Generating Fault-Tolerant Cluster States from Crystal Structures

Michael Newman1, Leonardo Andreta de Castro1,2, and Kenneth R. Brown1

1Departments of Electrical and Computer Engineering, Chemistry, and Physics, Duke University, Durham, NC, 27708, USA
2Q-CTRL Pty Ltd, Sydney, NSW, Australia

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Abstract

Measurement-based quantum computing (MBQC) is a promising alternative to traditional circuit-based quantum computing predicated on the construction and measurement of cluster states. Recent work has demonstrated that MBQC provides a more general framework for fault-tolerance that extends beyond foliated quantum error-correcting codes. We systematically expand on that paradigm, and use combinatorial tiling theory to study and construct new examples of fault-tolerant cluster states derived from crystal structures. Included among these is a robust self-dual cluster state requiring only degree-$3$ connectivity. We benchmark several of these cluster states in the presence of circuit-level noise, and find a variety of promising candidates whose performance depends on the specifics of the noise model. By eschewing the distinction between data and ancilla, this malleable framework lays a foundation for the development of creative and competitive fault-tolerance schemes beyond conventional error-correcting codes.

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► References

[1] Panos Aliferis, Daniel Gottesman, and John Preskill. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Information & Computation, 6 (2): 97–165, 2006.

[2] Emanuel Knill, Raymond Laflamme, and W Zurek. Threshold accuracy for quantum computation. arXiv preprint quant-ph/​9610011, 1996.
arXiv:quant-ph/9610011

[3] Dorit Aharonov and Michael Ben-Or. Fault-tolerant quantum computation with constant error. In Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, pages 176–188. ACM, 1997. https:/​/​doi.org/​10.1145/​258533.258579.
https:/​/​doi.org/​10.1145/​258533.258579

[4] Barbara M Terhal and Guido Burkard. Fault-tolerant quantum computation for local non-markovian noise. Physical Review A, 71 (1): 012336, 2005. https:/​/​doi.org/​10.1103/​physreva.71.012336.
https:/​/​doi.org/​10.1103/​physreva.71.012336

[5] Panos Aliferis and Barbara M Terhal. Fault-tolerant quantum computation for local leakage faults. Quantum Information & Computation, 7 (4): 139–156, 2007.

[6] Emanuel Knill. Quantum computing with realistically noisy devices. Nature, 434 (7029): 39, 2005. https:/​/​doi.org/​10.1038/​nature03350.
https:/​/​doi.org/​10.1038/​nature03350

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

[8] Robert Raussendorf and Jim Harrington. Fault-tolerant quantum computation with high threshold in two dimensions. Physical Review Letters, 98 (19): 190504, 2007. https:/​/​doi.org/​10.1103/​physrevlett.98.190504.
https:/​/​doi.org/​10.1103/​physrevlett.98.190504

[9] 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.
https:/​/​doi.org/​10.1016/​s0003-4916(02)00019-2

[10] Takuya Ohno, Gaku Arakawa, Ikuo Ichinose, and Tetsuo Matsui. Phase structure of the random-plaquette z2 gauge model: accuracy threshold for a toric quantum memory. Nuclear Physics B, 697 (3): 462–480, 2004. https:/​/​doi.org/​10.1016/​j.nuclphysb.2004.07.003.
https:/​/​doi.org/​10.1016/​j.nuclphysb.2004.07.003

[11] DS Wang, AG Fowler, AM Stephens, and LCL Hollenberg. Threshold error rates for the toric and planar codes. Quantum Information & Computation, 10 (5): 456–469, 2010.

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

[13] Austin G Fowler, Adam C Whiteside, and Lloyd CL Hollenberg. Towards practical classical processing for the surface code. Physical Review Letters, 108 (18): 180501, 2012. https:/​/​doi.org/​10.1103/​physrevlett.108.180501.
https:/​/​doi.org/​10.1103/​physrevlett.108.180501

[14] Martin Suchara, Andrew W Cross, and Jay M Gambetta. Leakage suppression in the toric code. Quantum Information & Computation, 15 (11-12): 997–1016, 2015a. https:/​/​doi.org/​10.1109/​isit.2015.7282629.
https:/​/​doi.org/​10.1109/​isit.2015.7282629

[15] David S Wang, Austin G Fowler, and Lloyd CL Hollenberg. Surface code quantum computing with error rates over 1%. Physical Review A, 83 (2): 020302, 2011. https:/​/​doi.org/​10.1103/​physreva.83.020302.
https:/​/​doi.org/​10.1103/​physreva.83.020302

[16] Ashley M Stephens, William J Munro, and Kae Nemoto. High-threshold topological quantum error correction against biased noise. Physical Review A, 88 (6): 060301, 2013. https:/​/​doi.org/​10.1103/​PhysRevA.88.060301.
https:/​/​doi.org/​10.1103/​PhysRevA.88.060301

[17] Peter W Shor. Fault-tolerant quantum computation. In Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on, pages 56–65. IEEE, 1996. https:/​/​doi.org/​10.1109/​sfcs.1996.548464.
https:/​/​doi.org/​10.1109/​sfcs.1996.548464

[18] Peter W Shor. Scheme for reducing decoherence in quantum computer memory. Physical Review A, 52 (4): R2493, 1995. https:/​/​doi.org/​10.1103/​physreva.52.r2493.
https:/​/​doi.org/​10.1103/​physreva.52.r2493

[19] A Robert Calderbank and Peter W Shor. Good quantum error-correcting codes exist. Physical Review A, 54 (2): 1098, 1996. https:/​/​doi.org/​10.1103/​PhysRevA.54.1098.
https:/​/​doi.org/​10.1103/​PhysRevA.54.1098

[20] Andrew M Steane. Error correcting codes in quantum theory. Physical Review Letters, 77 (5): 793, 1996. https:/​/​doi.org/​10.1103/​physrevlett.77.793.
https:/​/​doi.org/​10.1103/​physrevlett.77.793

[21] Andrew M Steane. Active stabilization, quantum computation, and quantum state synthesis. Physical Review Letters, 78 (11): 2252, 1997. https:/​/​doi.org/​10.1103/​physrevlett.78.2252.
https:/​/​doi.org/​10.1103/​physrevlett.78.2252

[22] Daniel Gottesman. Stabilizer codes and quantum error correction. arXiv preprint quant-ph/​9705052, 1997.
arXiv:quant-ph/9705052

[23] Robert Raussendorf, Daniel E Browne, and Hans J Briegel. Measurement-based quantum computation on cluster states. Physical Review A, 68 (2): 022312, 2003. https:/​/​doi.org/​10.1103/​physreva.68.022312.
https:/​/​doi.org/​10.1103/​physreva.68.022312

[24] Hans J Briegel, David E Browne, Wolfgang Dür, Robert Raussendorf, and Maarten Van den Nest. Measurement-based quantum computation. Nature Physics, 5 (1): 19, 2009. https:/​/​doi.org/​10.1038/​nphys1157.
https:/​/​doi.org/​10.1038/​nphys1157

[25] Robert Raussendorf, Jim Harrington, and Kovid Goyal. Topological fault-tolerance in cluster state quantum computation. New Journal of Physics, 9 (6): 199, 2007. https:/​/​doi.org/​10.1088/​1367-2630/​9/​6/​199.
https:/​/​doi.org/​10.1088/​1367-2630/​9/​6/​199

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

[27] Robert Raussendorf, Sergey Bravyi, and Jim Harrington. Long-range quantum entanglement in noisy cluster states. Physical Review A, 71 (6): 062313, 2005. https:/​/​doi.org/​10.1103/​physreva.71.062313.
https:/​/​doi.org/​10.1103/​physreva.71.062313

[28] A Bolt, G Duclos-Cianci, D Poulin, and TM Stace. Foliated quantum error-correcting codes. Physical Review Letters, 117 (7): 070501, 2016. https:/​/​doi.org/​10.1103/​physrevlett.117.070501.
https:/​/​doi.org/​10.1103/​physrevlett.117.070501

[29] A Bolt, D Poulin, and TM Stace. Decoding schemes for foliated sparse quantum error-correcting codes. Physical Review A, 98 (6): 062302, 2018. https:/​/​doi.org/​10.1103/​physreva.98.062302.
https:/​/​doi.org/​10.1103/​physreva.98.062302

[30] Benjamin Brown and Sam Roberts. Universal fault-tolerant measurement-based quantum computation. arXiv preprint arXiv:1811.11780, 2018.
arXiv:1811.11780

[31] Naomi Nickerson and Héctor Bombín. Measurement based fault tolerance beyond foliation. arXiv preprint arXiv:1810.09621, 2018.
arXiv:1810.09621

[32] John Nguyen, John Pezaris, Gill Pratt, and Steve Ward. Three-dimensional network topologies. In International Workshop on Parallel Computer Routing and Communication, pages 101–115. Springer, 1994. https:/​/​doi.org/​10.1007/​3-540-58429-3_31.
https:/​/​doi.org/​10.1007/​3-540-58429-3_31

[33] Behrooz Parhami and Ding-Ming Kwai. A unified formulation of honeycomb and diamond networks. IEEE Transactions on Parallel and Distributed Systems, 12 (1): 74–80, 2001. https:/​/​doi.org/​10.1109/​71.899940.
https:/​/​doi.org/​10.1109/​71.899940

[34] Sean D Barrett and Thomas M Stace. Fault tolerant quantum computation with very high threshold for loss errors. Physical Review Letters, 105 (20): 200502, 2010. https:/​/​doi.org/​10.1103/​physrevlett.105.200502.
https:/​/​doi.org/​10.1103/​physrevlett.105.200502

[35] Andreas WM Dress. Presentations of discrete groups, acting on simply connected manifolds, in terms of parametrized systems of coxeter matrices—a systematic approach. Advances in Mathematics, 63 (2): 196–212, 1987. https:/​/​doi.org/​10.1016/​0001-8708(87)90053-3.
https:/​/​doi.org/​10.1016/​0001-8708(87)90053-3

[36] Olaf Delgado-Friedrichs. Recognition of flat orbifolds and the classification of tilings in R3. Discrete & Computational Geometry, 26 (4): 549–571, 2001. https:/​/​doi.org/​10.1007/​s00454-001-0022-2.
https:/​/​doi.org/​10.1007/​s00454-001-0022-2

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

[38] Shilin Huang, Michael Newman, and Kenneth R Brown. Fault-tolerant weighted union-find decoding on the toric code. arXiv preprint arXiv:2004.04693, 2020.
arXiv:2004.04693

[39] Christopher T Chubb and Steven T Flammia. Statistical mechanical models for quantum codes with correlated noise. arXiv preprint arXiv:1809.10704, 2018.
arXiv:1809.10704

[40] Hector Bombín and Miguel A Martin-Delgado. Homological error correction: Classical and quantum codes. Journal of Mathematical Physics, 48 (5): 052105, 2007. https:/​/​doi.org/​10.1063/​1.2731356.
https:/​/​doi.org/​10.1063/​1.2731356

[41] A Yu Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303 (1): 2–30, 2003. https:/​/​doi.org/​10.1016/​s0003-4916(02)00018-0.
https:/​/​doi.org/​10.1016/​s0003-4916(02)00018-0

[42] Michael H Freedman, David A Meyer, and Feng Luo. Z2-systolic freedom and quantum codes. In Mathematics of Quantum Computation, pages 287–320. CRC Press, 2002. https:/​/​doi.org/​10.1201/​9781420035377.ch12.
https:/​/​doi.org/​10.1201/​9781420035377.ch12

[43] Allen Hatcher. Algebraic Topology. Cambridge Univ. Press, Cambridge, 2000. URL https:/​/​cds.cern.ch/​record/​478079.
https:/​/​cds.cern.ch/​record/​478079

[44] Sergey Bravyi and Matthew B Hastings. Homological product codes. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, pages 273–282. ACM, 2014. https:/​/​doi.org/​10.1145/​2591796.2591870.
https:/​/​doi.org/​10.1145/​2591796.2591870

[45] Matthew B Hastings. Quantum codes from high-dimensional manifolds. arXiv preprint arXiv:1608.05089, 2016.
arXiv:1608.05089

[46] Vivien Londe and Anthony Leverrier. Golden codes: quantum LDPC codes built from regular tessellations of hyperbolic 4-manifolds. Quantum Information & Computation, 19 (5-6): 361–391, 2019.

[47] Benjamin Audoux and Alain Couvreur. On tensor products of CSS codes. Annales de l’Institut Henri Poincaré D, 2019. https:/​/​doi.org/​10.4171/​aihpd/​71.
https:/​/​doi.org/​10.4171/​aihpd/​71

[48] Earl Campbell. A theory of single-shot error correction for adversarial noise. Quantum Science and Technology, 4: 025006, 2019. https:/​/​doi.org/​10.1088/​2058-9565/​aafc8f.
https:/​/​doi.org/​10.1088/​2058-9565/​aafc8f

[49] Héctor Bombín. Single-shot fault-tolerant quantum error correction. Physical Review X, 5 (3): 031043, 2015. https:/​/​doi.org/​10.1103/​physrevx.5.031043.
https:/​/​doi.org/​10.1103/​physrevx.5.031043

[50] Tomas Jochym-O'Connor. Fault-tolerant gates via homological product codes. Quantum, 3: Art–No, 2019. https:/​/​doi.org/​10.22331/​q-2019-02-04-120.
https:/​/​doi.org/​10.22331/​q-2019-02-04-120

[51] Anirudh Krishna and David Poulin. Fault-tolerant gates on hypergraph product codes. arXiv preprint arXiv:1909.07424, 2019.
arXiv:1909.07424

[52] Héctor Bombín, Ruben S Andrist, Masayuki Ohzeki, Helmut G Katzgraber, and Miguel A Martín-Delgado. Strong resilience of topological codes to depolarization. Physical Review X, 2 (2): 021004, 2012. https:/​/​doi.org/​10.1103/​physrevx.2.021004.
https:/​/​doi.org/​10.1103/​physrevx.2.021004

[53] Muyuan Li, Daniel Miller, Michael Newman, Yukai Wu, and Kenneth R Brown. 2D compass codes. Physical Review X, 9 (2): 021041, 2019. https:/​/​doi.org/​10.1103/​physrevx.9.021041.
https:/​/​doi.org/​10.1103/​physrevx.9.021041

[54] Aleksander Kubica and Beni Yoshida. Ungauging quantum error-correcting codes. arXiv preprint arXiv:1805.01836, 2018.
arXiv:1805.01836

[55] Austin K Daniel, Rafael N Alexander, and Akimasa Miyake. Computational universality of symmetry-protected topologically ordered cluster phases on 2d archimedean lattices. Quantum, 4: 228, 2020. https:/​/​doi.org/​10.22331/​q-2020-02-10-228.
https:/​/​doi.org/​10.22331/​q-2020-02-10-228

[56] Panos Aliferis, Frederico Brito, David P DiVincenzo, John Preskill, Matthias Steffen, and Barbara M Terhal. Fault-tolerant computing with biased-noise superconducting qubits: a case study. New Journal of Physics, 11 (1): 013061, 2009. https:/​/​doi.org/​10.1088/​1367-2630/​11/​1/​013061.
https:/​/​doi.org/​10.1088/​1367-2630/​11/​1/​013061

[57] Panos Aliferis and John Preskill. Fault-tolerant quantum computation against biased noise. Physical Review A, 78 (5): 052331, 2008. https:/​/​doi.org/​10.1103/​physreva.78.052331.
https:/​/​doi.org/​10.1103/​physreva.78.052331

[58] Colin J Trout, Muyuan Li, Mauricio Gutiérrez, Yukai Wu, Sheng-Tao Wang, Luming Duan, and Kenneth R Brown. Simulating the performance of a distance-3 surface code in a linear ion trap. New Journal of Physics, 20 (4): 043038, 2018. https:/​/​doi.org/​10.1088/​1367-2630/​aab341.
https:/​/​doi.org/​10.1088/​1367-2630/​aab341

[59] Nikolas P Breuckmann, Christophe Vuillot, Earl Campbell, Anirudh Krishna, and Barbara M Terhal. Hyperbolic and semi-hyperbolic surface codes for quantum storage. Quantum Science and Technology, 2 (3): 035007, 2017a. https:/​/​doi.org/​10.1088/​2058-9565/​aa7d3b.
https:/​/​doi.org/​10.1088/​2058-9565/​aa7d3b

[60] 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. https:/​/​doi.org/​10.1109/​tit.2016.2555700.
https:/​/​doi.org/​10.1109/​tit.2016.2555700

[61] Jonathan Conrad, Christopher Chamberland, Nikolas P Breuckmann, and Barbara M Terhal. The small stellated dodecahedron code and friends. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2123): 20170323, 2018. https:/​/​doi.org/​10.1098/​rsta.2017.0323.
https:/​/​doi.org/​10.1098/​rsta.2017.0323

[62] Andreas WM Dress. Regular polytopes and equivariant tessellations from a combinatorial point of view. In Algebraic Topology Göttingen 1984, pages 56–72. Springer, 1985. https:/​/​doi.org/​10.1007/​bfb0074423.
https:/​/​doi.org/​10.1007/​bfb0074423

[63] Andreas WM Dress and Daniel Huson. On tilings of the plane. Geometriae Dedicata, 24 (3): 295–310, 1987. https:/​/​doi.org/​10.1007/​bf00181602.
https:/​/​doi.org/​10.1007/​bf00181602

[64] Branko Grünbaum and Geoffrey Colin Shephard. Tilings and Patterns. Freeman, 1987. https:/​/​doi.org/​10.2307/​2323457.
https:/​/​doi.org/​10.2307/​2323457

[65] Daniel H Huson. The generation and classification of tile-k-transitive tilings of the euclidean plane, the sphere and the hyperbolic plane. Geometriae Dedicata, 47 (3): 269–296, 1993. https:/​/​doi.org/​10.1007/​bf01263661.
https:/​/​doi.org/​10.1007/​bf01263661

[66] Ludwig Balke and Daniel H Huson. Two-dimensional groups, orbifolds and tilings. Geometriae Dedicata, 60 (1): 89–106, 1996. https:/​/​doi.org/​10.1007/​bf00150869.
https:/​/​doi.org/​10.1007/​bf00150869

[67] Olaf Delgado-Friedrichs, Michael O'Keeffe, and Omar M. Yaghi. Three-periodic nets and tilings: regular and quasiregular nets. Acta Crystallographica Section A, 59 (1): 22–27, 2003a. https:/​/​doi.org/​10.1107/​S0108767302018494.
https:/​/​doi.org/​10.1107/​S0108767302018494

[68] Olaf Delgado-Friedrichs, Andreas WM Dress, Daniel H Huson, Jacek Klinowski, and Alan L Mackay. Systematic enumeration of crystalline networks. Nature, 400 (6745): 644, 1999. https:/​/​doi.org/​10.1038/​23210.
https:/​/​doi.org/​https:/​/​doi.org/​10.1038/​23210

[69] Olaf Delgado Friedrichs and Daniel H Huson. Orbifold triangulations and crystallographic groups. Periodica Mathematica Hungarica, 34 (1-2): 29–55, 1997.

[70] William P Thurston, Steve Kerckhoff, WJ Floyd, and John Willard Milnor. The Geometry and Topology of Three-Manifolds. Unpublished, 1997.

[71] ST Hyde, O Delgado-Friedrichs, SJ Ramsden, and Vanessa Robins. Towards enumeration of crystalline frameworks: the 2D hyperbolic approach. Solid State Sciences, 8 (7): 740–752, 2006. https:/​/​doi.org/​10.1016/​j.solidstatesciences.2006.04.001.
https:/​/​doi.org/​10.1016/​j.solidstatesciences.2006.04.001

[72] VA Blatov, M O'Keeffe, and DM Proserpio. Vertex-, face-, point-, Schläfli-, and Delaney-symbols in nets, polyhedra and tilings: recommended terminology. CrystEngComm, 12 (1): 44–48, 2010. https:/​/​doi.org/​10.1039/​b910671e.
https:/​/​doi.org/​10.1039/​b910671e

[73] Ludwig Balke and Alba Valverde. Chamber systems, coloured graphs and orbifolds. Contributions to Algebra and Geometry, 37 (1): 17–29, 1996.

[74] Ludwig Bieberbach. Über die bewegungsgruppen der euklidischen räume. Mathematische Annalen, 70 (3): 297–336, 1911. https:/​/​doi.org/​10.1007/​bf01564500.
https:/​/​doi.org/​10.1007/​bf01564500

[75] Ludwig Bieberbach. Über die bewegungsgruppen der euklidischen räume (zweite abhandlung.) die gruppen mit einem endlichen fundamentalbereich. Mathematische Annalen, 72 (3): 400–412, 1912. https:/​/​doi.org/​10.1007/​bf01456724.
https:/​/​doi.org/​10.1007/​bf01456724

[76] Geoffery Hemion. The Classification of Knots and 3-Dimensional Spaces. Oxford University Press, 1992.

[77] Olaf Delgado-Friedrichs. Gavrog. https:/​/​github.com/​odf/​gavrog.
https:/​/​github.com/​odf/​gavrog

[78] Olaf Delgado-Friedrichs, Michael O'Keeffe, and Omar M. Yaghi. Three-periodic nets and tilings: semiregular nets. Acta Crystallographica Section A, 59 (6): 515–525, 2003b. https:/​/​doi.org/​10.1107/​s0108767302018494.
https:/​/​doi.org/​10.1107/​s0108767302018494

[79] Natalie C Brown, Michael Newman, and Kenneth R Brown. Handling leakage with subsystem codes. New Journal of Physics, 21 (1): 073055, 2019. https:/​/​doi.org/​10.1088/​1367-2630/​ab3372.
https:/​/​doi.org/​10.1088/​1367-2630/​ab3372

[80] 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. Physical Review X, 10 (1): 011022, 2020a. https:/​/​doi.org/​10.1103/​physrevx.10.011022.
https:/​/​doi.org/​10.1103/​physrevx.10.011022

[81] Christopher Chamberland, Aleksander Kubica, Ted Yoder, and Guanyu Zhu. Triangular color codes on trivalent graphs with flag qubits. New Journal of Physics, 22 (2): 023019, 2020b. https:/​/​doi.org/​10.1088/​1367-2630/​ab68fd.
https:/​/​doi.org/​10.1088/​1367-2630/​ab68fd

[82] Koichi Momma and Fujio Izumi. Vesta 3 for three-dimensional visualization of crystal, volumetric and morphology data. Journal of Applied Crystallography, 44 (6): 1272–1276, 2011. https:/​/​doi.org/​10.1107/​s0021889811038970.
https:/​/​doi.org/​10.1107/​s0021889811038970

[83] Olaf Delgado-Friedrichs and Michael O'Keeffe. Identification of and symmetry computation for crystal nets. Acta Crystallographica Section A: Foundations of Crystallography, 59 (4): 351–360, 2003. https:/​/​doi.org/​10.1107/​s0108767303012017.
https:/​/​doi.org/​10.1107/​s0108767303012017

[84] Olaf Delgado-Friedrichs. Barycentric drawings of periodic graphs. In International Symposium on Graph Drawing, pages 178–189. Springer, 2003. https:/​/​doi.org/​10.1007/​978-3-540-24595-7_17.
https:/​/​doi.org/​10.1007/​978-3-540-24595-7_17

[85] Olaf Delgado-Friedrichs and Michael O'Keeffe. Crystal nets as graphs: Terminology and definitions. Journal of Solid State Chemistry, 178 (8): 2480–2485, 2005. https:/​/​doi.org/​10.1016/​j.jssc.2005.06.011.
https:/​/​doi.org/​10.1016/​j.jssc.2005.06.011

[86] Olaf Delgado-Friedrichs. Equilibrium placement of periodic graphs and convexity of plane tilings. Discrete & Computational Geometry, 33 (1): 67–81, 2005. https:/​/​doi.org/​10.1007/​s00454-004-1147-x.
https:/​/​doi.org/​10.1007/​s00454-004-1147-x

[87] GAP. GAP – Groups, Algorithms, and Programming, Version 4.10.2. The GAP Group, 2019. URL https:/​/​www.gap-system.org.
https:/​/​www.gap-system.org

[88] Tinkercad. URL http:/​/​www.tinkercad.com.
http:/​/​www.tinkercad.com

[89] Michael O'Keeffe, Maxim A Peskov, Stuart J Ramsden, and Omar M Yaghi. The reticular chemistry structure resource (RCSR) database of, and symbols for, crystal nets. Accounts of Chemical Research, 41 (12): 1782–1789, 2008. https:/​/​doi.org/​10.1021/​ar800124u.
https:/​/​doi.org/​10.1021/​ar800124u

[90] Jack Edmonds. Paths, trees, and flowers. In Classic Papers in Combinatorics, pages 361–379. Springer, 2009. https:/​/​doi.org/​10.1007/​978-0-8176-4842-8_26.
https:/​/​doi.org/​10.1007/​978-0-8176-4842-8_26

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

[92] Hidetoshi Nishimori. Geometry-induced phase transition in the$\pm$J ising model. Journal of the Physical Society of Japan, 55 (10): 3305–3307, 1986. https:/​/​doi.org/​10.1143/​jpsj.55.3305.
https:/​/​doi.org/​10.1143/​jpsj.55.3305

[93] Aleksander Kubica, Michael E Beverland, Fernando Brandão, John Preskill, and Krysta M Svore. Three-dimensional color code thresholds via statistical-mechanical mapping. Physical Review Letters, 120 (18): 180501, 2018. https:/​/​doi.org/​10.1103/​physrevlett.120.180501.
https:/​/​doi.org/​10.1103/​physrevlett.120.180501

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

[95] Pierre L'Eculier. Good parameter sets for combined multiple recursive random number generators. Operations Research, 47: 159–164, 1999. https:/​/​doi.org/​10.1287/​opre.47.1.159.
https:/​/​doi.org/​10.1287/​opre.47.1.159

[96] Pierre L'Eculier, R. Simard, E. J. Chen, and W. D. Kelton. An objected-oriented random-number package with many long streams and substreams. Operations Research, 50: 1073–1075, 2002. https:/​/​doi.org/​10.1287/​opre.50.6.1073.358.
https:/​/​doi.org/​10.1287/​opre.50.6.1073.358

[97] Thomas M Stace, Sean D Barrett, and Andrew C Doherty. Thresholds for topological codes in the presence of loss. Physical Review Letters, 102 (20): 200501, 2009. https:/​/​doi.org/​10.1103/​physrevlett.102.200501.
https:/​/​doi.org/​10.1103/​physrevlett.102.200501

[98] Terry Rudolph. Why I am optimistic about the silicon-photonic route to quantum computing. APL Photonics, 2 (3): 030901, 2017. https:/​/​doi.org/​10.1063/​1.4976737.
https:/​/​doi.org/​10.1063/​1.4976737

[99] 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, 2018. https:/​/​doi.org/​10.1103/​physrevlett.120.050505.
https:/​/​doi.org/​10.1103/​physrevlett.120.050505

[100] David K Tuckett, Andrew S Darmawan, Christopher T Chubb, Sergey Bravyi, Stephen D Bartlett, and Steven T Flammia. Tailoring surface codes for highly biased noise. Physical Review X, 9 (4): 041031, 2019. https:/​/​doi.org/​10.1103/​physrevx.9.041031.
https:/​/​doi.org/​10.1103/​physrevx.9.041031

[101] 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. https:/​/​doi.org/​10.1103/​physrevlett.124.130501.
https:/​/​doi.org/​10.1103/​physrevlett.124.130501

[102] Xiaosi Xu, Qi Zhao, Xiao Yuan, and Simon C Benjamin. High-threshold code for modular hardware with asymmetric noise. Physical Review Applied, 12 (6): 064006, 2019. https:/​/​doi.org/​10.1103/​physrevapplied.12.064006.
https:/​/​doi.org/​10.1103/​physrevapplied.12.064006

[103] Shruti Puri, Lucas St-Jean, Jonathan A Gross, Alexander Grimm, NE Frattini, Pavithran S Iyer, Anirudh Krishna, Steven Touzard, Liang Jiang, Alexandre Blais, et al. Bias-preserving gates with stabilized cat qubits. arXiv preprint arXiv:1905.00450, 2019.
arXiv:1905.00450

[104] Paul Webster, Stephen D Bartlett, and David Poulin. Reducing the overhead for quantum computation when noise is biased. Physical Review A, 92 (6): 062309, 2015. https:/​/​doi.org/​10.1103/​physreva.92.062309.
https:/​/​doi.org/​10.1103/​physreva.92.062309

[105] Peter Brooks and John Preskill. Fault-tolerant quantum computation with asymmetric Bacon-Shor codes. Physical Review A, 87 (3): 032310, 2013. https:/​/​doi.org/​10.1103/​physreva.87.032310.
https:/​/​doi.org/​10.1103/​physreva.87.032310

[106] Sergey Bravyi, Guillaume Duclos-Cianci, David Poulin, and Martin Suchara. Subsystem surface codes with three-qubit check operators. Quantum Information & Computation, 13 (11-12): 963–985, 2013.

[107] Martin Suchara, Andrew W Cross, and Jay M Gambetta. Leakage suppression in the toric code. In Information Theory (ISIT), 2015 IEEE International Symposium on, pages 1119–1123. IEEE, 2015b. https:/​/​doi.org/​10.1109/​isit.2015.7282629.
https:/​/​doi.org/​10.1109/​isit.2015.7282629

[108] Michael Varnava, Daniel E Browne, and Terry Rudolph. Loss tolerant linear optical quantum memory by measurement-based quantum computing. New Journal of Physics, 9 (6): 203, 2007. https:/​/​doi.org/​10.1088/​1367-2630/​9/​6/​203.
https:/​/​doi.org/​10.1088/​1367-2630/​9/​6/​203

[109] Nikolas P Breuckmann and Vivien Londe. Single-shot decoding of linear rate LDPC quantum codes with high performance. arXiv preprint arXiv:2001.03568, 2020.
arXiv:2001.03568

[110] Kasper Duivenvoorden, Nikolas P Breuckmann, and Barbara M Terhal. Renormalization group decoder for a four-dimensional toric code. IEEE Transactions on Information Theory, 65 (4): 2545–2562, 2018. https:/​/​doi.org/​10.1109/​tit.2018.2879937.
https:/​/​doi.org/​10.1109/​tit.2018.2879937

[111] Nikolas P Breuckmann, Kasper Duivenvoorden, Dominik Michels, and Barbara M Terhal. Local decoders for the 2D and 4D toric code. Quantum Information & Computation, 17 (3-4): 181–208, 2017b.

[112] Aleksander Kubica and John Preskill. Cellular-automaton decoders with provable thresholds for topological codes. Physical Review Letters, 123 (2): 020501, 2019. https:/​/​doi.org/​10.1103/​physrevlett.123.020501.
https:/​/​doi.org/​10.1103/​physrevlett.123.020501

Cited by

[1] 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).

[2] Benjamin J. Brown and Sam Roberts, "Universal fault-tolerant measurement-based quantum computation", arXiv:1811.11780, Physical Review Research 2 3, 033305 (2018).

[3] Hayata Yamasaki, Kosuke Fukui, Yuki Takeuchi, Seiichiro Tani, and Masato Koashi, "Polylog-overhead highly fault-tolerant measurement-based quantum computation: all-Gaussian implementation with Gottesman-Kitaev-Preskill code", arXiv:2006.05416.

[4] Ye Wang, Mu Qiao, Zhengyang Cai, Kuan Zhang, Naijun Jin, Pengfei Wang, Wentao Chen, Chunyang Luan, Haiyan Wang, Yipu Song, Dahyun Yum, and Kihwan Kim, "Realization of two-dimensional crystal of ions in a monolithic Paul trap", arXiv:1912.04262.

[5] 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.

[6] Armanda O. Quintavalle, Michael Vasmer, Joschka Roffe, and Earl T. Campbell, "Single-shot error correction of three-dimensional homological product codes", arXiv:2009.11790.

The above citations are from Crossref's cited-by service (last updated successfully 2020-10-20 20:05:45) and SAO/NASA ADS (last updated successfully 2020-10-20 20:05:46). The list may be incomplete as not all publishers provide suitable and complete citation data.