Decoding algorithms for surface codes
Open accessAbstract
Quantum technologies have the potential to solve certain computationally hard problems with polynomial or super-polynomial speedups when compared to classical methods. Unfortunately, the unstable nature of quantum information makes it prone to errors. For this reason, quantum error correction is an invaluable tool to make quantum information reliable and enable the ultimate goal of fault-tolerant quantum computing. Surface codes currently stand as the most promising candidates to build near term error corrected qubits given their two-dimensional architecture, the requirement of only local operations, and high tolerance to quantum noise. Decoding algorithms are an integral component of any error correction scheme, as they are tasked with producing accurate estimates of the errors that affect quantum information, so that they can subsequently be corrected. A critical aspect of decoding algorithms is their speed, since the quantum state will suffer additional errors with the passage of time. This poses a connundrum, where decoding performance is improved at the expense of complexity and viceversa. In this review, a thorough discussion of state-of-the-art decoding algorithms for surface codes is provided. The target audience of this work are both readers with an introductory understanding of the field as well as those seeking to further their knowledge of the decoding paradigm of surface codes. We describe the core principles of these decoding methods as well as existing variants that show promise for improved results. In addition, both the decoding performance, in terms of error correction capability, and decoding complexity, are compared. A review of the existing software tools regarding surface codes decoding is also provided.
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Quantum codes may be degenerate, meaning distinct physical errors can have the same action on the codespace, up to an irrelevant global phase.
“They may be degenerate, meaning that distinct physical errors can have the same action on the codespace, up to an irrelevant global phase [173].”
From Quantum error correction and fault tolerance: A comprehensive tutorial (2026) · cites this paper as [173]
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Quantum Computing in the NISQ era and beyond
Preskill, J. (2018) · Quantum · doi:10.22331/q-2018-08-06-79 · Cited by 8,830 · see citing works
Algorithms for quantum computation: discrete logarithms and factoring
Shor, P. W. (2002) · Proceedings 35th Annual Symposium on Foundations of Computer Science · doi:10.1109/sfcs.1994.365700 · Cited by 8,713 · see citing works
Quantum Computation and Quantum Information: 10th Anniversary Edition
Nielsen, M. A., & Chuang, I. L. (2012) · doi:10.1017/cbo9780511976667 · Cited by 8,436 · see citing works
Quantum cryptography: Public key distribution and coin tossing
Bennett, C. H., & Brassard, G. (2014) · Theoretical Computer Science · doi:10.1016/j.tcs.2014.05.025 · Cited by 8,299 · see citing works
Quantum supremacy using a programmable superconducting processor
Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J. C., Barends, R., Biswas, R., Boixo, S., Brandao, F. G. S. L., Buell, D. A., et al. (2019) · Nature · doi:10.1038/s41586-019-1666-5 · Cited by 7,720 · see citing works
On the generators of quantum dynamical semigroups
Lindblad, G. (1976) · Communications in Mathematical Physics · doi:10.1007/bf01608499 · Cited by 7,548 · cited 1× · see citing works
“…ways in which this set of physical interactions can be mathematically described such as the GoriniKossakowski-Lindblad-Sudarshan (GKLS) master equation (Gorini et al., 1976; Lindblad, 1976; Turchette et al., 2000) or the quantum channel formulation (Choi, 1975; Etxezarreta Martinez et al., 2020).”
BackgroundFactor graphs and the sum-product algorithm
Kschischang, F. R., Frey, B. J., & Loeliger, H.-A. (2001) · IEEE Transactions on Information Theory · doi:10.1109/18.910572 · Cited by 6,486 · see citing works
Completely positive dynamical semigroups of <i>N</i> -level systems
Gorini, V., Kossakowski, A., & Sudarshan, E. C. G. (1976) · Journal of Mathematical Physics · doi:10.1063/1.522979 · Cited by 3,998 · cited 1× · see citing works
“…ways in which this set of physical interactions can be mathematically described such as the GoriniKossakowski-Lindblad-Sudarshan (GKLS) master equation (Gorini et al., 1976; Lindblad, 1976; Turchette et al., 2000) or the quantum channel formulation (Choi, 1975; Etxezarreta Martinez et al., 2020).”
BackgroundSurface codes: Towards practical large-scale quantum computation
Fowler, A. G., Mariantoni, M., Martinis, J. M., & Cleland, A. N. (2012) · Physical Review A · doi:10.1103/physreva.86.032324 · Cited by 3,059 · cited 21× · see citing works
“Additionally, even when the multiple measurement protocol is applied, the code threshold significantly decreases (usually an order of magnitude) in comparison to thresholds that only consider data qubit noise (Fowler et al., 2012; Higgott et al., 2023).”
BackgroundMethodInfluentialCompletely positive linear maps on complex matrices
Choi, M.-D. (1975) · Linear Algebra and its Applications · doi:10.1016/0024-3795(75)90075-0 · Cited by 2,659 · see citing works
Good quantum error-correcting codes exist
Calderbank, A. R., & Shor, P. W. (1996) · Physical Review A · doi:10.1103/physreva.54.1098 · Cited by 2,516 · see citing works
Edmonds, J. (1965) · Canadian Journal of Mathematics · doi:10.4153/cjm-1965-045-4 · Cited by 2,428 · see citing works
Quantum computational advantage using photons
Zhong, H.-S., Wang, H., Deng, Y.-H., Chen, M.-C., Peng, L.-C., Luo, Y.-H., Qin, J., Wu, D., Ding, X., Hu, Y., et al. (2020) · Science · doi:10.1126/science.abe8770 · Cited by 2,405 · see citing works
Noisy intermediate-scale quantum algorithms
Bharti, K., Cervera-Lierta, A., Kyaw, T. H., Haug, T., Alperin-Lea, S., Anand, A., Degroote, M., Heimonen, H., Kottmann, J. S., Menke, T., et al. (2022) · Reviews of Modern Physics · doi:10.1103/revmodphys.94.015004 · Cited by 1,726 · see citing works
Quantum computations: algorithms and error correction
Kitaev, A. Y. (1997) · Russian Mathematical Surveys · doi:10.1070/rm1997v052n06abeh002155 · Cited by 1,424 · see citing works
Quantum error correction for quantum memories
Terhal, B. M. (2015) · Reviews of Modern Physics · doi:10.1103/revmodphys.87.307 · Cited by 1,354 · see citing works
Logical quantum processor based on reconfigurable atom arrays
Bluvstein, D., Evered, S. J., Geim, A. A., Li, S. H., Zhou, H., Manovitz, T., Ebadi, S., Cain, M., Kalinowski, M., Hangleiter, D., et al. (2024) · Nature · doi:10.1038/s41586-023-06927-3 · Cited by 1,119 · see citing works
Suppressing quantum errors by scaling a surface code logical qubit
Google Quantum, A. I., Acharya, R., Aleiner, I., Allen, R., Andersen, T. I., Ansmann, M., Arute, F., Arya, K., Asfaw, A., Atalaya, J., et al. (2023) · Nature · doi:10.1038/s41586-022-05434-1 · Cited by 1,103 · cited 11× · see citing works
“The Google team in (Acharya et al., 2023) studied the tensor network decoding process in a distance-5 code for 25 syndrome extraction rounds by considering a Tanner graph between the syndrome elements and the circuit-level error mechanisms in a similar manner than the method used in (Higgott et…”
BackgroundMethodInfluentialQuantum algorithms: an overview
Montanaro, A. (2016) · npj Quantum Information · doi:10.1038/npjqi.2015.23 · Cited by 1,071 · cited 2× · see citing works
“This technology presents an unprecedented potential to solve problems that are computationally hard since algorithms that present an exponential speedup in terms of computational complexity have been theoretically proposed (Montanaro, 2016; Shor, 1994).”
BackgroundQuantum computational advantage with a programmable photonic processor
Madsen, L. S., Laudenbach, F., Askarani, M. F., Rortais, F., Vincent, T., Bulmer, J. F. F., Miatto, F. M., Neuhaus, L., Helt, L. G., Collins, M. J., et al. (2022) · Nature · doi:10.1038/s41586-022-04725-x · Cited by 1,031 · see citing works
Realizing repeated quantum error correction in a distance-three surface code
Krinner, S., Lacroix, N., Remm, A., Di Paolo, A., Genois, E., Leroux, C., Hellings, C., Lazar, S., Swiadek, F., Herrmann, J., et al. (2022) · Nature · doi:10.1038/s41586-022-04566-8 · Cited by 686 · see citing works
Photonic quantum information processing: a concise review
Slussarenko, S., & Pryde, G. J. (2019) · Applied Physics Reviews · doi:10.1063/1.5115814 · Cited by 644 · see citing works
Surface code quantum computing by lattice surgery
Horsman, D., Fowler, A. G., Devitt, S., & Meter, R. V. (2012) · New Journal of Physics · doi:10.1088/1367-2630/14/12/123011 · Cited by 598 · cited 3× · see citing works
“Usually, it is considered that a distance 𝑑 surface code requires O(𝑑) syndrome extraction rounds per decoding round (Chamberland and Campbell, 2022; Higgott et al., 2023; Horsman et al., 2012) in order to preserve the distance property32.”
BackgroundQuantum advantage in learning from experiments
Huang, H.-Y., Broughton, M., Cotler, J., Chen, S., Li, J., Mohseni, M., Neven, H., Babbush, R., Kueng, R., Preskill, J., et al. (2022) · Science · doi:10.1126/science.abn7293 · Cited by 519 · see citing works
High-threshold universal quantum computation on the surface code
Fowler, A. G., Stephens, A. M., & Groszkowski, P. (2009) · Physical Review A · doi:10.1103/physreva.80.052312 · Cited by 486 · cited 1× · see citing works
“Surface codes are one of the most promising families of codes for constructing primitive fault-tolerant quantum computers in the near-term (Bombin and Martin- Delgado, 2007; Bravyi et al., 2018; Dennis et al., 2002; Fowler et al., 2012, 2009; Kitaev, 2003; Raussendorf and Harrington, 2007).”
BackgroundHigh-threshold and low-overhead fault-tolerant quantum memory
Bravyi, S., Cross, A. W., Gambetta, J. M., Maslov, D., Rall, P., & Yoder, T. J. (2024) · Nature · doi:10.1038/s41586-024-07107-7 · Cited by 456 · see citing works
Stim: a fast stabilizer circuit simulator
Gidney, C. (2021) · Quantum · doi:10.22331/q-2021-07-06-497 · Cited by 415 · see citing works
Codes for the quantum erasure channel
Grassl, M., Beth, T., & Pellizzari, T. (1997) · Physical Review A · doi:10.1103/physreva.56.33 · Cited by 374 · see citing works
Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors
van den Berg, E., Minev, Z. K., Kandala, A., & Temme, K. (2023) · Nature Physics · doi:10.1038/s41567-023-02042-2 · Cited by 305 · see citing works
Quantum-channel capacity of very noisy channels
DiVincenzo, D. P., Shor, P. W., & Smolin, J. A. (1998) · Physical Review A · doi:10.1103/physreva.57.830 · Cited by 297 · see citing works
A programmable qudit-based quantum processor
Chi, Y., Huang, J., Zhang, Z., Mao, J., Zhou, Z., Chen, X., Zhai, C., Bao, J., Dai, T., Yuan, H., et al. (2022) · Nature Communications · doi:10.1038/s41467-022-28767-x · Cited by 273 · see citing works
Bonilla Ataides, J. P., Tuckett, D. K., Bartlett, S. D., Flammia, S. T., & Brown, B. J. (2021) · Nature Communications · doi:10.1038/s41467-021-22274-1 · Cited by 262 · see citing works
Optimal resources for topological two-dimensional stabilizer codes: Comparative study
Bombin, H., & Martin-Delgado, M. A. (2007) · Physical Review A · doi:10.1103/physreva.76.012305 · Cited by 178 · see citing works
Correcting coherent errors with surface codes
Bravyi, S., Englbrecht, M., König, R., & Peard, N. (2018) · npj Quantum Information · doi:10.1038/s41534-018-0106-y · Cited by 157 · see citing works
Leverrier, A., & Zemor, G. (2022) · 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) · doi:10.1109/focs54457.2022.00117 · Cited by 140 · see citing works
Quantum Turbo Decoding for Quantum Channels Exhibiting Memory
Mohd Izhar, M. A., Babar, Z., Nguyen, H. V., Botsinis, P., Alanis, D., Chandra, D., Ng, S. X., & Hanzo, L. (2018) · IEEE Access · doi:10.1109/access.2018.2808373 · Cited by 9 · see citing works
Flagged first, then by citations
Research works printed but not linked: 155
- Richard P Feynman. "Simulating physics with computers". International journal of theoretical physics 21, 467-488 (1982). · DOI · Paper page
- John Preskill. "Quantum computing and the entanglement frontier" (2012). url: https:// doi.org/10.48550/arXiv.1203.5813. · DOI · Paper page
- Artur K. Ekert. "Quantum cryptography based on bell's theorem". Phys. Rev. Lett. 67, 661- 663 (1991). · DOI · Paper page
- Carlos Outeiral, Martin Strahm, Jiye Shi, Gar- rett M. Morris, Simon C. Benjamin, and Char- lotte M. Deane. "The prospects of quantum computing in computational molecular biology". WIREs Computational Molecular Science 11, e1481 (2021). · DOI · Paper page
- Daniel Jafferis, Alexander Zlokapa, Joseph D. Lykken, David K. Kolchmeyer, Samantha I. Davis, Nikolai Lauk, Hartmut Neven, and Maria Spiropulu. "Traversable wormhole dynamics on a quantum processor". Nature 612, 51- 55 (2022). · DOI
- Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, Tung- Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, Ming Gong, et al. "Strong quantum com- putational advantage using a superconducting quantum processor". Phys. Rev. Lett. 127, 180501 (2021). · DOI · Paper page
- Robert Raussendorf and Hans J. Briegel. "A one-way quantum computer". Phys. Rev. Lett. 86, 5188-5191 (2001). · DOI · Paper page
- H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest. "Measurement-based quantum computation". Nature Physics 5, 19-26 (2009). · DOI · Paper page
- Anjan K. Chandra, Arnab Das, and Bikas K. Chakrabarti. "Quantum quenching, anneal- ing and computation". Springer-Verlag Berlin. DE (2010). 1 edition. · DOI
- Atharv Joshi, Kyungjoo Noh, and Yvonne Y Gao. "Quantum information processing with bosonic qubits in circuit qed". Quantum Sci- ence and Technology 6, 033001 (2021). · DOI
- He-Liang Huang, Dachao Wu, Daojin Fan, and Xiaobo Zhu. "Superconducting quantum com- puting: a review". Science China Information Sciences 63, 180501 (2020). · DOI
- H. Häffner, C.F. Roos, and R. Blatt. "Quantum computing with trapped ions". Physics Reports 469, 155-203 (2008). · DOI · Paper page
- Peter W. Shor. "Scheme for reducing decoher- ence in quantum computer memory". Phys. Rev. A 52, R2493-R2496 (1995). · DOI · Paper page
- Josu Etxezarreta Martinez, Patricio Fuentes, Pedro M. Crespo, and J. Garcia-Frias. "Ap- proximating decoherence processes for the de- sign and simulation of quantum error correction codes on classical computers". IEEE Access 8, 172623-172643 (2020). · DOI
- Jonathan J. Burnett, Andreas Bengtsson, Marco Scigliuzzo, David Niepce, Marina Kudra, Per Delsing, and Jonas Bylander. "Decoherence benchmarking of superconducting qubits". npj Quantum Information 5, 54 (2019). · DOI · Paper page
- Josu Etxezarreta Martinez, Patricio Fuentes, Pedro Crespo, and Javier Garcia-Frias. "Time- varying quantum channel models for supercon- ducting qubits". npj Quantum Information 7, 115 (2021). · DOI
- Daniel Gottesman. "Stabilizer codes and quan- tum error correction". Phd thesis. California Institute of Technology. Pasadena, CA (1997). · DOI · Paper page
- A.Yu. Kitaev. "Fault-tolerant quantum compu- tation by anyons". Annals of Physics 303, 2- 30 (2003). · DOI · Paper page
- David K. Tuckett, Stephen D. Bartlett, and Steven T. Flammia. "Ultrahigh error threshold for surface codes with biased noise". Phys. Rev. Lett. 120, 050505 (2018). · DOI · Paper page
- Christopher Chamberland, Guanyu Zhu, Theodore J. Yoder, Jared B. Hertzberg, and Andrew W. Cross. "Topological and subsystem codes on low-degree graphs with flag qubits". Phys. Rev. X 10, 011022 (2020). · DOI · Paper page
- H. Bombin and M. A. Martin-Delgado. "Topo- logical quantum distillation". Phys. Rev. Lett. 97, 180501 (2006). · DOI · Paper page
- D.J.C. MacKay, G. Mitchison, and P.L. Mc- Fadden. "Sparse-graph codes for quantum error correction". IEEE Transactions on Information Theory 50, 2315-2330 (2004). · DOI
- Pavel Panteleev and Gleb Kalachev. "Degener- ate Quantum LDPC Codes With Good Finite Length Performance". Quantum 5, 585 (2021). · DOI · Paper page
- Pavel Panteleev and Gleb Kalachev. "Asymp- totically good quantum and locally testable classical ldpc codes". In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. Page 375-388. STOC 2022New York, NY, USA (2022). Association for Com- puting Machinery. · DOI · Paper page
- Nikolas P. Breuckmann and Jens Niklas Eber- hardt. "Quantum low-density parity-check codes". PRX Quantum 2, 040101 (2021). · DOI · Paper page
- David Poulin, Jean-Pierre Tillich, and Harold Ollivier. "Quantum serial turbo codes". IEEE Transactions on Information Theory 55, 2776- 2798 (2009). · DOI
- Patricio Fuentes, Josu Etxezarreta Martinez, Pedro M. Crespo, and Javier Garcia-Frías. "De- generacy and its impact on the decoding of sparse quantum codes". IEEE Access 9, 89093- 89119 (2021). · DOI
- Pavithran Iyer and David Poulin. "Hardness of decoding quantum stabilizer codes". IEEE Transactions on Information Theory 61, 5209- 5223 (2015). · DOI
- Oscar Higgott and Craig Gidney. "Sparse blos- som: correcting a million errors per core sec- ond with minimum-weight matching" (2023). url: https://doi.org/10.48550/arXiv.2303. 15933. · DOI · Paper page
- Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. "Topological quantum mem- ory". Journal of Mathematical Physics 43, 4452-4505 (2002). · DOI · Paper page
- Robert Raussendorf and Jim Harrington. "Fault-tolerant quantum computation with high threshold in two dimensions". Phys. Rev. Lett. 98, 190504 (2007). · DOI · Paper page
- Todd A. Brun, Igor Devetak, and Min-Hsiu Hsieh. "Catalytic quantum error correction". IEEE Transactions on Information Theory 60, 3073-3089 (2014). · DOI
- Misty A. Wahl, Andrea Mari, Nathan Shammah, William J. Zeng, and Gokul Sub- ramanian Ravi. "Zero noise extrapolation on logical qubits by scaling the error correction code distance". In 2023 IEEE International Conference on Quantum Computing and En- gineering (QCE). Volume 01, pages 888-897. (2023). · DOI
- Zhenyu Cai, Ryan Babbush, Simon C. Ben- jamin, Suguru Endo, William J. Huggins, Ying Li, Jarrod R. McClean, and Thomas E. O'Brien. "Quantum error mitigation". Rev. Mod. Phys. 95, 045005 (2023). · DOI
- Todd Brun, Igor Devetak, and Min-Hsiu Hsieh. "Correcting quantum errors with entanglement". Science 314, 436-439 (2006). · DOI
- Josu Etxezarreta Martinez. "Decoherence and quantum error correction for quantum com- puting and communications". Phd thesis.
- Tecnun -Universidad de Navarra. Donos- tia, Spain (2022). url: https://doi.org/10. 15581/10171/63013. · DOI
- Emanuel Knill and Raymond Laflamme. "The- ory of quantum error-correcting codes". Phys. Rev. A 55, 900-911 (1997). · DOI · Paper page
- Sergey Bravyi, Martin Suchara, and Alexander Vargo. "Efficient algorithms for maximum likeli- hood decoding in the surface code". Phys. Rev. A 90, 032326 (2014). · DOI · Paper page
- Sergey Bravyi and Alexei Y. Kitaev. "Quantum codes on a lattice with boundary" (1998). url: https://doi.org/10.48550/arXiv. quant-ph/9811052. · DOI · Paper page
- Andrew Steane. "Multiple-particle interference and quantum error correction". Proceedings: Mathematical, Physical and Engineering Sci- ences 452, 2551-2577 (1996).
- Oscar Higgott, Thomas C. Bohdanowicz, Alek- sander Kubica, Steven T. Flammia, and Earl T. Campbell. "Improved decoding of circuit noise and fragile boundaries of tailored surface codes". Phys. Rev. X 13, 031007 (2023). · DOI
- Antonio deMarti iOlius, Josu Etxezarreta Mar- tinez, Patricio Fuentes, and Pedro M. Crespo. "Performance enhancement of surface codes via recursive minimum-weight perfect-match de- coding". Phys. Rev. A 108, 022401 (2023). · DOI
- Yuuki Tokunaga and Keisuke Fujii. "Thresholds of surface codes on the general lattice structures suffering biased error and loss". AIP Conference Proceedings 1633, 198-200 (2014). · DOI
- Dorit Aharonov and Michael Ben-Or. "Fault- tolerant quantum computation with constant error rate". SIAM Journal on Computing 38, 1207-1282 (2008). · DOI · Paper page
- Emanuel Knill, Raymond Laflamme, and Wo- jciech H. Zurek. "Resilient quantum computa- tion". Science 279, 342-345 (1998). · DOI · Paper page
- Dave Bacon. "Operator quantum error- correcting subsystems for self-correcting quantum memories". Phys. Rev. A 73, 012340 (2006). · DOI · Paper page
- John M. Martinis. "Saving superconducting quantum processors from decay and correlated errors generated by gamma and cosmic rays". npj Quantum Information 7, 90 (2021). · DOI
- Ognyan Oreshkov and Todd A. Brun. "Continu- ous quantum error correction for non-markovian decoherence". Phys. Rev. A 76, 022318 (2007). · DOI
- Steffen Schlör. "Intrinsic decoherence in super- conducting quantum circuits". Phd thesis. Karl- sruher Institut für Technologie. Karlsruhe, Ger- many (2019). · DOI
- Angel Rivas and Susana F. Huelga. "Open quan- tum systems -an introduction". Springer Berlin, Heidelberg. DE (2011). 1 edition. · DOI
- Sumeet Khatri, Kunal Sharma, and Mark M. Wilde. "Information-theoretic aspects of the generalized amplitude-damping channel". Phys. Rev. A 102, 012401 (2020). · DOI
- Pradeep Kiran Sarvepalli, Andreas Klappe- necker, and Martin Rötteler. "Asymmetric quantum codes: constructions, bounds and per- formance". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sci- ences 465, 1645-1672 (2009). · DOI
- Joseph Emerson, Marcus Silva, Osama Moussa, Colm Ryan, Martin Laforest, Jonathan Baugh, David G. Cory, and Raymond Laflamme. "Sym- metrized characterization of noisy quantum pro- cesses". Science 317, 1893-1896 (2007). · DOI
- M. Silva, E. Magesan, D. W. Kribs, and J. Emerson. "Scalable protocol for identifica- tion of correctable codes". Phys. Rev. A 78, 012347 (2008). · DOI
- D Gottesman. "The heisenberg representation of quantum computers". Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics (1998). · DOI · Paper page
- Antonio deMarti iOlius, Josu Etxezarreta Mar- tinez, Patricio Fuentes, Pedro M. Crespo, and Javier Garcia-Frias. "Performance of surface codes in realistic quantum hardware". Phys. Rev. A 106, 062428 (2022). · DOI
- Robin Harper and Steven T. Flammia. "Learn- ing correlated noise in a 39-qubit quantum pro- cessor". PRX Quantum 4, 040311 (2023). · DOI
- Konstantin Tiurev, Peter-Jan H. S. Derks, Joschka Roffe, Jens Eisert, and Jan-Michael Reiner. "Correcting non-independent and non-identically distributed errors with surface codes". Quantum 7, 1123 (2023). · DOI
- Christopher Chamberland and Earl T. Camp- bell. "Universal quantum computing with twist- free and temporally encoded lattice surgery". PRX Quantum 3, 010331 (2022). · DOI
- Stefano Mangini, Lorenzo Maccone, and Chiara Macchiavello. "Qubit noise deconvolution". EPJ Quantum Technology 9, 29 (2022). · DOI
- Nicolas Delfosse and Naomi H. Nickerson. "Almost-linear time decoding algorithm for topological codes". Quantum 5, 595 (2021). · DOI · Paper page
- Francesco Battistel. "Mitigating leakage and noise in superconducting quantum comput- ing". Phd thesis. TU Delft. Delft, Nether- lands (2022). · DOI
- Roman Stricker, Davide Vodola, Alexander Er- hard, Lukas Postler, Michael Meth, Martin Ringbauer, Philipp Schindler, Thomas Monz, Markus Müller, and Rainer Blatt. "Experimen- tal deterministic correction of qubit loss". Na- ture 585, 207-210 (2020). · DOI
- Joydip Ghosh, Austin G. Fowler, John M. Mar- tinis, and Michael R. Geller. "Understand- ing the effects of leakage in superconducting quantum-error-detection circuits". Phys. Rev. A 88, 062329 (2013). · DOI
- Ayumu Nakayama and Manabu Hagiwara. "Sin- gle quantum deletion error-correcting codes". In 2020 International Symposium on Information Theory and Its Applications (ISITA). Pages 329-333. (2020). · DOI
- Y. Wu and L. Zhong. "Fusion blossom: Fast mwpm decoders for qec". In 2023 IEEE Interna- tional Conference on Quantum Computing and Engineering (QCE). Pages 928-938. Los Alami- tos, CA, USA (2023). IEEE Computer Society. · DOI · Paper page
- Oscar Higgott. "Pymatching: A python pack- age for decoding quantum codes with minimum- weight perfect matching". ACM Transactions on Quantum Computing3 (2022). · DOI · Paper page
- Joschka Roffe, David R. White, Simon Burton, and Earl Campbell. "Decoding across the quan- tum low-density parity-check code landscape". Phys. Rev. Res. 2, 043423 (2020). · DOI · Paper page
- Guillaume Duclos-Cianci and David Poulin. "Fast decoders for topological quantum codes". Phys. Rev. Lett. 104, 050504 (2010). · DOI
- Michael Herold, Earl T. Campbell, Jens Eis- ert, and Michael J. Kastoryano. "Cellular- automaton decoders for topological quantum memories". npj Quantum Information 1, 15010 (2015). · DOI
- Savvas Varsamopoulos, Ben Criger, and Koen Bertels. "Decoding small surface codes with feedforward neural networks". Quantum Science and Technology 3, 015004 (2017). · DOI
- Lucas Berent, Lukas Burgholzer, Peter-Jan H. S. Derks, Jens Eisert, and Robert Wille. "Decoding quantum color codes with maxsat" (2023). url: https://doi.org/10. 48550/arXiv.2303.14237. · DOI
- Austin G. Fowler. "Minimum weight perfect matching of fault-tolerant topological quantum error correction in average o(1) parallel time". Quantum Info. Comput. 15, 145-158 (2015). · DOI
- Yue Wu, Namitha Liyanage, and Lin Zhong. "An interpretation of union-find decoder on weighted graphs" (2022). url: https://doi. org/10.48550/arXiv.2211.03288. · DOI
- Patricio Fuentes, Josu Etxezarreta Martinez, Pedro M. Crespo, and Javier Garcia-Frías. "On the logical error rate of sparse quantum codes". IEEE Transactions on Quantum Engineering 3, 1-12 (2022). · DOI
- Jonghyun Lee, Jooyoun Park, and Jun Heo. "Rectangular surface code under biased noise". Quantum Information Processing 20, 231 (2021). · DOI
- David K. Tuckett, Stephen D. Bartlett, Steven T. Flammia, and Benjamin J. Brown. "Fault-tolerant thresholds for the surface code in excess of 5% under biased noise". Phys. Rev. Lett. 124, 130501 (2020). · DOI
- Josu Etxezarreta Martinez, Patricio Fuentes, Antonio deMarti iOlius, Javier Garcia-Frias, Javier Rodríguez Fonollosa, and Pedro M. Cre- spo. "Multiqubit time-varying quantum chan- nels for nisq-era superconducting quantum pro- cessors". Phys. Rev. Res. 5, 033055 (2023). · DOI
- Nicolas Delfosse and Jean-Pierre Tillich. "A de- coding algorithm for css codes using the x/z cor- relations". In 2014 IEEE International Sympo- sium on Information Theory. Pages 1071-1075. (2014). · DOI
- Austin G. Fowler. "Optimal complexity cor- rection of correlated errors in the surface code" (2013). url: https://doi.org/10. 48550/arXiv.1310.0863. · DOI · Paper page
- Nicolas Delfosse and Gilles Zémor. "Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel". Phys. Rev. Res. 2, 033042 (2020). · DOI
- Robert Endre Tarjan. "Efficiency of a good but not linear set union algorithm". J. ACM 22, 215-225 (1975). · DOI
- Shilin Huang, Michael Newman, and Ken- neth R. Brown. "Fault-tolerant weighted union- find decoding on the toric code". Phys. Rev. A 102, 012419 (2020). · DOI
- Nicolas Delfosse, Vivien Londe, and Michael E. Beverland. "Toward a union-find decoder for quantum ldpc codes". IEEE Transactions on Information Theory 68, 3187-3199 (2022). · DOI
- Judea Pearl. "Probabilistic reasoning in in- telligent systems". Morgan Kaufmann. CA, USA (1998). 1 edition. · DOI
- David Poulin and Yeojin Chung. "On the itera- tive decoding of sparse quantum codes". Quan- tum Info. Comput. 8, 987-1000 (2008). · DOI
- Sae-Young Chung, G.D. Forney, T.J. Richard- son, and R. Urbanke. "On the design of low- density parity-check codes within 0.0045 db of the shannon limit". IEEE Communications Let- ters 5, 58-60 (2001). · DOI
- Matthew B. Hastings, Jeongwan Haah, and Ryan O'Donnell. "Fiber bundle codes: break- ing the n1/2 polylog(n) barrier for quantum ldpc codes". In Proceedings of the 53rd An- nual ACM SIGACT Symposium on Theory of Computing. Page 1276-1288. STOC 2021New York, NY, USA (2021). Association for Com- puting Machinery. · DOI
- Nikolas P. Breuckmann and Jens N. Eberhardt. "Balanced product quantum codes". IEEE Transactions on Information Theory 67, 6653- 6674 (2021). · DOI
- Qian Xu, J. Pablo Bonilla Ataides, Christo- pher A. Pattison, Nithin Raveendran, Dolev Bluvstein, Jonathan Wurtz, Bane Vasić, Mikhail D. Lukin, Liang Jiang, and Hengyun Zhou. "Constant-overhead fault-tolerant quan- tum computation with reconfigurable atom ar- rays". Nature Physics (2024). · DOI
- Nithin Raveendran and Bane Vasić. "Trapping Sets of Quantum LDPC Codes". Quantum 5, 562 (2021). · DOI
- M.P.C. Fossorier and Shu Lin. "Soft-decision decoding of linear block codes based on ordered statistics". IEEE Transactions on Information Theory 41, 1379-1396 (1995). · DOI
- M.P.C. Fossorier. "Iterative reliability-based de- coding of low-density parity check codes". IEEE Journal on Selected Areas in Communications 19, 908-917 (2001). · DOI
- Joschka Roffe, Lawrence Z. Cohen, Armanda O. Quintavalle, Daryus Chandra, and Earl T. Campbell. "Bias-tailored quantum LDPC codes". Quantum 7, 1005 (2023). · DOI
- Ching-Yi Lai and Kao-Yueh Kuo. "Log-domain decoding of quantum ldpc codes over binary fi- nite fields". IEEE Transactions on Quantum En- gineering 2, 1-15 (2021). · DOI
- M.C. Davey and D.J.C. MacKay. "Low density parity check codes over gf(q)". In 1998 Infor- mation Theory Workshop (Cat. No.98EX131). Pages 70-71. (1998). · DOI
- Javier Valls, Francisco Garcia-Herrero, Nithin Raveendran, and Bane Vasić. "Syndrome-based min-sum vs osd-0 decoders: Fpga implemen- tation and analysis for quantum ldpc codes". IEEE Access 9, 138734-138743 (2021). · DOI
- 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. (2022). · DOI
- Ben Criger and Imran Ashraf. "Multi-path Sum- mation for Decoding 2D Topological Codes". Quantum 2, 102 (2018). · DOI
- Nithin Raveendran, Narayanan Rengaswamy, Asit Kumar Pradhan, and Bane Vasić. "Soft syndrome decoding of quantum ldpc codes for joint correction of data and syndrome errors". In 2022 IEEE International Conference on Quan- tum Computing and Engineering (QCE). Pages 275-281. (2022). · DOI
- Christopher A. Pattison, Michael E. Beverland, Marcus P. da Silva, and Nicolas Delfosse. "Im- proved quantum error correction using soft in- formation" (2021). url: https://doi.org/10. 48550/arXiv.2107.13589. · DOI
- Oscar Higgott and Nikolas P. Breuckmann. "Improved single-shot decoding of higher- dimensional hypergraph-product codes". PRX Quantum 4, 020332 (2023). · DOI
- Héctor Bombín. "Single-shot fault-tolerant quantum error correction". Phys. Rev. X 5, 031043 (2015). · DOI
- Nikolas P. Breuckmann and Vivien Londe. "Single-shot decoding of linear rate ldpc quan- tum codes with high performance". IEEE Transactions on Information Theory 68, 272- 286 (2022). · DOI
- R. Orús. "A practical introduction to tensor networks: Matrix product states and projected entangled pair states". Annals of Physics 349, 117-158 (2014). · DOI · Paper page
- Andrew J. Ferris and David Poulin. "Tensor networks and quantum error correction". Phys. Rev. Lett. 113, 030501 (2014). · DOI
- David K. Tuckett, Andrew S. Darmawan, Christopher T. Chubb, Sergey Bravyi, Stephen D. Bartlett, and Steven T. Flam- mia. "Tailoring surface codes for highly biased noise". Phys. Rev. X 9, 041031 (2019). · DOI
- Christopher T. Chubb. "General tensor network decoding of 2d pauli codes" (2021). url: https: //doi.org/10.48550/arXiv.2101.04125. · DOI · Paper page
- David Kingsley Tuckett. "Tailoring surface codes: Improvements in quantum error correc- tion with biased noise". PhD thesis. University of Sydney. (2020). · DOI
- Norbert Schuch, Michael M. Wolf, Frank Ver- straete, and J. Ignacio Cirac. "Computational complexity of projected entangled pair states". Phys. Rev. Lett. 98, 140506 (2007). · DOI · Paper page
- Stephen Wolfram. "Statistical mechanics of cel- lular automata". Rev. Mod. Phys. 55, 601- 644 (1983). · DOI
- Martin Gardner. "Mathematical games". Scien- tific American 223, 120-123 (1970). url: http: //www.jstor.org/stable/24927642.
- James W. Harrington. "Analysis of quantum error-correcting codes: Symplectic lattice codes and toric codes". Phd thesis. California Institute of Technology. Pasadena, CA (2004). · DOI
- Aleksander M. Kubica. "The abcs of the color code: A study of topological quantum codes as toy models for fault-tolerant quantum computa- tion and quantum phases of matter". Phd thesis. California Institute of Technology. Pasadena, CA (2018). · DOI
- Aleksander Kubica and John Preskill. "Cellular- automaton decoders with provable thresholds for topological codes". Phys. Rev. Lett. 123, 020501 (2019). · DOI
- Michael Vasmer, Dan E. Browne, and Alek- sander Kubica. "Cellular automaton decoders for topological quantum codes with noisy mea- surements and beyond". Scientific Reports 11, 2027 (2021). · DOI
- Aleksander Kubica and Nicolas Delfosse. "Ef- ficient color code decoders in d ≥ 2 dimen- sions from toric code decoders". Quantum 7, 929 (2023). · DOI
- Chenyang Wang, Jim Harrington, and John Preskill. "Confinement-higgs transition in a dis- ordered gauge theory and the accuracy thresh- old for quantum memory". Annals of Physics 303, 31-58 (2003). · DOI
- Pradeep Sarvepalli and Robert Raussendorf. "Efficient decoding of topological color codes". Phys. Rev. A 85, 022317 (2012). · DOI
- Sergey Bravyi and Jeongwan Haah. "Quan- tum self-correction in the 3d cubic code model". Phys. Rev. Lett. 111, 200501 (2013). · DOI
- Fern H. E. Watson, Hussain Anwar, and Dan E. Browne. "Fast fault-tolerant decoder for qubit and qudit surface codes". Phys. Rev. A 92, 032309 (2015). · DOI
- James Wootton. "A simple decoder for topolog- ical codes". Entropy 17, 1946-1957 (2015). · DOI
- Hussain Anwar, Benjamin J Brown, Earl T Campbell, and Dan E Browne. "Fast decoders for qudit topological codes". New Journal of Physics 16, 063038 (2014). · DOI
- Adrian Hutter, Daniel Loss, and James R Woot- ton. "Improved hdrg decoders for qudit and non- abelian quantum error correction". New Journal of Physics 17, 035017 (2015). · DOI
- K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal. "Renormalization group de- coder for a four-dimensional toric code". IEEE Transactions on Information Theory 65, 2545- 2562 (2019). · DOI
- Pierre Sermanet, David Eigen, Xiang Zhang, Michael Mathieu, Rob Fergus, and Yann Le- Cun. "Overfeat: Integrated recognition, local- ization and detection using convolutional net- works". In 2nd International Conference on Learning Representations, ICLR 2014. (2014). url: https://doi.org/10.48550/arXiv.1312. 6229. · DOI · Paper page
- Christopher Chamberland and Pooya Ronagh. "Deep neural decoders for near term fault- tolerant experiments". Quantum Science and Technology 3, 044002 (2018). · DOI
- Savvas Varsamopoulos, Koen Bertels, and Car- men Garcia Almudever. "Comparing neural net- work based decoders for the surface code". IEEE Transactions on Computers 69, 300-311 (2020). · DOI
- Ryan Sweke, Markus S Kesselring, Evert P L van Nieuwenburg, and Jens Eisert. "Reinforce- ment learning decoders for fault-tolerant quan- tum computation". Machine Learning: Science and Technology 2, 025005 (2020). · DOI
- Savvas Varsamopoulos, Koen Bertels, and Car- men G. Almudever. "Decoding surface code with a distributed neural network-based decoder". Quantum Machine Intelligence 2, 3 (2020). · DOI
- Boris Mihailov Varbanov, Marc Serra-Peralta, David Byfield, and Barbara Maria Terhal. "Neu- ral network decoder for near-term surface-code experiments" (2023). url: https://doi.org/ 10.48550/arXiv.2307.03280. · DOI
- Moritz Lange, Pontus Havstrom, Basudha Sri- vastava, Valdemar Bergentall, Karl Hammar, Olivia Heuts, Evert van Nieuwenburg, and Mats Granath. "Data-driven decoding of quan- tum error correcting codes using graph neural networks" (2023). url: https://doi.org/10. 48550/arXiv.2307.01241. · DOI
- Giacomo Torlai and Roger G. Melko. "Neural decoder for topological codes". Phys. Rev. Lett. 119, 030501 (2017). · DOI
- Stefan Krastanov and Liang Jiang. "Deep neu- ral network probabilistic decoder for stabilizer codes". Scientific Reports 7, 11003 (2017). · DOI
- Ye-Hua Liu and David Poulin. "Neural belief-propagation decoders for quantum error- correcting codes". Phys. Rev. Lett. 122, 200501 (2019). · DOI
- Philip Andreasson, Joel Johansson, Simon Lil- jestrand, and Mats Granath. "Quantum error correction for the toric code using deep rein- forcement learning". Quantum 3, 183 (2019). · DOI
- Thomas Wagner, Hermann Kampermann, and Dagmar Bruß. "Symmetries for a high-level neu- ral decoder on the toric code". Phys. Rev. A 102, 042411 (2020). · DOI
- David Fitzek, Mattias Eliasson, Anton Frisk Kockum, and Mats Granath. "Deep q-learning decoder for depolarizing noise on the toric code". Phys. Rev. Res. 2, 023230 (2020). · DOI
- Laia Domingo Colomer, Michalis Skotiniotis, and Ramon Muñoz-Tapia. "Reinforcement learning for optimal error correction of toric codes". Physics Letters A 384, 126353 (2020). · DOI
- Xiaotong Ni. "Neural Network Decoders for Large-Distance 2D Toric Codes". Quantum 4, 310 (2020). · DOI
- Kai Meinerz, Chae-Yeun Park, and Simon Trebst. "Scalable neural decoder for topolog- ical surface codes". Phys. Rev. Lett. 128, 080505 (2022). · DOI
- P Baireuther, M D Caio, B Criger, C W J Beenakker, and T E O'Brien. "Neural net- work decoder for topological color codes with circuit level noise". New Journal of Physics 21, 013003 (2019). · DOI
- S. Varona and M. A. Martin-Delgado. "Determi- nation of the semion code threshold using neural decoders". Phys. Rev. A 102, 032411 (2020). · DOI
- Christopher Chamberland, Luis Goncalves, Prasahnt Sivarajah, Eric Peterson, and Se- bastian Grimberg. "Techniques for combining fast local decoders with global decoders under circuit-level noise". Quantum Science and Tech- nology 8, 045011 (2023). · DOI
- Mark W. Krentel. "The complexity of optimiza- tion problems". Journal of Computer and Sys- tem Sciences 36, 490-509 (1988). · DOI
- Lucas Berent, Lukas Burgholzer, and Robert Wille. "Software tools for decoding quantum low-density parity-check codes". In Proceedings of the 28th Asia and South Pacific Design Au- tomation Conference. Page 709-714. ASPDAC '23New York, NY, USA (2023). Association for Computing Machinery. · DOI
- Luka Skoric, Dan E. Browne, Kenton M. Barnes, Neil I. Gillespie, and Earl T. Campbell. "Parallel window decoding enables scalable fault tolerant quantum computation". Nature Com- munications 14, 7040 (2023). · DOI
- Xinyu Tan, Fang Zhang, Rui Chao, Yaoyun Shi, and Jianxin Chen. "Scalable surface-code de- coders with parallelization in time". PRX Quan- tum 4, 040344 (2023). · DOI
- P Aliferis, F Brito, D P DiVincenzo, J Preskill, M Steffen, and B M Terhal. "Fault-tolerant computing with biased-noise superconducting qubits: a case study". New Journal of Physics 11, 013061 (2009). · DOI
- J. Tolar. "On Clifford groups in quantum computing". In Journal of Physics Conference Series. Volume 1071 of Journal of Physics Conference Series, page 012022. (2018). arXiv:1810.10259. · DOI
- Qian Xu, Nam Mannucci, Alireza Seif, Alek- sander Kubica, Steven T. Flammia, and Liang Jiang. "Tailored xzzx codes for biased noise". Phys. Rev. Res. 5, 013035 (2023). · DOI
- Yaping Yuan and Chung-Chin Lu. "A modified mwpm decoding algorithm for quantum surface codes over depolarizing channels". 2022 Interna- tional Symposium on Information Theory and Its Applications (ISITA)Pages 204-208 (2022). url: https://doi.org/10.48550/arXiv.2202. 11239. · DOI
- David Tse and Pramod Viswanath. "Funda- mentals of wireless communication". Cambridge University Press. USA (2005). · DOI · Paper page
- Thomas Wagner, Hermann Kampermann, Dag- mar Bruß, and Martin Kliesch. "Pauli chan- nels can be estimated from syndrome measure- ments in quantum error correction". Quantum 6, 809 (2022). · DOI
Cited by
Sorted by citations
Quantum error correction and fault tolerance: A comprehensive tutorial
Spencer, D. J., Jain, S. P., Tanggara, A., Sun, Z., Haug, T., Khu, D., & Bharti, K. (2026) · arXiv
- Authors
- deMarti iOlius, A., Fuentes, P., Orús, R., Crespo, P. M., & Etxezarreta Martinez, J.
- Affiliations
- Department of Basic Sciences, Tecnun - University of Navarra, 20018 San Sebastian, Spain.
- Photonic Inc., Vancouver, British Columbia, Canada.
- Ikerbasque (ES)
- Universidad de Navarra (ES)
- Venue
- Quantum
- Published
- 2024-10-10
- Volume
- 8
- Pages
- 1498
- ISSN
- 2521-327X (ISSN-L)
- OpenAlex
- W4403310488
- Cited by
- 60
- References
- 36
- Citations by year
- 2026: 26 · 2025: 25 · 2024: 7 · 2023: 2
- FWCI
- 7.53
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Locations
- Quantum (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion, cc-by)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- arXiv (Cornell University) (cc-by)
- Funders
- Spanish Ministry of Economy and Competitiveness (PID2019-104958RB-C44, PID2022-137099NBC44)
- Spanish Ministry of Science and Innovation (PLEC2021-008251)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- surface code, quantum error correction, decoding algorithms, fault-tolerant quantum computing, decoding complexity, error correction capability, quantum noise
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