A Hierarchy of Anyon Models Realised by Twists in Stacked Surface Codes

T. R. Scruby and D. E. Browne

Dept. of Physics and Astronomy, University College London, London WC1E 6BT, UK

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Abstract

Braiding defects in topological stabiliser codes can be used to fault-tolerantly implement logical operations. Twists are defects corresponding to the end-points of domain walls and are associated with symmetries of the anyon model of the code. We consider twists in multiple copies of the 2d surface code and identify necessary and sufficient conditions for considering these twists as anyons: namely that they must be self-inverse and that all charges which can be localised by the twist must be invariant under its associated symmetry. If both of these conditions are satisfied the twist and its set of localisable anyonic charges reproduce the behaviour of an anyonic model belonging to a hierarchy which generalises the Ising anyons. We show that the braiding of these twists results in either (tensor products of) the S gate or (tensor products of) the CZ gate. We also show that for any number of copies of the 2d surface code the application of H gates within a copy and CNOT gates between copies is sufficient to generate all possible twists.

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► References

[1] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, New York, NY, USA, 10th edition, 2011. ISBN 978-1-107-00217-3.

[2] Sergey Bravyi and Robert Koenig. Classification of topologically protected gates for local stabilizer codes. Physical Review Letters, 110 (17): 170503, April 2013. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.110.170503. arXiv: 1206.1609.
https:/​/​doi.org/​10.1103/​PhysRevLett.110.170503

[3] Paul Webster and Stephen D. Bartlett. Locality-preserving logical operators in topological stabilizer codes. Physical Review A, 97 (1): 012330, January 2018a. 10.1103/​PhysRevA.97.012330.
https:/​/​doi.org/​10.1103/​PhysRevA.97.012330

[4] Jonathan E. Moussa. Transversal Clifford gates on folded surface codes. Physical Review A, 94 (4): 042316, October 2016. 10.1103/​PhysRevA.94.042316.
https:/​/​doi.org/​10.1103/​PhysRevA.94.042316

[5] Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, and James R. Wootton. Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code. Physical Review X, 7 (2): 021029, May 2017. 10.1103/​PhysRevX.7.021029.
https:/​/​doi.org/​10.1103/​PhysRevX.7.021029

[6] Markus S. Kesselring, Fernando Pastawski, Jens Eisert, and Benjamin J. Brown. The boundaries and twist defects of the color code and their applications to topological quantum computation. Quantum, 2: 101, October 2018. 10.22331/​q-2018-10-19-101.
https:/​/​doi.org/​10.22331/​q-2018-10-19-101

[7] H. Bombin. Topological Order with a Twist: Ising Anyons from an Abelian Model. Physical Review Letters, 105 (3): 030403, July 2010. 10.1103/​PhysRevLett.105.030403.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.030403

[8] Maissam Barkeshli, Parsa Bonderson, Meng Cheng, and Zhenghan Wang. Symmetry, Defects, and Gauging of Topological Phases. arXiv:1410.4540 [cond-mat, physics:hep-th, physics:math-ph, physics:quant-ph], October 2014. 10.1103/​PhysRevB.100.115147. arXiv: 1410.4540.
https:/​/​doi.org/​10.1103/​PhysRevB.100.115147
arXiv:1410.4540

[9] Daniel Gottesman. Stabilizer Codes and Quantum Error Correction. arXiv:quant-ph/​9705052, May 1997. URL http:/​/​arxiv.org/​abs/​quant-ph/​9705052. arXiv: quant-ph/​9705052.
arXiv:quant-ph/9705052

[10] H. Bombin and M. A. Martin-Delgado. Topological Quantum Distillation. Physical Review Letters, 97 (18): 180501, October 2006. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.97.180501. arXiv: quant-ph/​0605138.
https:/​/​doi.org/​10.1103/​PhysRevLett.97.180501
arXiv:quant-ph/0605138

[11] 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. 2018. 10.7907/​059V-MG69.
https:/​/​doi.org/​10.7907/​059V-MG69

[12] H. Bombin. An Introduction to Topological Quantum Codes. arXiv:1311.0277 [quant-ph], November 2013. URL http:/​/​arxiv.org/​abs/​1311.0277. arXiv: 1311.0277.
arXiv:1311.0277

[13] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Physical Review A, 86 (3): 032324, September 2012. 10.1103/​PhysRevA.86.032324.
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[14] F. A. Bais and J. K. Slingerland. Condensate induced transitions between topologically ordered phases. Physical Review B, 79 (4): 045316, January 2009. ISSN 1098-0121, 1550-235X. 10.1103/​PhysRevB.79.045316. arXiv: 0808.0627.
https:/​/​doi.org/​10.1103/​PhysRevB.79.045316

[15] Philippe Francesco, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997. ISBN 978-0-387-94785-3. URL https:/​/​www.springer.com/​gp/​book/​9780387947853.
https:/​/​www.springer.com/​gp/​book/​9780387947853

[16] Jiannis K. Pachos. Introduction to Topological Quantum Computation by Jiannis K. Pachos, April 2012.

[17] Adrian Hutter, James R. Wootton, and Daniel Loss. Parafermions in a Kagome Lattice of Qubits for Topological Quantum Computation. Physical Review X, 5 (4): 041040, December 2015. 10.1103/​PhysRevX.5.041040.
https:/​/​doi.org/​10.1103/​PhysRevX.5.041040

[18] Daisuke Tambara and Shigeru Yamagami. Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups. Journal of Algebra, 209 (2): 692–707, November 1998. ISSN 0021-8693. 10.1006/​jabr.1998.7558.
https:/​/​doi.org/​10.1006/​jabr.1998.7558

[19] J. J. Sylvester. LX. Thoughts on inverse orthogonal matrices, simultaneous signsuccessions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34 (232): 461–475, December 1867. ISSN 1941-5982. 10.1080/​14786446708639914.
https:/​/​doi.org/​10.1080/​14786446708639914

[20] A. Hedayat and W. D. Wallis. Hadamard Matrices and Their Applications. The Annals of Statistics, 6 (6): 1184–1238, November 1978. ISSN 0090-5364, 2168-8966. 10.1214/​aos/​1176344370.
https:/​/​doi.org/​10.1214/​aos/​1176344370

[21] Emil Artin. Galois theory: lectures delivered at the University of Notre Dame /​ by Dr. Emil Artin ; edited and supplemented with a section on applications by Dr. Arthur N. Milgram. Notre Dame mathematical lectures ; no. 2. University of Notre Dame, University of Notre Dame Press, Notre Dame, Ind., Indiana, 2nd ed., with additions and revisions. edition, 1959.

[22] Jerzy Rozanski. Bicharacters, braids and Jacobi identity. arXiv:q-alg/​9611029, November 1996. URL http:/​/​arxiv.org/​abs/​q-alg/​9611029. arXiv: q-alg/​9611029.
arXiv:q-alg/9611029

[23] R. Craigen. Trace, Symmetry and Orthogonality. Canadian Mathematical Bulletin, 37 (4): 461–467, December 1994. ISSN 0008-4395, 1496-4287. 10.4153/​CMB-1994-067-1.
https:/​/​doi.org/​10.4153/​CMB-1994-067-1

[24] Jacob A. Siehler. Braided Near-group Categories. arXiv:math/​0011037, November 2000. URL http:/​/​arxiv.org/​abs/​math/​0011037. arXiv: math/​0011037.
arXiv:math/0011037

[25] Aleksander Kubica, Beni Yoshida, and Fernando Pastawski. Unfolding the color code. New Journal of Physics, 17 (8): 083026, August 2015. ISSN 1367-2630. 10.1088/​1367-2630/​17/​8/​083026. arXiv: 1503.02065.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​8/​083026

[26] Wilhelm Magnus, Abraham Karrass, and Donald Solitar. Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Courier Corporation, January 2004. ISBN 978-0-486-43830-6. Google-Books-ID: 1LW4s1RDRHQC.

[27] Paul Webster and Stephen D. Bartlett. Braiding defects in topological stabiliser codes of any dimension cannot be universal. arXiv:1811.11789 [quant-ph], November 2018b. URL http:/​/​arxiv.org/​abs/​1811.11789. arXiv: 1811.11789.
arXiv:1811.11789

[28] Paul Webster and Stephen D. Bartlett. Fault-Tolerant Quantum Gates with Defects in Topological Stabiliser Codes. arXiv:1906.01045 [quant-ph], June 2019. URL http:/​/​arxiv.org/​abs/​1906.01045. arXiv: 1906.01045.
arXiv:1906.01045

[29] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321 (1): 2–111, January 2006. ISSN 0003-4916. 10.1016/​j.aop.2005.10.005.
https:/​/​doi.org/​10.1016/​j.aop.2005.10.005

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