Analysing correlated noise on the surface code using adaptive decoding algorithms

Naomi H. Nickerson1 and Benjamin J. Brown2,3

1Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom
2Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
3Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia

Laboratory hardware is rapidly progressing towards a state where quantum error-correcting codes can be realised. As such, we must learn how to deal with the complex nature of the noise that may occur in real physical systems. Single qubit Pauli errors are commonly used to study the behaviour of error-correcting codes, but in general we might expect the environment to introduce correlated errors to a system. Given some knowledge of structures that errors commonly take, it may be possible to adapt the error-correction procedure to compensate for this noise, but performing full state tomography on a physical system to analyse this structure quickly becomes impossible as the size increases beyond a few qubits. Here we develop and test new methods to analyse blue a particular class of spatially correlated errors by making use of parametrised families of decoding algorithms. We demonstrate our method numerically using a diffusive noise model. We show that information can be learnt about the parameters of the noise model, and additionally that the logical error rates can be improved. We conclude by discussing how our method could be utilised in a practical setting blue and propose extensions of our work to study more general error models.

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[1] J. Chiaverini, D. Leibfried, T. Schaetz, M. D. Barrett, R. B. Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, R. Ozeri, and D. J. Wineland. Realization of quantum error correction. Nature, 432: 602 EP -, 12 2004. URL http:/​/​dx.doi.org/​10.1038/​nature03074.
https://doi.org/10.1038/nature03074

[2] M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf. Realization of three-qubit quantum error correction with superconducting circuits. Nature, 482: 382 EP -, 02 2012. URL http:/​/​dx.doi.org/​10.1038/​nature10786.
https://doi.org/10.1038/nature10786

[3] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O'Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and John M. Martinis. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature, 508: 500 EP -, 04 2014. URL http:/​/​dx.doi.org/​10.1038/​nature13171.
https://doi.org/10.1038/nature13171

[4] D. Nigg, M. Müller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, and R. Blatt. Quantum computations on a topologically encoded qubit. Science, 345 (6194): 302-305, 2014. 10.1126/​science.1253742. URL http:/​/​science.sciencemag.org/​content/​345/​6194/​302.
https://doi.org/10.1126/science.1253742
http:/​/​science.sciencemag.org/​content/​345/​6194/​302

[5] A. D. Córcoles, Easwar Magesan, Srikanth J. Srinivasan, Andrew W. Cross, M. Steffen, Jay M. Gambetta, and Jerry M. Chow. Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nature Communications, 6: 6979 EP -, 04 2015. URL http:/​/​dx.doi.org/​10.1038/​ncomms7979.
https://doi.org/10.1038/ncomms7979

[6] J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I. C. Hoi, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and John M. Martinis. State preservation by repetitive error detection in a superconducting quantum circuit. Nature, 519: 66 EP -, 03 2015. URL http:/​/​dx.doi.org/​10.1038/​nature14270.
https://doi.org/10.1038/nature14270

[7] Maika Takita, A. D. Córcoles, Easwar Magesan, Baleegh Abdo, Markus Brink, Andrew Cross, Jerry M. Chow, and Jay M. Gambetta. Demonstration of weight-four parity measurements in the surface code architecture. Physical Review Letters, 117 (21): 210505-, 11 2016. 10.1103/​PhysRevLett.117.210505. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.117.210505.
https://doi.org/10.1103/PhysRevLett.117.210505

[8] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Physical Review A, 52 (4): R2493-R2496, 10 1995. 10.1103/​PhysRevA.52.R2493. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.52.R2493.
https://doi.org/10.1103/PhysRevA.52.R2493

[9] A. M. Steane. Error correcting codes in quantum theory. Physical Review Letters, 77 (5): 793-797, 07 1996. 10.1103/​PhysRevLett.77.793. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.77.793.
https://doi.org/10.1103/PhysRevLett.77.793

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

[11] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics, 43 (9): 4452-4505, 2018/​10/​10 2002. 10.1063/​1.1499754. URL https:/​/​doi.org/​10.1063/​1.1499754.
https://doi.org/10.1063/1.1499754

[12] Barbara M. Terhal. Quantum error correction for quantum memories. Reviews of Modern Physics, 87 (2): 307-346, 04 2015. 10.1103/​RevModPhys.87.307. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.87.307.
https://doi.org/10.1103/RevModPhys.87.307

[13] Benjamin J. Brown, Daniel Loss, Jiannis K. Pachos, Chris N. Self, and James R. Wootton. Quantum memories at finite temperature. Reviews of Modern Physics, 88 (4): 045005-, 11 2016a. 10.1103/​RevModPhys.88.045005. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.88.045005.
https://doi.org/10.1103/RevModPhys.88.045005

[14] Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot. Roads towards fault-tolerant universal quantum computation. Nature, 549: 172 EP -, 09 2017. URL http:/​/​dx.doi.org/​10.1038/​nature23460.
https://doi.org/10.1038/nature23460

[15] David P. DiVincenzo. Fault-tolerant architectures for superconducting qubits. Phys. Scr., T137: 014020, 2009. URL https:/​/​doi.org/​10.1088/​0031-8949/​2009/​T137/​014020.
https://doi.org/10.1088/0031-8949/2009/T137/014020

[16] Martin Wosnitzka, Fabio L. Pedrocchi, and David P. DiVincenzo. Methodology for bus layout for topological quantum error correcting codes. EPJ Quantum Technology, 3 (1): 4, 2016. 10.1140/​epjqt/​s40507-016-0042-8. URL https:/​/​doi.org/​10.1140/​epjqt/​s40507-016-0042-8.
https://doi.org/10.1140/epjqt/s40507-016-0042-8

[17] Sergey Bravyi and Alexander Vargo. Simulation of rare events in quantum error correction. Physical Review A, 88 (6): 062308-, 12 2013. 10.1103/​PhysRevA.88.062308. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.88.062308.
https://doi.org/10.1103/PhysRevA.88.062308

[18] Fern H. E. Watson and Sean D. Barrett. Logical error rate scaling of the toric code. New J. Phys., 16: 093045, 2014. URL https:/​/​doi.org/​10.1088/​1367-2630/​16/​9/​093045.
https://doi.org/10.1088/1367-2630/16/9/093045

[19] Benjamin J. Brown, Naomi H. Nickerson, and Dan E. Browne. Fault-tolerant error correction with the gauge color code. Nature Communications, 7: 12302 EP -, 07 2016b. URL http:/​/​dx.doi.org/​10.1038/​ncomms12302.
https://doi.org/10.1038/ncomms12302

[20] Andrew J. Landahl, Jonas T. Anderson, and Patrick R. Rice. Fault-tolerant quantum computing with color codes. arXiv:1108.5738, 2011. URL https:/​/​arxiv.org/​abs/​1108.5738.
arXiv:1108.5738

[21] H. G. Katzgraber and R. S. Andrist. Stability of topologically-protected quantum computing proposals as seen through spin glasses. J. Phys.: Conf. Ser., 473: 012019, 2013. URL https:/​/​doi.org/​10.1088/​1742-6596/​473/​1/​012019.
https://doi.org/10.1088/1742-6596/473/1/012019

[22] Andrew S. Darmawan and David Poulin. Tensor-network simulations of the surface code under realistic noise. Physical Review Letters, 119 (4): 040502-, 07 2017. 10.1103/​PhysRevLett.119.040502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.119.040502.
https://doi.org/10.1103/PhysRevLett.119.040502

[23] Sergey Bravyi, Matthias Englbrecht, Robert König, and Nolan Peard. Correcting coherent errors with surface codes. npj Quantum Information, 4: 55, 2018. 10.1038/​s41534-018-0106-y. URL https:/​/​doi.org/​10.1038/​s41534-018-0106-y.
https://doi.org/10.1038/s41534-018-0106-y

[24] Pavithran S. Iyer and David Poulin. A small quantum computer is needed to optimize fault-tolerant protocols. arXiv:1711.04736, 2017. URL https:/​/​arxiv.org/​abs/​1711.04736.
arXiv:1711.04736

[25] Dorit Aharonov, Alexei Kitaev, and John Preskill. Fault-tolerant quantum computation with long-range correlated noise. Physical Review Letters, 96 (5): 050504-, 02 2006. 10.1103/​PhysRevLett.96.050504. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.96.050504.
https://doi.org/10.1103/PhysRevLett.96.050504

[26] Hui Khoon Ng and John Preskill. Fault-tolerant quantum computation versus gaussian noise. Physical Review A, 79 (3): 032318-, 03 2009. 10.1103/​PhysRevA.79.032318. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.79.032318.
https://doi.org/10.1103/PhysRevA.79.032318

[27] John Preskill. Sufficient condition on noise correlations for scalable quantum computing. Quant. Inf. Comp., 13: 181, 2013. URL https:/​/​doi.org/​10.26421/​QIC13.3-4.
https://doi.org/10.26421/QIC13.3-4

[28] Pejman Jouzdani, E. Novais, I. S. Tupitsyn, and Eduardo R. Mucciolo. Fidelity threshold of the surface code beyond single-qubit error models. Physical Review A, 90 (4): 042315-, 10 2014. 10.1103/​PhysRevA.90.042315. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.90.042315.
https://doi.org/10.1103/PhysRevA.90.042315

[29] Adrian Hutter and Daniel Loss. Breakdown of surface-code error correction due to coupling to a bosonic bath. Physical Review A, 89 (4): 042334-, 04 2014. 10.1103/​PhysRevA.89.042334. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.89.042334.
https://doi.org/10.1103/PhysRevA.89.042334

[30] Austin G. Fowler and John M. Martinis. Quantifying the effects of local many-qubit errors and nonlocal two-qubit errors on the surface code. Physical Review A, 89 (3): 032316-, 03 2014. 10.1103/​PhysRevA.89.032316. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.89.032316.
https://doi.org/10.1103/PhysRevA.89.032316

[31] E. Novais, A. J. Stanforth, and Eduardo R. Mucciolo. Surface code fidelity at finite temperatures. Physical Review A, 95 (4): 042339-, 04 2017. 10.1103/​PhysRevA.95.042339. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.95.042339.
https://doi.org/10.1103/PhysRevA.95.042339

[32] Joe O'Gorman, Naomi H Nickerson, Philipp Ross, John JL Morton, and Simon C Benjamin. A silicon-based surface code quantum computer. Npj Quantum Information, 2: 15019 EP -, 02 2016. URL https:/​/​doi.org/​10.1038/​npjqi.2015.19.
https://doi.org/10.1038/npjqi.2015.19

[33] R Alicki, M Fannes, and M Horodecki. On thermalization in kitaev's 2d model. Journal of Physics A: Mathematical and Theoretical, 42 (6): 065303, 2009. 10.1088/​1751-8113/​42/​6/​065303. URL https:/​/​doi.org/​10.1088/​1751-8113/​42/​6/​065303.
https://doi.org/10.1088/1751-8113/42/6/065303

[34] Stefano Chesi, Beat Röthlisberger, and Daniel Loss. Self-correcting quantum memory in a thermal environment. Physical Review A, 82 (2): 022305-, 08 2010. 10.1103/​PhysRevA.82.022305. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.82.022305.
https://doi.org/10.1103/PhysRevA.82.022305

[35] E. Novais and Eduardo R. Mucciolo. Surface code threshold in the presence of correlated errors. Physical Review Letters, 110 (1): 010502-, 01 2013. 10.1103/​PhysRevLett.110.010502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.110.010502.
https://doi.org/10.1103/PhysRevLett.110.010502

[36] P. Jouzdani, E. Novais, and E. R. Mucciolo. Fidelity of the surface code in the presence of a bosonic bath. Physical Review A, 88 (1): 012336-, 07 2013. 10.1103/​PhysRevA.88.012336. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.88.012336.
https://doi.org/10.1103/PhysRevA.88.012336

[37] C. Daniel Freeman, C. M. Herdman, D. J. Gorman, and K. B. Whaley. Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low-temperature regime. Physical Review B, 90 (13): 134302-, 10 2014. 10.1103/​PhysRevB.90.134302. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.90.134302.
https://doi.org/10.1103/PhysRevB.90.134302

[38] D. A. López-Delgado, E. Novais, E. R. Mucciolo, and A. O. Caldeira. Long-time efficacy of the surface code in the presence of a super-ohmic environment. Physical Review A, 95 (6): 062328-, 06 2017. 10.1103/​PhysRevA.95.062328. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.95.062328.
https://doi.org/10.1103/PhysRevA.95.062328

[39] Dara P. S. McCutcheon, Netanel H. Lindner, and Terry Rudolph. Error distributions on large entangled states with non-markovian dynamics. Physical Review Letters, 113 (26): 260503-, 12 2014. 10.1103/​PhysRevLett.113.260503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.113.260503.
https://doi.org/10.1103/PhysRevLett.113.260503

[40] R. Raussendorf, J. Harrington, and K. Goyal. A fault-tolerant one-way quantum computer. Annals of Physics, 321 (9): 2242-2270, 2006. 10.1016/​j.aop.2006.01.012. URL https:/​/​doi.org/​10.1016/​j.aop.2006.01.012.
https://doi.org/10.1016/j.aop.2006.01.012

[41] Austin G. Fowler, Ashley M. Stephens, and Peter Groszkowski. High-threshold universal quantum computation on the surface code. Physical Review A, 80 (5): 052312-, 11 2009. 10.1103/​PhysRevA.80.052312. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.80.052312.
https://doi.org/10.1103/PhysRevA.80.052312

[42] Austin G. Fowler, Adam C. Whiteside, Angus L. McInnes, and Alimohammad Rabbani. Topological code autotune. Physical Review X, 2 (4): 041003-, 10 2012. 10.1103/​PhysRevX.2.041003. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.2.041003.
https://doi.org/10.1103/PhysRevX.2.041003

[43] Naomi H. Nickerson, Ying Li, and Simon C. Benjamin. Topological quantum computing with a very noisy network and local error rates approaching one percent. Nature Communications, 4: 1756 EP -, 04 2013. URL https:/​/​doi.org/​10.1038/​ncomms2773.
https://doi.org/10.1038/ncomms2773

[44] Yu Tomita and Krysta M. Svore. Low-distance surface codes under realistic quantum noise. Physical Review A, 90 (6): 062320-, 12 2014. 10.1103/​PhysRevA.90.062320. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.90.062320.
https://doi.org/10.1103/PhysRevA.90.062320

[45] Sergey Bravyi and Andrew Cross. Doubled color codes. arXiv:1509.03239, 2015. URL https:/​/​arxiv.org/​abs/​1509.03239.
arXiv:1509.03239

[46] Christopher T. Chubb and Steven T. Flammia. Statistical mechanical models for quantum codes with correlated noise. arXiv:1809.10704, 2018. URL https:/​/​arxiv.org/​abs/​1809.10704.
arXiv:1809.10704

[47] Jack Edmonds. Paths, trees and flowers. Canad. J. Math., 17: 449, 1965. URL https:/​/​doi.org/​10.4153/​CJM-1965-045-4.
https://doi.org/10.4153/CJM-1965-045-4

[48] 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. URL https:/​/​doi.org/​10.1007/​s12532-009-0002-8.
https://doi.org/10.1007/s12532-009-0002-8

[49] Joshua Combes, Christopher Ferrie, Chris Cesare, Markus Tiersch, Gerard J. Milburn, Hans J. Briegel, and Carlton M. Caves. In-situ characterization of quantum devices with error correction. arXiv:1405.5656, 2014. URL https:/​/​arxiv.org/​abs/​1405.5656.
arXiv:1405.5656

[50] David P. DiVincenzo and Peter W. Shor. Fault-tolerant error correction with efficient quantum codes. Physical Review Letters, 77 (15): 3260-3263, 10 1996. 10.1103/​PhysRevLett.77.3260. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.77.3260.
https://doi.org/10.1103/PhysRevLett.77.3260

[51] David P. DiVincenzo, Peter W. Shor, and John A. Smolin. Quantum-channel capacity of very noisy channels. Physical Review A, 57 (2): 830-839, 02 1998. 10.1103/​PhysRevA.57.830. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.57.830.
https://doi.org/10.1103/PhysRevA.57.830

[52] Ming-Xia Huo and Ying Li. Learning time-dependent noise to reduce logical errors: real time error rate estimation in quantum error correction. New Journal of Physics, 19: 123032, 2017. 10.1088/​1367-2630/​aa916e. URL https:/​/​doi.org/​10.1088/​1367-2630/​aa916e.
https://doi.org/10.1088/1367-2630/aa916e

[53] J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Lucero, M. Neeley, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, and John M. Martinis. Scalable in situ qubit calibration during repetitive error detection. Physical Review A, 94 (3): 032321-, 09 2016. 10.1103/​PhysRevA.94.032321. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.94.032321.
https://doi.org/10.1103/PhysRevA.94.032321

[54] S. T. Spitz, B. Tarasinkski, C. W. J. Beenakker, and T. E. O'Brien. Adaptive weight estimator for quantum error correction in a time-dependent environment. Adv. Quant. Tech., 1: 1800012, 2018. URL https:/​/​doi.org/​10.1002/​qute.201800012.
https://doi.org/10.1002/qute.201800012

[55] Guillaume Duclos-Cianci and David Poulin. Fast decoders for topological quantum codes. Physical Review Letters, 104 (5): 050504-, 02 2010. 10.1103/​PhysRevLett.104.050504. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.104.050504.
https://doi.org/10.1103/PhysRevLett.104.050504

[56] James R. Wootton and Daniel Loss. High threshold error correction for the surface code. Physical Review Letters, 109 (16): 160503-, 10 2012. 10.1103/​PhysRevLett.109.160503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.109.160503.
https://doi.org/10.1103/PhysRevLett.109.160503

[57] Austin G. Fowler. Optimal complexity correction of correlated errors in the surface code. arXiv:1310.0863, 2013.
arXiv:1310.0863

[58] N. Delfosse and J. Tillich. A decoding algorithm for css codes using the x/​z correlations. In 2014 IEEE International Symposium on Information Theory, pages 1071-1075, 2014. ISBN 2157-8117. 10.1109/​ISIT.2014.6874997.
https://doi.org/10.1109/ISIT.2014.6874997

[59] Ben Criger and Imran Ashraf. Multi-path summation for deocding 2D topological codes. Quantum, 2: 102, 2017. 10.22331/​q-2018-10-19-102. URL https:/​/​doi.org/​10.22331/​q-2018-10-19-102.
https://doi.org/10.22331/q-2018-10-19-102

[60] Savvas Varsamopoulos, Ben Criger, and Koen Bertels. Decoding small surface codes with feedforward neural networks. Quantum Science and Technology, 3 (1): 015004, 2017. 10.1088/​2058-9565/​aa955a. URL http:/​/​dx.doi.org/​10.1088/​2058-9565/​aa955a.
https://doi.org/10.1088/2058-9565/aa955a

[61] Stefan Krastanov and Liang Jiang. Deep neural network probabilistic decoder for stabilizer codes. Scientific Reports, 7 (1): 11003, 2017. 10.1038/​s41598-017-11266-1. URL https:/​/​doi.org/​10.1038/​s41598-017-11266-1.
https://doi.org/10.1038/s41598-017-11266-1

[62] P. Baireuther, T. E. O'Brien, B. Tarasinkski, and C. W. J. Beenakker. Machine-learning-assisted correction of correlated qubit errors in a topological code. Quantum, 2: 48, 2018. 10.22331/​q-2018-01-29-48. URL https:/​/​doi.org/​10.22331/​q-2018-01-29-48.
https://doi.org/10.22331/q-2018-01-29-48

[63] Mishad Maskara, Aleksander Kubica, and Tomas Jochym-O'Connor. Advantages of versatile neural-network decoding for topological codes. arXiv:1802.08680, 2018.
arXiv:1802.08680

[64] Daniel Gottesman. Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology, 1997.

[65] Thomas M. Stace and Sean D. Barrett. Error correction and degeneracy in surface codes suffering loss. Physical Review A, 81 (2): 022317-, 02 2010. 10.1103/​PhysRevA.81.022317. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.81.022317.
https://doi.org/10.1103/PhysRevA.81.022317

[66] Sergey Bravyi, Martin Suchara, and Alexander Vargo. Efficient algorithms for maximum likelihood decoding in the surface code. Physical Review A, 90 (3): 032326-, 09 2014. 10.1103/​PhysRevA.90.032326. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.90.032326.
https://doi.org/10.1103/PhysRevA.90.032326

[67] H. Nishimori. Geometry-induced phase transition in the $\pm j$ Ising model. J. Phys. Soc. Jpn., 55: 3305, 1986. 10.1143/​JPSJ.55.3305. URL https:/​/​doi.org/​10.1143/​JPSJ.55.3305.
https://doi.org/10.1143/JPSJ.55.3305

[68] Héctor Bombín. Resilience to time-correlated noise in quantum computation. Physical Review X, 6 (4): 041034-, 11 2016. 10.1103/​PhysRevX.6.041034. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.6.041034.
https://doi.org/10.1103/PhysRevX.6.041034

[69] Héctor Bombín. Single-shot fault-tolerant quantum error correction. Phys. Rev. X, 5: 031043, 2015. 10.1103/​PhysRevX.5.031043. URL https:/​/​doi.org/​10.1103/​PhysRevX.5.031043.
https://doi.org/10.1103/PhysRevX.5.031043

[70] Shota Nagayama, Austin G Fowler, Dominic Horsman, Simon J Devitt, and Rodney Van Meter. Surface code error correction on a defective lattice. New Journal of Physics, 19 (2): 023050, 2017. 10.1088/​1367-2630/​aa5918. URL http:/​/​dx.doi.org/​10.1088/​1367-2630/​aa5918.
https://doi.org/10.1088/1367-2630/aa5918

[71] James M. Auger, Hussain Anwar, Mercedes Gimeno-Segovia, Thomas M. Stace, and Dan E. Browne. Fault-tolerance thresholds for the surface code with fabrication errors. Physical Review A, 96 (4): 042316-, 10 2017. 10.1103/​PhysRevA.96.042316. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.96.042316.
https://doi.org/10.1103/PhysRevA.96.042316

[72] Adrian Hutter, James R. Wootton, and Daniel Loss. Efficient markov chain monte carlo algorithm for the surface code. Physical Review A, 89 (2): 022326-, 02 2014. 10.1103/​PhysRevA.89.022326. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.89.022326.
https://doi.org/10.1103/PhysRevA.89.022326

[73] 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. URL http:/​/​dx.doi.org/​10.1088/​1367-2630/​16/​6/​063038.
https://doi.org/10.1088/1367-2630/16/6/063038

[74] 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. URL http:/​/​dx.doi.org/​10.1088/​1367-2630/​17/​3/​035017.
https://doi.org/10.1088/1367-2630/17/3/035017

[75] Nicolas Delfosse and Naomi H. Nickerson. Almost-linear time decoding algorithm for topological codes. arXiv:1709.06218, 2017.
arXiv:1709.06218

[76] David K. Tuckett, Stephen D. Bartlett, and Steven T. Flammia. Ultrahigh error threshold for surface codes with biased noise. Physical Review Letters, 120 (5): 050505-, 01 2018. 10.1103/​PhysRevLett.120.050505. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.120.050505.
https://doi.org/10.1103/PhysRevLett.120.050505

[77] David S. Wang, Austin G. Fowler, Charles D. Hill, and Lloyd C. L. Hollenberg. Graphical algorithms and threshold error rates for the 2D color code. Quant. Inf. Comp., 10: 0780, 2010. 10.26421/​QIC10.9-10. URL https:/​/​doi.org/​10.26421/​QIC10.9-10.
https://doi.org/10.26421/QIC10.9-10

[78] Nicolas Delfosse. Decoding color codes by projection onto surface codes. Physical Review A, 89 (1): 012317-, 01 2014. 10.1103/​PhysRevA.89.012317. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.89.012317.
https://doi.org/10.1103/PhysRevA.89.012317

[79] Ashley M. Stephens. Efficient fault-tolerant decoding of topological color codes. arXiv:1402.3037, 2014.
arXiv:1402.3037

[80] 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. https:/​/​doi.org/​10.1016/​S0003-4916(02)00019-2. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0003491602000192.
https://doi.org/https://doi.org/10.1016/S0003-4916(02)00019-2
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0003491602000192

[81] Giacomo Torlai and Roger G. Melko. Neural decoder for topological codes. Physical Review Letters, 119 (3): 030501-, 07 2017. 10.1103/​PhysRevLett.119.030501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.119.030501.
https://doi.org/10.1103/PhysRevLett.119.030501

Cited by

[1] Muyuan Li, Daniel Miller, Michael Newman, Yukai Wu, and Kenneth R. Brown, "2D Compass Codes", arXiv:1809.01193, Physical Review X 9 2, 021041 (2019).

[2] Nishad Maskara, Aleksander Kubica, and Tomas Jochym-O'Connor, "Advantages of versatile neural-network decoding for topological codes", arXiv:1802.08680, Physical Review A 99 5, 052351 (2019).

[3] Aleksander Kubica and John Preskill, "Cellular-automaton decoders with provable thresholds for topological codes", arXiv:1809.10145.

[4] Benjamin J. Brown and Dominic J. Williamson, "Parallelized quantum error correction with fracton topological codes", arXiv:1901.08061.

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

[6] Christopher T. Chubb and Steven T. Flammia, "Statistical mechanical models for quantum codes with correlated noise", arXiv:1809.10704.

[7] Benjamin J. Brown, "A fault-tolerant non-Clifford gate for the surface code in two dimensions", arXiv:1903.11634.

The above citations are from Crossref's cited-by service (last updated 2019-06-18 13:00:02) and SAO/NASA ADS (last updated 2019-06-18 13:00:03). The list may be incomplete as not all publishers provide suitable and complete citation data.