Almost-linear time decoding algorithm for topological codes

Nicolas Delfosse; Naomi H. Nickerson

doi:10.22331/q-2021-12-02-595

Almost-linear time decoding algorithm for topological codes

Nicolas Delfosse^1,2,3 and Naomi H. Nickerson⁴

¹IQIM, California Institute of Technology, Pasadena, CA, USA
²Department of Physics and Astronomy, University of California, Riverside, CA, USA
³Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA
⁴Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom

Published:	2021-12-02, volume 5, page 595
Eprint:	arXiv:1709.06218v3
Doi:	https://doi.org/10.22331/q-2021-12-02-595
Citation:	Quantum 5, 595 (2021).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

In order to build a large scale quantum computer, one must be able to correct errors extremely fast. We design a fast decoding algorithm for topological codes to correct for Pauli errors and erasure and combination of both errors and erasure. Our algorithm has a worst case complexity of $O(n \alpha(n))$, where $n$ is the number of physical qubits and $\alpha$ is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, $\alpha(n) \leq 3$. We prove that our algorithm performs optimally for errors of weight up to $(d-1)/2$ and for loss of up to $d-1$ qubits, where $d$ is the minimum distance of the code. Numerically, we obtain a threshold of $9.9\%$ for the 2d-toric code with perfect syndrome measurements and $2.6\%$ with faulty measurements.

► BibTeX data

@article{Delfosse2021almostlineartime,
  doi = {10.22331/q-2021-12-02-595},
  url = {https://doi.org/10.22331/q-2021-12-02-595},
  title = {Almost-linear time decoding algorithm for topological codes},
  author = {Delfosse, Nicolas and Nickerson, Naomi H.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {595},
  month = dec,
  year = {2021}
}

► References

[1] Hussain Anwar, Benjamin J Brown, Earl T Campbell, and Dan E Browne. Fast decoders for qudit topological codes. New Journal of Physics, 16 (6): 063038, 2014. 10.1088/1367-2630/16/6/063038.
https://doi.org/10.1088/1367-2630/16/6/063038

[2] CJ Ballance, TP Harty, NM Linke, and DM Lucas. High-fidelity two-qubit quantum logic gates using trapped calcium-43 ions. arXiv preprint arXiv:1406.5473, 2014. 10.1103/PhysRevLett.117.060504.
https://doi.org/10.1103/PhysRevLett.117.060504
arXiv:1406.5473

[3] Rami Barends, Julian Kelly, Anthony Megrant, Andrzej Veitia, Daniel Sank, Evan Jeffrey, Ted C White, Josh Mutus, Austin G Fowler, Brooks Campbell, et al. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature, 508 (7497): 500–503, 2014. 10.1038/nature13171.
https://doi.org/10.1038/nature13171

[4] Sean D Barrett and Thomas M Stace. Fault tolerant quantum computation with very high threshold for loss errors. Physical review letters, 105 (20): 200502, 2010a. 10.1103/PhysRevLett.105.200502.
https://doi.org/10.1103/PhysRevLett.105.200502

[5] Sean D Barrett and Thomas M Stace. Fault tolerant quantum computation with very high threshold for loss errors. Physical review letters, 105 (20): 200502, 2010b. 10.1103/PhysRevLett.105.200502.
https://doi.org/10.1103/PhysRevLett.105.200502

[6] Hector Bombin and Miguel Angel Martin-Delgado. Topological quantum distillation. Physical review letters, 97 (18): 180501, 2006. 10.1103/PhysRevLett.97.180501.
https://doi.org/10.1103/PhysRevLett.97.180501

[7] Hector Bombin, Guillaume Duclos-Cianci, and David Poulin. Universal topological phase of two-dimensional stabilizer codes. New Journal of Physics, 14 (7): 073048, 2012. 10.1088/1367-2630/14/7/073048.
https://doi.org/10.1088/1367-2630/14/7/073048

[8] S. B. Bravyi and A. Y. Kitaev. Quantum codes on a lattice with boundary. arXiv preprint arXiv:9811052, 1998.
arXiv:quant-ph/9811052

[9] Sergey Bravyi and Jeongwan Haah. Quantum self-correction in the 3d cubic code model. Physical review letters, 111 (20): 200501, 2013. 10.1103/PhysRevLett.111.200501.
https://doi.org/10.1103/PhysRevLett.111.200501

[10] Sergey Bravyi, Martin Suchara, and Alexander Vargo. Efficient algorithms for maximum likelihood decoding in the surface code. Physical Review A, 90 (3): 032326, 2014. 10.1103/PhysRevA.90.032326.
https://doi.org/10.1103/PhysRevA.90.032326

[11] Nikolas P Breuckmann and Barbara M Terhal. Constructions and noise threshold of hyperbolic surface codes. IEEE Transactions on Information Theory, 62 (6): 3731–3744, 2016. 10.1109/TIT.2016.2555700.
https://doi.org/10.1109/TIT.2016.2555700

[12] Nikolas P Breuckmann, Christophe Vuillot, Earl Campbell, Anirudh Krishna, and Barbara M Terhal. Hyperbolic and semi-hyperbolic surface codes for quantum storage. arXiv preprint arXiv:1703.00590, 2017. 10.1088/2058-9565/aa7d3b.
https://doi.org/10.1088/2058-9565/aa7d3b
arXiv:1703.00590

[13] Daniel E Browne and Terry Rudolph. Resource-efficient linear optical quantum computation. Physical Review Letters, 95 (1): 010501, 2005. 10.1103/PhysRevLett.95.010501.
https://doi.org/10.1103/PhysRevLett.95.010501

[14] Jacques Carolan, Christopher Harrold, Chris Sparrow, Enrique Martín-López, Nicholas J Russell, Joshua W Silverstone, Peter J Shadbolt, Nobuyuki Matsuda, Manabu Oguma, Mikitaka Itoh, et al. Universal linear optics. Science, 349 (6249): 711–716, 2015. 10.1126/science.aab3642.
https://doi.org/10.1126/science.aab3642

[15] S Debnath, NM Linke, C Figgatt, KA Landsman, K Wright, and C Monroe. Demonstration of a small programmable quantum computer with atomic qubits. Nature, 536 (7614): 63–66, 2016. 10.1038/nature18648.
https://doi.org/10.1038/nature18648

[16] Nicolas Delfosse. Tradeoffs for reliable quantum information storage in surface codes and color codes. In Information Theory Proceedings (ISIT), 2013 IEEE International Symposium on, pages 917–921. IEEE, 2013. 10.1109/ISIT.2013.6620360.
https://doi.org/10.1109/ISIT.2013.6620360

[17] Nicolas Delfosse. Decoding color codes by projection onto surface codes. Physical Review A, 89 (1): 012317, 2014. 10.1103/PhysRevA.89.012317.
https://doi.org/10.1103/PhysRevA.89.012317

[18] Nicolas Delfosse and Gilles Zémor. Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel. arXiv preprint arXiv:1703.01517, 2017. 10.1103/PhysRevResearch.2.033042.
https://doi.org/10.1103/PhysRevResearch.2.033042
arXiv:1703.01517

[19] Eric Dennis. Purifying quantum states: Quantum and classical algorithms. arXiv preprint quant-ph/0503169, 2005.
arXiv:quant-ph/0503169

[20] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics, 43 (9): 4452–4505, 2002. 10.1063/1.1499754.
https://doi.org/10.1063/1.1499754

[21] Guillaume Duclos-Cianci and David Poulin. Fast decoders for topological quantum codes. Physical review letters, 104 (5): 050504, 2010. 10.1103/PhysRevLett.104.050504.
https://doi.org/10.1103/PhysRevLett.104.050504

[22] Guillaume Duclos-Cianci and David Poulin. Kitaev's z d-code threshold estimates. Physical Review A, 87 (6): 062338, 2013. 10.1103/PhysRevA.87.062338.
https://doi.org/10.1103/PhysRevA.87.062338

[23] Guillaume Duclos-Cianci and David Poulin. Fault-tolerant renormalization group decoder for abelian topological codes. Quantum Info. Comput., 14 (9 & 10): 721–740, jul 2014. ISSN 1533-7146. 10.5555/2638670.2638671.
https://doi.org/10.5555/2638670.2638671

[24] Austin Fowler. Towards sufficiently fast quantum error correction. Conference QEC 2017, 2017.
https://qec2017.gatech.edu/

[25] Austin G Fowler. Minimum weight perfect matching of fault-tolerant topological quantum error correction in average $ o (1) $ parallel time. Quantum Information and Computation, 15 (1&2): 0145–0158, 2015. 10.5555/2685188.2685197.
https://doi.org/10.5555/2685188.2685197

[26] Austin G Fowler, Adam C Whiteside, and Lloyd CL Hollenberg. Towards practical classical processing for the surface code. Physical review letters, 108 (18): 180501, 2012a. 10.1103/PhysRevLett.108.180501.
https://doi.org/10.1103/PhysRevLett.108.180501

[27] Austin G Fowler, Adam C Whiteside, Angus L McInnes, and Alimohammad Rabbani. Topological code autotune. Physical Review X, 2 (4): 041003, 2012b. 10.1103/PhysRevX.2.041003.
https://doi.org/10.1103/PhysRevX.2.041003

[28] Michael Fredman and Michael Saks. The cell probe complexity of dynamic data structures. In Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 345–354. ACM, 1989. 10.1145/73007.73040.
https://doi.org/10.1145/73007.73040

[29] M. H. Freedman and D. A. Meyer. Projective plane and planar quantum codes. Foundations of Computational Mathematics, 1 (3): 325–332, 2001. 10.1007/s102080010013.
https://doi.org/10.1007/s102080010013

[30] M.H. Freedman, D.A. Meyer, F. Luo, and Computer Society (IEEE) Washington DC. Z(2)-Systolic Freedom and Quantum Codes. Defense Technical Information Center, 2002. 10.2140/gtm.1999.2.113. URL https://books.google.com/books?id=KieaDAEACAAJ.
https://doi.org/10.2140/gtm.1999.2.113
https://books.google.com/books?id=KieaDAEACAAJ

[31] Bernard A Galler and Michael J Fisher. An improved equivalence algorithm. Communications of the ACM, 7 (5): 301–303, 1964. 10.1145/364099.364331.
https://doi.org/10.1145/364099.364331

[32] James William Harrington. Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes. PhD thesis, California Institute of Technology, 2004.

[33] Reinier W Heeres, Philip Reinhold, Nissim Ofek, Luigi Frunzio, Liang Jiang, Michel H Devoret, and Robert J Schoelkopf. Implementing a universal gate set on a logical qubit encoded in an oscillator. Nature Communications, 8, 2017. 10.1038/s41467-017-00045-1.
https://doi.org/10.1038/s41467-017-00045-1

[34] Michael Herold, Michael J Kastoryano, Earl T Campbell, and Jens Eisert. Fault tolerant dynamical decoders for topological quantum memories. arXiv preprint arXiv:1511.05579, 2015. 10.1088/1367-2630/aa7099.
https://doi.org/10.1088/1367-2630/aa7099
arXiv:1511.05579

[35] Charles D Hill, Eldad Peretz, Samuel J Hile, Matthew G House, Martin Fuechsle, Sven Rogge, Michelle Y Simmons, and Lloyd CL Hollenberg. A surface code quantum computer in silicon. Science advances, 1 (9): e1500707, 2015. 10.1038/npjqi.2015.19.
https://doi.org/10.1038/npjqi.2015.19

[36] John Hopcroft and Robert Tarjan. Algorithm 447: efficient algorithms for graph manipulation. Communications of the ACM, 16 (6): 372–378, 1973. 10.1145/362248.362272.
https://doi.org/10.1145/362248.362272

[37] Adrian Hutter, James R Wootton, and Daniel Loss. Efficient markov chain monte carlo algorithm for the surface code. Physical Review A, 89 (2): 022326, 2014. 10.1103/PhysRevA.89.022326.
https://doi.org/10.1103/PhysRevA.89.022326

[38] Adrian Hutter, Daniel Loss, and James R Wootton. Improved hdrg decoders for qudit and non-abelian quantum error correction. New Journal of Physics, 17 (3): 035017, 2015. 10.1088/1367-2630/17/3/035017.
https://doi.org/10.1088/1367-2630/17/3/035017

[39] IBM. Quantum experience API. https://github.com/QISKit, 2017.
https://github.com/QISKit

[40] Norbert Kalb, Andreas A Reiserer, Peter C Humphreys, Jacob JW Bakermans, Sten J Kamerling, Naomi H Nickerson, Simon C Benjamin, Daniel J Twitchen, Matthew Markham, and Ronald Hanson. Entanglement distillation between solid-state quantum network nodes. Science, 356 (6341): 928–932, 2017. 10.1126/science.aan0070.
https://doi.org/10.1126/science.aan0070

[41] Konrad Kieling, Terry Rudolph, and Jens Eisert. Percolation, renormalization, and quantum computing with nondeterministic gates. Physical Review Letters, 99 (13): 130501, 2007. 10.1103/PhysRevLett.99.130501.
https://doi.org/10.1103/PhysRevLett.99.130501

[42] A Yu Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303 (1): 2–30, 2003. 10.1016/S0003-4916(02)00018-0.
https://doi.org/10.1016/S0003-4916(02)00018-0

[43] Emanuel Knill, Raymond Laflamme, and Gerald J Milburn. A scheme for efficient quantum computation with linear optics. nature, 409 (6816): 46–52, 2001. 10.1038/35051009.
https://doi.org/10.1038/35051009

[44] Vladimir Kolmogorov. Blossom v: a new implementation of a minimum cost perfect matching algorithm. Mathematical Programming Computation, 1 (1): 43–67, 2009. 10.1007/s12532-009-0002-8.
https://doi.org/10.1007/s12532-009-0002-8

[45] Aleksander Marek Kubica. The ABCs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-Tolerant Quantum Computation and Quantum Phases Of Matter. PhD thesis, California Institute of Technology, 2018.

[46] Andrew J Landahl, Jonas T Anderson, and Patrick R Rice. Fault-tolerant quantum computing with color codes. arXiv preprint arXiv:1108.5738, 2011.
arXiv:1108.5738

[47] Bjoern Lekitsch, Sebastian Weidt, Austin G Fowler, Klaus Mølmer, Simon J Devitt, Christof Wunderlich, and Winfried K Hensinger. Blueprint for a microwave trapped ion quantum computer. Science Advances, 3 (2): e1601540, 2017. 10.1126/sciadv.1601540.
https://doi.org/10.1126/sciadv.1601540

[48] Thomas Monz, Daniel Nigg, Esteban A Martinez, Matthias F Brandl, Philipp Schindler, Richard Rines, Shannon X Wang, Isaac L Chuang, and Rainer Blatt. Realization of a scalable shor algorithm. Science, 351 (6277): 1068–1070, 2016. 10.1126/science.aad9480.
https://doi.org/10.1126/science.aad9480

[49] Michael A Nielsen. Optical quantum computation using cluster states. Physical review letters, 93 (4): 040503, 2004. 10.1103/PhysRevLett.93.040503.
https://doi.org/10.1103/PhysRevLett.93.040503

[50] R. Raussendorf and J. Harrington. Fault-tolerant quantum computation with high threshold in two dimensions. Physical Review Letters, 98 (19): 190504, 2007. 10.1103/PhysRevLett.98.190504.
https://doi.org/10.1103/PhysRevLett.98.190504

[51] R. Raussendorf, J. Harrington, and K. Goyal. Topological fault-tolerance in cluster state quantum computation. New Journal of Physics, 9: 199, 2007. 10.1088/1367-2630/9/6/199.
https://doi.org/10.1088/1367-2630/9/6/199

[52] Robert Raussendorf, Jim Harrington, and Kovid Goyal. A fault-tolerant one-way quantum computer. Annals of physics, 321 (9): 2242–2270, 2006. 10.1016/j.aop.2006.01.012.
https://doi.org/10.1016/j.aop.2006.01.012

[53] Pradeep Sarvepalli and Robert Raussendorf. Efficient decoding of topological color codes. Physical Review A, 85 (2): 022317, 2012. 10.1103/PhysRevA.85.022317.
https://doi.org/10.1103/PhysRevA.85.022317

[54] Thomas M Stace and Sean D Barrett. Error correction and degeneracy in surface codes suffering loss. Physical Review A, 81 (2): 022317, 2010a. 10.1103/PhysRevA.81.022317.
https://doi.org/10.1103/PhysRevA.81.022317

[55] Thomas M Stace and Sean D Barrett. Error correction and degeneracy in surface codes suffering loss. Physical Review A, 81 (2): 022317, 2010b. 10.1103/PhysRevA.81.022317.
https://doi.org/10.1103/PhysRevA.81.022317

[56] T.M. Stace, S.D. Barrett, and A.C. Doherty. Thresholds for topological codes in the presence of loss. Physical Review Letters, 102 (20): 200501, 2009. 10.1103/PhysRevLett.102.200501.
https://doi.org/10.1103/PhysRevLett.102.200501

[57] Robert Endre Tarjan. Efficiency of a good but not linear set union algorithm. Journal of the ACM (JACM), 22 (2): 215–225, 1975. 10.1145/321879.321884.
https://doi.org/10.1145/321879.321884

[58] Robert Endre Tarjan. A class of algorithms which require nonlinear time to maintain disjoint sets. Journal of computer and system sciences, 18 (2): 110–127, 1979. 10.1016/0022-0000(79)90042-4.
https://doi.org/10.1016/0022-0000(79)90042-4

[59] Giacomo Torlai and Roger G Melko. Neural decoder for topological codes. Physical Review Letters, 119 (3): 030501, 2017. 10.1103/PhysRevLett.119.030501.
https://doi.org/10.1103/PhysRevLett.119.030501

[60] David K Tuckett, Stephen D Bartlett, and Steven T Flammia. Ultra-high error threshold for surface codes with biased noise. arXiv preprint arXiv:1708.08474, 2017. 10.1103/PhysRevLett.120.050505.
https://doi.org/10.1103/PhysRevLett.120.050505
arXiv:1708.08474

[61] Savvas Varsamopoulos, Ben Criger, and Koen Bertels. Decoding small surface codes with feedforward neural networks. arXiv preprint arXiv:1705.00857, 2017. 10.1088/2058-9565/aa955a.
https://doi.org/10.1088/2058-9565/aa955a
arXiv:1705.00857

[62] Chenyang Wang, Jim Harrington, and John Preskill. Confinement-higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory. Annals of Physics, 303 (1): 31–58, 2003. 10.1016/S0003-4916(02)00019-2.
https://doi.org/10.1016/S0003-4916(02)00019-2

[63] David S Wang, Austin G Fowler, Charles D Hill, and Lloyd Christopher L Hollenberg. Graphical algorithms and threshold error rates for the 2d colour code. Quantum Info. Comput., 10 (9): 780–802, sep 2009. ISSN 1533-7146. 10.5555/2011464.2011469.
https://doi.org/10.5555/2011464.2011469

[64] Fern HE Watson, Hussain Anwar, and Dan E Browne. Fast fault-tolerant decoder for qubit and qudit surface codes. Physical Review A, 92 (3): 032309, 2015. 10.1103/PhysRevA.92.032309.
https://doi.org/10.1103/PhysRevA.92.032309

[65] Adam C. Whiteside and Austin G. Fowler. Upper bound for loss in practical topological-cluster-state quantum computing. Phys. Rev. A, 90: 052316, Nov 2014. 10.1103/PhysRevA.90.052316.
https://doi.org/10.1103/PhysRevA.90.052316

[66] James Wootton. A simple decoder for topological codes. Entropy, 17 (4): 1946–1957, 2015. 10.3390/e17041946.
https://doi.org/10.3390/e17041946

[67] James R Wootton and Daniel Loss. High threshold error correction for the surface code. Physical review letters, 109 (16): 160503, 2012. 10.1103/PhysRevLett.109.160503.
https://doi.org/10.1103/PhysRevLett.109.160503

[68] Gilles Zémor. On cayley graphs, surface codes, and the limits of homological coding for quantum error correction. In IWCC, pages 259–273. Springer, 2009. 10.1007/978-3-642-01877-0_21.
https://doi.org/10.1007/978-3-642-01877-0_21

Cited by

[1] Shayan Srinivasa Garani, Priya J. Nadkarni, and Ankur Raina, "Theory Behind Quantum Error Correcting Codes: An Overview", Journal of the Indian Institute of Science 103 2, 449 (2023).

[2] Sara Bartolucci, Patrick Birchall, Hector Bombín, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Terry Rudolph, and Chris Sparrow, "Fusion-based quantum computation", Nature Communications 14 1, 912 (2023).

[3] Karl Hammar, Alexei Orekhov, Patrik Wallin Hybelius, Anna Katariina Wisakanto, Basudha Srivastava, Anton Frisk Kockum, and Mats Granath, "Error-rate-agnostic decoding of topological stabilizer codes", Physical Review A 105 4, 042616 (2022).

[4] Zhaoyi Li, Isaac Kim, and Patrick Hayden, "Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction", Quantum 7, 1089 (2023).

[5] Isaac H. Kim, Ye-Hua Liu, Sam Pallister, William Pol, Sam Roberts, and Eunseok Lee, "Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules", Physical Review Research 4 2, 023019 (2022).

[6] Oscar Higgott, Thomas C. Bohdanowicz, Aleksander Kubica, Steven T. Flammia, and Earl T. Campbell, "Improved Decoding of Circuit Noise and Fragile Boundaries of Tailored Surface Codes", Physical Review X 13 3, 031007 (2023).

[7] Antonio deMarti iOlius, Josu Etxezarreta Martinez, Patricio Fuentes, and Pedro M. Crespo, "Performance enhancement of surface codes via recursive minimum-weight perfect-match decoding", Physical Review A 108 2, 022401 (2023).

[8] Lucas Berent, Lukas Burgholzer, and Robert Wille, Proceedings of the 28th Asia and South Pacific Design Automation Conference 709 (2023) ISBN:9781450397834.

[9] Yun-Jia Xue, Hao-Wen Wang, Yan-Bing Tian, Yi-Nuo Wang, Yu-Xuan Wang, Shu-Mei Wang, and Liuguo Yin, "Quantum Information Protection Scheme Based on Reinforcement Learning for Periodic Surface Codes", Quantum Engineering 2022, 1 (2022).

[10] Yue Wu, Shimon Kolkowitz, Shruti Puri, and Jeff D. Thompson, "Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays", Nature Communications 13 1, 4657 (2022).

[11] Kao-Yueh Kuo and Ching-Yi Lai, "Exploiting degeneracy in belief propagation decoding of quantum codes", npj Quantum Information 8 1, 111 (2022).

[12] Seok-Hyung Lee, Srikrishna Omkar, Yong Siah Teo, and Hyunseok Jeong, "Parity-encoding-based quantum computing with Bayesian error tracking", npj Quantum Information 9 1, 39 (2023).

[13] Benjamin J. Brown, "Conservation Laws and Quantum Error Correction: Toward a Generalized Matching Decoder", IEEE BITS the Information Theory Magazine 2 3, 5 (2022).

[14] Ramon W. J. Overwater, Masoud Babaie, and Fabio Sebastiano, "Neural-Network Decoders for Quantum Error Correction Using Surface Codes: A Space Exploration of the Hardware Cost-Performance Tradeoffs", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[15] Haowen Wang, Yunjia Xue, Yingjie Qu, Xiaoyi Mu, and Hongyang Ma, "Multidimensional Bose quantum error correction based on neural network decoder", npj Quantum Information 8 1, 134 (2022).

[16] Shilin Huang, Tomas Jochym-O’Connor, and Theodore J. Yoder, "Homomorphic Logical Measurements", PRX Quantum 4 3, 030301 (2023).

[17] Aleksander Kubica and Nicolas Delfosse, "Efficient color code decoders in d≥2 dimensions from toric code decoders", Quantum 7, 929 (2023).

[18] Mingyu Kang, Wesley C. Campbell, and Kenneth R. Brown, "Quantum Error Correction with Metastable States of Trapped Ions Using Erasure Conversion", PRX Quantum 4 2, 020358 (2023).

[19] Josias Old and Manuel Rispler, "Generalized Belief Propagation Algorithms for Decoding of Surface Codes", Quantum 7, 1037 (2023).

[20] Irit Dinur, Min-Hsiu Hsieh, Ting-Chun Lin, and Thomas Vidick, Proceedings of the 55th Annual ACM Symposium on Theory of Computing 905 (2023) ISBN:9781450399135.

[21] Jun Fujisaki, Hirotaka Oshima, Shintaro Sato, and Keisuke Fujii, "Practical and scalable decoder for topological quantum error correction with an Ising machine", Physical Review Research 4 4, 043086 (2022).

[22] F Battistel, C Chamberland, K Johar, R W J Overwater, F Sebastiano, L Skoric, Y Ueno, and M Usman, "Real-time decoding for fault-tolerant quantum computing: progress, challenges and outlook", Nano Futures 7 3, 032003 (2023).

[23] Oscar Higgott, "PyMatching: A Python Package for Decoding Quantum Codes with Minimum-Weight Perfect Matching", ACM Transactions on Quantum Computing 3 3, 1 (2022).

[24] Tzu-Hsuan Huang, Ting-An Hu, and Yeong-Luh Ueng, "Branch-Assisted Sign-Flipping Belief Propagation Decoding for Topological Quantum Codes Based on Hypergraph Product Structure", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[25] Maximilian Jakob Heer, Emanuele Del Sozzo, Keisuke Fujii, and Kentaro Sano, 2023 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW) 524 (2023) ISBN:979-8-3503-1199-0.

[26] Poulami Das, Aditya Locharla, and Cody Jones, Proceedings of the 27th ACM International Conference on Architectural Support for Programming Languages and Operating Systems 541 (2022) ISBN:9781450392051.

[27] Christopher Chamberland, Luis Goncalves, Prasahnt Sivarajah, Eric Peterson, and Sebastian Grimberg, "Techniques for combining fast local decoders with global decoders under circuit-level noise", Quantum Science and Technology 8 4, 045011 (2023).

[28] Samuel C. Smith, Benjamin J. Brown, and Stephen D. Bartlett, "Local Predecoder to Reduce the Bandwidth and Latency of Quantum Error Correction", Physical Review Applied 19 3, 034050 (2023).

[29] Nicolas Delfosse, Vivien Londe, and Michael E. Beverland, "Toward a Union-Find Decoder for Quantum LDPC Codes", IEEE Transactions on Information Theory 68 5, 3187 (2022).

[30] Kai Meinerz, Chae-Yeun Park, and Simon Trebst, "Scalable Neural Decoder for Topological Surface Codes", Physical Review Letters 128 8, 080505 (2022).

[31] Spiro Gicev, Lloyd C. L. Hollenberg, and Muhammad Usman, "A scalable and fast artificial neural network syndrome decoder for surface codes", Quantum 7, 1058 (2023).

[32] Wouter Rozendaal and Gilles Zémor, 2023 IEEE International Symposium on Information Theory (ISIT) 625 (2023) ISBN:978-1-6654-7554-9.

[33] John R. Scott and Krishna C. Balram, "Timing Constraints Imposed by Classical Digital Control Systems on Photonic Implementations of Measurement-Based Quantum Computing", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[34] Pedro Parrado-Rodríguez, Manuel Rispler, and Markus Müller, "Rescaling decoder for two-dimensional topological quantum color codes on 4.8.8 lattices", Physical Review A 106 3, 032431 (2022).

[35] Yasunari Suzuki, Suguru Endo, Keisuke Fujii, and Yuuki Tokunaga, "Quantum error mitigation as a universal error-minimization technique: applications from NISQ to FTQC eras", arXiv:2010.03887, (2020).

[36] Hector Bombin, Chris Dawson, Terry Farrelly, Yehua Liu, Naomi Nickerson, Mihir Pant, Fernando Pastawski, and Sam Roberts, "Fault-tolerant complexes", arXiv:2308.07844, (2023).

[37] J. Eli Bourassa, Rafael N. Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q. Baragiola, Saikat Guha, Guillaume Dauphinais, Krishna K. Sabapathy, Nicolas C. Menicucci, and Ish Dhand, "Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer", Quantum 5, 392 (2021).

[38] Kyungjoo Noh, Christopher Chamberland, and Fernando G. S. L. Brandão, "Low-Overhead Fault-Tolerant Quantum Error Correction with the Surface-GKP Code", PRX Quantum 3 1, 010315 (2022).

[39] Nikolas P. Breuckmann and Jens Niklas Eberhardt, "Quantum Low-Density Parity-Check Codes", PRX Quantum 2 4, 040101 (2021).

[40] 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", Physical Review Letters 124 13, 130501 (2020).

[41] Héctor Bombín, Chris Dawson, Ryan V. Mishmash, Naomi Nickerson, Fernando Pastawski, and Sam Roberts, "Logical Blocks for Fault-Tolerant Topological Quantum Computation", PRX Quantum 4 2, 020303 (2023).

[42] Oscar Higgott, "PyMatching: A Python package for decoding quantum codes with minimum-weight perfect matching", arXiv:2105.13082, (2021).

[43] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G. S. L. Brandão, "Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes", PRX Quantum 3 1, 010329 (2022).

[44] Poulami Das, Christopher A. Pattison, Srilatha Manne, Douglas Carmean, Krysta Svore, Moinuddin Qureshi, and Nicolas Delfosse, "A Scalable Decoder Micro-architecture for Fault-Tolerant Quantum Computing", arXiv:2001.06598, (2020).

[45] Christopher Chamberland and Pooya Ronagh, "Deep neural decoders for near term fault-tolerant experiments", Quantum Science and Technology 3 4, 044002 (2018).

[46] Christopher Chamberland, Aleksander Kubica, Theodore J. Yoder, and Guanyu Zhu, "Triangular color codes on trivalent graphs with flag qubits", New Journal of Physics 22 2, 023019 (2020).

[47] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G. S. L. Brandão, "Building a fault-tolerant quantum computer using concatenated cat codes", arXiv:2012.04108, (2020).

[48] Muyuan Li, Daniel Miller, Michael Newman, Yukai Wu, and Kenneth R. Brown, "2D Compass Codes", Physical Review X 9 2, 021041 (2019).

[49] Michael J. Gullans, Stefan Krastanov, David A. Huse, Liang Jiang, and Steven T. Flammia, "Quantum Coding with Low-Depth Random Circuits", Physical Review X 11 3, 031066 (2021).

[50] Hector Bombin, Isaac H. Kim, Daniel Litinski, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Sam Roberts, and Terry Rudolph, "Interleaving: Modular architectures for fault-tolerant photonic quantum computing", arXiv:2103.08612, (2021).

[51] Michael Vasmer and Dan E. Browne, "Three-dimensional surface codes: Transversal gates and fault-tolerant architectures", Physical Review A 100 1, 012312 (2019).

[52] Benjamin J. Brown and Dominic J. Williamson, "Parallelized quantum error correction with fracton topological codes", Physical Review Research 2 1, 013303 (2020).

[53] Yasunari Suzuki, Suguru Endo, Keisuke Fujii, and Yuuki Tokunaga, "Quantum Error Mitigation as a Universal Error Reduction Technique: Applications from the NISQ to the Fault-Tolerant Quantum Computing Eras", PRX Quantum 3 1, 010345 (2022).

[54] Nishad Maskara, Aleksander Kubica, and Tomas Jochym-O'Connor, "Advantages of versatile neural-network decoding for topological codes", Physical Review A 99 5, 052351 (2019).

[55] Andrew S. Darmawan and David Poulin, "Linear-time general decoding algorithm for the surface code", Physical Review E 97 5, 051302 (2018).

[56] Markus S. Kesselring, Julio C. Magdalena de la Fuente, Felix Thomsen, Jens Eisert, Stephen D. Bartlett, and Benjamin J. Brown, "Anyon condensation and the color code", arXiv:2212.00042, (2022).

[57] Aleksander Kubica and John Preskill, "Cellular-Automaton Decoders with Provable Thresholds for Topological Codes", Physical Review Letters 123 2, 020501 (2019).

[58] Hendrik Poulsen Nautrup, Nicolas Delfosse, Vedran Dunjko, Hans J. Briegel, and Nicolai Friis, "Optimizing Quantum Error Correction Codes with Reinforcement Learning", Quantum 3, 215 (2019).

[59] Hendrik Poulsen Nautrup, Nicolas Delfosse, Vedran Dunjko, Hans J. Briegel, and Nicolai Friis, "Optimizing Quantum Error Correction Codes with Reinforcement Learning", arXiv:1812.08451, (2018).

[60] Namitha Liyanage, Yue Wu, Alexander Deters, and Lin Zhong, "Scalable Quantum Error Correction for Surface Codes using FPGA", arXiv:2301.08419, (2023).

[61] Shilin Huang, Michael Newman, and Kenneth R. Brown, "Fault-tolerant weighted union-find decoding on the toric code", Physical Review A 102 1, 012419 (2020).

[62] Naomi Nickerson and Héctor Bombín, "Measurement based fault tolerance beyond foliation", arXiv:1810.09621, (2018).

[63] Nicolas Delfosse, "Hierarchical decoding to reduce hardware requirements for quantum computing", arXiv:2001.11427, (2020).

[64] Xinyu Tan, Fang Zhang, Rui Chao, Yaoyun Shi, and Jianxin Chen, "Scalable surface code decoders with parallelization in time", arXiv:2209.09219, (2022).

[65] Christopher T. Chubb, "General tensor network decoding of 2D Pauli codes", arXiv:2101.04125, (2021).

[66] Christopher A. Pattison, Michael E. Beverland, Marcus P. da Silva, and Nicolas Delfosse, "Improved quantum error correction using soft information", arXiv:2107.13589, (2021).

[67] Robert J. Harris, Nathan A. McMahon, Gavin K. Brennen, and Thomas M. Stace, "Calderbank-Shor-Steane holographic quantum error-correcting codes", Physical Review A 98 5, 052301 (2018).

[68] Cupjin Huang, Xiaotong Ni, Fang Zhang, Michael Newman, Dawei Ding, Xun Gao, Tenghui Wang, Hui-Hai Zhao, Feng Wu, Gengyan Zhang, Chunqing Deng, Hsiang-Sheng Ku, Jianxin Chen, and Yaoyun Shi, "Alibaba Cloud Quantum Development Platform: Surface Code Simulations with Crosstalk", arXiv:2002.08918, (2020).

[69] Ye-Hua Liu and David Poulin, "Neural Belief-Propagation Decoders for Quantum Error-Correcting Codes", Physical Review Letters 122 20, 200501 (2019).

[70] Oscar Higgott and Nikolas P. Breuckmann, "Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead", Physical Review X 11 3, 031039 (2021).

[71] Agustin Di Paolo, Arne L. Grimsmo, Peter Groszkowski, Jens Koch, and Alexandre Blais, "Control and coherence time enhancement of the 0-π qubit", New Journal of Physics 21 4, 043002 (2019).

[72] Sam J. Griffiths and Dan E. Browne, "Union-find quantum decoding without union-find", arXiv:2306.09767, (2023).

[73] Yosuke Ueno, Masaaki Kondo, Masamitsu Tanaka, Yasunari Suzuki, and Yutaka Tabuchi, "NEO-QEC: Neural Network Enhanced Online Superconducting Decoder for Surface Codes", arXiv:2208.05758, (2022).

[74] Earl Campbell, Ankur Khurana, and Ashley Montanaro, "Applying quantum algorithms to constraint satisfaction problems", Quantum 3, 167 (2019).

[75] Michael Vasmer and Aleksander Kubica, "Morphing Quantum Codes", PRX Quantum 3 3, 030319 (2022).

[76] Nishad Maskara, Aleksander Kubica, and Tomas Jochym-O'Connor, "Advantages of versatile neural-network decoding for topological codes", arXiv:1802.08680, (2018).

[77] Michael Vasmer, Dan E. Browne, and Aleksander Kubica, "Cellular automaton decoders for topological quantum codes with noisy measurements and beyond", Scientific Reports 11, 2027 (2021).

[78] Oscar Higgott and Nikolas P. Breuckmann, "Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes", PRX Quantum 4 2, 020332 (2023).

[79] Rui Chao, Michael E. Beverland, Nicolas Delfosse, and Jeongwan Haah, "Optimization of the surface code design for Majorana-based qubits", Quantum 4, 352 (2020).

[80] Tomas Jochym-O'Connor and Theodore J. Yoder, "Four-dimensional toric code with non-Clifford transversal gates", Physical Review Research 3 1, 013118 (2021).

[81] Savvas Varsamopoulos, Koen Bertels, and Carmen G. Almudever, "Comparing neural network based decoders for the surface code", arXiv:1811.12456, (2018).

[82] G. Dauphinais, L. Ortiz, S. Varona, and M. A. Martin-Delgado, "Quantum error correction with the semion code", New Journal of Physics 21 5, 053035 (2019).

[83] Milap Sheth, Sara Zafar Jafarzadeh, and Vlad Gheorghiu, "Neural ensemble decoding for topological quantum error-correcting codes", Physical Review A 101 3, 032338 (2020).

[84] Shilin Huang and Kenneth R. Brown, "Fault-tolerant compass codes", Physical Review A 101 4, 042312 (2020).

[85] Suhas Vittal, Poulami Das, and Moinuddin Qureshi, "ERASER: Towards Adaptive Leakage Suppression for Fault-Tolerant Quantum Computing", arXiv:2309.13143, (2023).

[86] Adam Holmes, Mohammad Reza Jokar, Ghasem Pasandi, Yongshan Ding, Massoud Pedram, and Frederic T. Chong, "NISQ+: Boosting quantum computing power by approximating quantum error correction", arXiv:2004.04794, (2020).

[87] Armanda O. Quintavalle and Earl T. Campbell, "ReShape: a decoder for hypergraph product codes", arXiv:2105.02370, (2021).

[88] Robert J. Harris, Elliot Coupe, Nathan A. McMahon, Gavin K. Brennen, and Thomas M. Stace, "Decoding Holographic Codes with an Integer Optimisation Decoder", arXiv:2008.10206, (2020).

[89] Christopher A. Pattison, Anirudh Krishna, and John Preskill, "Hierarchical memories: Simulating quantum LDPC codes with local gates", arXiv:2303.04798, (2023).

[90] Naomi H. Nickerson and Benjamin J. Brown, "Analysing correlated noise on the surface code using adaptive decoding algorithms", Quantum 3, 131 (2019).

[91] Kaavya Sahay and Benjamin J. Brown, "Decoder for the Triangular Color Code by Matching on a Möbius Strip", PRX Quantum 3 1, 010310 (2022).

[92] Poulami Das, Aditya Locharla, and Cody Jones, "LILLIPUT: A Lightweight Low-Latency Lookup-Table Based Decoder for Near-term Quantum Error Correction", arXiv:2108.06569, (2021).

[93] Michael Newman, Leonardo Andreta de Castro, and Kenneth R. Brown, "Generating Fault-Tolerant Cluster States from Crystal Structures", Quantum 4, 295 (2020).

[94] Héctor Bombín, Chris Dawson, Ye-Hua Liu, Naomi Nickerson, Fernando Pastawski, and Sam Roberts, "Modular decoding: parallelizable real-time decoding for quantum computers", arXiv:2303.04846, (2023).

[95] Nicolas Delfosse, Vivien Londe, and Michael Beverland, "Toward a Union-Find decoder for quantum LDPC codes", arXiv:2103.08049, (2021).

[96] Vivien Londe and Anthony Leverrier, "Golden codes: quantum LDPC codes built from regular tessellations of hyperbolic 4-manifolds", arXiv:1712.08578, (2017).

[97] Arun B. Aloshious and Pradeep Kiran Sarvepalli, "Decoding toric codes on three dimensional simplical complexes", arXiv:1911.06056, (2019).

[98] Arun B. Aloshious and Pradeep Kiran Sarvepalli, "Projecting three-dimensional color codes onto three-dimensional toric codes", Physical Review A 98 1, 012302 (2018).

[99] Eric Sabo, Arun B. Aloshious, and Kenneth R. Brown, "Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes", arXiv:2106.08251, (2021).

[100] Robert J. Harris, Elliot Coupe, Nathan A. McMahon, Gavin K. Brennen, and Thomas M. Stace, "Decoding holographic codes with an integer optimization decoder", Physical Review A 102 6, 062417 (2020).

[101] Naomi H. Nickerson and Benjamin J. Brown, "Analysing correlated noise on the surface code using adaptive decoding algorithms", arXiv:1712.00502, (2017).

[102] Yosuke Ueno, Masaaki Kondo, Masamitsu Tanaka, Yasunari Suzuki, and Yutaka Tabuchi, "QECOOL: On-Line Quantum Error Correction with a Superconducting Decoder for Surface Code", arXiv:2103.14209, (2021).

[103] Nicolas Delfosse and Matthew B. Hastings, "Union-Find Decoders For Homological Product Codes", Quantum 5, 406 (2021).

[104] Shilin Huang and Kenneth R. Brown, "Between Shor and Steane: A Unifying Construction for Measuring Error Syndromes", Physical Review Letters 127 9, 090505 (2021).

[105] Muyuan Li and Theodore J. Yoder, "A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes", arXiv:2002.06257, (2020).

[106] James R. Cruise, Neil I. Gillespie, and Brendan Reid, "Practical Quantum Computing: The value of local computation", arXiv:2009.08513, (2020).

[107] Yaping Yuan and Chung-Chin Lu, "A Modified MWPM Decoding Algorithm for Quantum Surface Codes Over Depolarizing Channels", arXiv:2202.11239, (2022).

[108] Omar Fawzi, Antoine Grospellier, and Anthony Leverrier, "Efficient decoding of random errors for quantum expander codes", arXiv:1711.08351, (2017).

[109] Shilin Huang and Kenneth R. Brown, "Constructions for measuring error syndromes in Calderbank-Shor-Steane codes between Shor and Steane methods", Physical Review A 104 2, 022429 (2021).

[110] David Amaro, Jemma Bennett, Davide Vodola, and Markus Müller, "Analytical percolation theory for topological color codes under qubit loss", Physical Review A 101 3, 032317 (2020).

[111] Ramon Overwater, Masoud Babaie, and Fabio Sebastiano, "Neural-Network Decoders for Quantum Error Correction using Surface Codes:A Space Exploration of the Hardware Cost-Performance Trade-Offs", arXiv:2202.05741, (2022).

[112] Georgia M. Nixon and Benjamin J. Brown, "Correcting spanning errors with a fractal code", arXiv:2002.11738, (2020).

[113] Jun Fujisaki, Kazunori Maruyama, Hirotaka Oshima, Shintaro Sato, Tatsuya Sakashita, Yusaku Takeuchi, and Keisuke Fujii, "Quantum error correction with an Ising machine under circuit-level noise", arXiv:2308.00369, (2023).

[114] Alex Fischer and Akimasa Miyake, "Hardness results for decoding the surface code with Pauli noise", arXiv:2309.10331, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2023-09-30 19:12:48) and SAO/NASA ADS (last updated successfully 2023-09-30 19:12:49). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.