconstrainedquantumcodesbounds,
      title={Check-weight-constrained quantum codes: Bounds and examples}, 
      author={Lily Wang and Andy Zeyi Liu and Ray Li and Aleksander Kubica and Shouzhen Gu},
      year={2026},
      eprint={2601.15446},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2601.15446}, 
}

@software{feroz_ahmed_mian_2026_18396531,
  author       = {Feroz Ahmed Mian and
                  Gwilliam, Owen and
                  Krastanov, Stefan},
  title        = {Fe-r-oz/Multivariate-Multicycle-Codes-For-
                   Complete-Single-Shot-Decoding: Multivariate
                   Multicycle Codes for Complete Single-Shot Decoding
                  },
  month        = jan,
  year         = 2026,
  publisher    = {Zenodo},
  version      = {v.1.0.0},
  doi          = {10.5281/zenodo.18396531},
  url          = {https://doi.org/10.5281/zenodo.18396531},
  swhid        = {swh:1:dir:b3bc67b9f90b40553d4d4a505c042157c2d87462
                   ;origin=https://doi.org/10.5281/zenodo.18396530;vi
                   sit=swh:1:snp:637a19ff3fa76dce0e995385e95b0e26976a
                   9035;anchor=swh:1:rel:1a5a7ba05e98e7442beff33f881d
                   69607d9a518a;path=Fe-r-oz-Multivariate-Multicycle-
                   Codes-For-Complete-Single-Shot-Decoding-6e453d9
                  },
}

@phdthesis{guemard2025lifts,
  title={Lifts of Quantum CSS Codes-New Constructions Beyond Product Codes},
  author={Gu{\'e}mard, Virgile},
  year={2025},
  school={Aix-Marseile Universit{\'e}}
}

@article{liang2025planar,
  title={Planar quantum low-density parity-check codes with open boundaries},
  author={Liang, Zijian and Eberhardt, Jens Niklas and Chen, Yu-An},
  journal={PRX Quantum},
  volume={6},
  number={4},
  pages={040330},
  year={2025},
  publisher={APS}
}

@article{Guo_2026,
   title={Toward Self-Correcting Quantum Codes for Neutral Atom Arrays},
   volume={7},
   ISSN={2691-3399},
   url={http://dx.doi.org/10.1103/mfmt-fwkg},
   DOI={10.1103/mfmt-fwkg},
   number={1},
   journal={PRX Quantum},
   publisher={American Physical Society (APS)},
   author={Guo, Jinkang and Hong, Yifan and Kaufman, Adam and Lucas, Andrew},
   year={2026},
   month=jan }


@misc{li2026quantumbootstrapproductcodes,
      title={Quantum bootstrap product codes}, 
      author={Meng-Yuan Li},
      year={2026},
      eprint={2601.22363},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2601.22363}, 
}

@article{breuckmann2024cups,
  title={Cups and Gates I: Cohomology invariants and logical quantum operations},
  author={Breuckmann, Nikolas P and Davydova, Margarita and Eberhardt, Jens N and Tantivasadakarn, Nathanan},
  journal={arXiv preprint arXiv:2410.16250},
  year={2024}
}

@article{guemard2025moderate,
  title={Moderate-length lifted quantum Tanner codes},
  author={Guemard, Virgile and Z{\'e}mor, Gilles},
  journal={arXiv preprint arXiv:2502.20297},
  year={2025}
}

@article{radebold2025explicit,
  title={Explicit Instances of Quantum Tanner Codes},
  author={Radebold, Rebecca Katharina and Bartlett, Stephen D and Doherty, Andrew C},
  journal={arXiv preprint arXiv:2508.05095},
  year={2025}
}

@article{wang2026check,
  title={Check-weight-constrained quantum codes: Bounds and examples},
  author={Wang, Lily and Liu, Andy Zeyi and Li, Ray and Kubica, Aleksander and Gu, Shouzhen},
  journal={arXiv preprint arXiv:2601.15446},
  year={2026}
}

@article{eberhardt2024pruning,
  title={Pruning qLDPC codes: Towards bivariate bicycle codes with open boundary conditions},
  author={Eberhardt, Jens Niklas and Pereira, Francisco Revson F and Steffan, Vincent},
  journal={arXiv preprint arXiv:2412.04181},
  year={2024}
}

@article{breuckmann2025logical,
  title={Logical Operators and Derived Automorphisms of Tile Codes},
  author={Breuckmann, Nikolas P and Choe, Shin Ho and Eberhardt, Jens Niklas and Pereira, Francisco Revson Fernandes and Steffan, Vincent},
  journal={arXiv preprint arXiv:2511.14589},
  year={2025}
}

@article{gulshen2025quantum,
  title={Quantum Tanner Color Codes on Qubits with Transversal Gates},
  author={Gulshen, Kyle and Kaufman, Tali},
  journal={arXiv preprint arXiv:2510.07864},
  year={2025}
}

@article{li2025poincar,
  title={Poincar$\backslash$'e Duality and Multiplicative Structures on Quantum Codes},
  author={Li, Yiming and Li, Zimu and Liu, Zi-Wen and Nguyen, Quynh T},
  journal={arXiv preprint arXiv:2512.21922},
  year={2025}
}

@article{aydin2025cyclic,
  title={Cyclic Hypergraph Product Code},
  author={Aydin, Arda and Delfosse, Nicolas and Tham, Edwin},
  journal={arXiv preprint arXiv:2511.09683},
  year={2025}
}

% new single shot decoding works that provide some additional insights
@article{b9r5-xhcp,
  title = {Improved energy barrier in higher-dimensional hypergraph product codes},
  author = {Zhao, Guangqi},
  journal = {Phys. Rev. A},
  volume = {113},
  issue = {1},
  pages = {012440},
  numpages = {21},
  year = {2026},
  month = {Jan},
  publisher = {American Physical Society},
  doi = {10.1103/b9r5-xhcp},
  url = {https://link.aps.org/doi/10.1103/b9r5-xhcp}
}


@misc{scruby2024highthresholdlowoverheadsingleshotdecodable,
      title={High-threshold, low-overhead and single-shot decodable fault-tolerant quantum memory}, 
      author={Thomas R. Scruby and Timo Hillmann and Joschka Roffe},
      year={2024},
      eprint={2406.14445},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2406.14445}, 
}

@misc{bastioni2026climbingcliffordhierarchy,
      title={Climbing the Clifford Hierarchy}, 
      author={Luca Bastioni and Samuel Glandon and Tefjol Pllaha and Madison Stewart and Phillip Waitkevich},
      year={2026},
      eprint={2603.12088},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2603.12088}, 
}

@article{tremblay2022constant,
  title={Constant-overhead quantum error correction with thin planar connectivity},
  author={Tremblay, Maxime A and Delfosse, Nicolas and Beverland, Michael E},
  journal={Physical Review Letters},
  volume={129},
  number={5},
  pages={050504},
  year={2022},
  publisher={APS}
}

@article{scruby2026high,
  title={High-threshold, low-overhead and single-shot decodable fault-tolerant quantum memory},
  author={Scruby, Thomas R and Hillmann, Timo and Roffe, Joschka},
  journal={PRX Quantum},
  volume={7},
  number={2},
  pages={020310},
  year={2026},
  publisher={APS}
}

@article{optimization2020gurobi,
  title={Gurobi optimizer reference manual, 2020},
  author={Optimization, LLC Gurobi and others},
  journal={URL http://www. gurobi. com},
  volume={12},
  year={2020}
}

@article{strikis2026high,
  title={High-performance syndrome extraction circuits for quantum codes},
  author={Strikis, Armands and Browne, Dan E and Beverland, Michael E},
  journal={arXiv preprint arXiv:2603.05481},
  year={2026}
}

@inproceedings{shor1996fault,
  title={Fault-tolerant quantum computation},
  author={Shor, Peter W},
  booktitle={Proceedings of 37th conference on foundations of computer science},
  pages={56--65},
  year={1996},
  organization={IEEE}
}

@article{chao2018quantum,
  title = {Quantum Error Correction with Only Two Extra Qubits},
  author = {Chao, Rui and Reichardt, Ben W.},
  journal = {Phys. Rev. Lett.},
  volume = {121},
  issue = {5},
  pages = {050502},
  numpages = {5},
  year = {2018},
  month = {Aug},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevLett.121.050502},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.121.050502}
}

@article{chamberland2018flag,
  doi = {10.22331/q-2018-02-08-53},
  url = {https://doi.org/10.22331/q-2018-02-08-53},
  title = {Flag fault-tolerant error correction with arbitrary distance codes},
  author = {Chamberland, Christopher and Beverland, Michael E.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {53},
  month = feb,
  year = {2018}
}


@article{lingling2020faulttolerant,
  title = {Fault-tolerant quantum error correction on near-term quantum processors using flag and bridge qubits},
  author = {Lao, Lingling and Almudever, Carmen G.},
  journal = {Phys. Rev. A},
  volume = {101},
  issue = {3},
  pages = {032333},
  numpages = {11},
  year = {2020},
  month = {Mar},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevA.101.032333},
  url = {https://link.aps.org/doi/10.1103/PhysRevA.101.032333}
}

@article{xu2024constant,
  title={Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays},
  author={Xu, Qian and Bonilla Ataides, J Pablo and Pattison, Christopher A and Raveendran, Nithin and Bluvstein, Dolev and Wurtz, Jonathan and Vasi{\'c}, Bane and Lukin, Mikhail D and Jiang, Liang and Zhou, Hengyun},
  journal={Nature Physics},
  volume={20},
  number={7},
  pages={1084--1090},
  year={2024},
  publisher={Nature Publishing Group UK London}
}

@article{terhal2015quantum,
  title={Quantum error correction for quantum memories},
  author={Terhal, Barbara M},
  journal={Reviews of Modern Physics},
  volume={87},
  number={2},
  pages={307--346},
  year={2015},
  publisher={APS}
}

@article{skoric2023parallel,
  title={Parallel window decoding enables scalable fault tolerant quantum computation},
  author={Skoric, Luka and Browne, Dan E and Barnes, Kenton M and Gillespie, Neil I and Campbell, Earl T},
  journal={Nature Communications},
  volume={14},
  number={1},
  pages={7040},
  year={2023},
  publisher={Nature Publishing Group UK London}
}