Communications in Mathematical Physics

, Volume 369, Issue 3, pp 1167–1185 | Cite as

Domain Walls in Topological Phases and the Brauer–Picard Ring for \({{\rm Vec} (\mathbb{Z}/p\mathbb{Z})}\)

  • Daniel Barter
  • Jacob C. BridgemanEmail author
  • Corey Jones


We show how to calculate the relative tensor product of bimodule categories (not necessarily invertible) using ladder string diagrams. As an illustrative example, we compute the Brauer–Picard ring for the fusion category \({{\bf Vec} (\mathbb{Z}/p\mathbb{Z})}\) . Moreover, we provide a physical interpretation of all indecomposable bimodule categories in terms of domain walls in the associated topological phase. We show how this interpretation can be used to compute the Brauer–Picard ring from a physical perspective.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work is supported by the Australian Research Council (ARC) via the Centre of Excellence in Engineered Quantum Systems (EQuS) Project Number CE170100009, Discovery Project “Subfactors and symmetries“ DP140100732 and Discovery Project “Low dimensional categories“ DP16010347. We thank Andrew Doherty, Steve Flammia, Dominic Williamson, Cain Edie-Michell, Scott Morrison, and Christoph Schweigert. This work would not have been possible without their input and feedback. We thank Paul Wedrich and Christopher Chubb for feedback on the draft manuscript. We thank Noah Snyder for explaining how to prove Proposition 11.


  1. 1.
    Dennis E., Kitaev A., Landahl A., Preskill J.: Topological quantum memory. J. Math. Phys. 43, 4452 (2002) arXiv:quant-ph/0110143 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2 (2003) arXiv:quant-ph/9707021 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brown B.J., Loss D., Pachos J.K., Self C.N., Wootton J.R.: Quantum memories at finite temperature. Rev. Mod. Phys. 88, 045005 (2016) arXiv:1411.6643 ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Terhal B.M.: Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307 (2015) arXiv:1302.3428 ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Raussendorf R., Harrington J.: Fault-tolerant quantum computation with high threshold in two dimensions. Phys. Rev. Lett. 98, 190504 (2007) arXiv:quant-ph/0610082 ADSCrossRefGoogle Scholar
  6. 6.
    Bombin H., Martin-Delgado M.: Quantum measurements and gates by code deformation. J. Phys. A: Math. Theor. 42, 095302 (2009) arXiv:0704.2540 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bombin H.: Topological order with a twist: Ising anyons from an Abelian model. Physical Review Letters 105, 030403 (2010) arXiv:1004.1838 ADSCrossRefGoogle Scholar
  8. 8.
    Brown B.J., Al-Shimary A., Pachos J.K.: Entropic barriers for two-dimensional quantum memories. Phys. Rev. Lett. 112, 120503 (2014) arXiv:1307.6222 ADSCrossRefGoogle Scholar
  9. 9.
    Pastawski F., Yoshida B.: Fault-tolerant logical gates in quantum error-correcting codes. Phys. Rev. A 91, 012305 (2015) arXiv:1408.1720 ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Yoshida B.: Topological color code and symmetry-protected topological phases. Phys. Rev. B 91, 245131 (2015) arXiv:1503.07208 ADSCrossRefGoogle Scholar
  11. 11.
    Brown B.J., Laubscher K., Kesselring M.S., Wootton J.R.: Poking holes and cutting corners to achieve Clifford gates with the surface code. Phys. Rev. X 7, 021029 (2017) arXiv:1609.04673 Google Scholar
  12. 12.
    Cong, I., Cheng, M., Wang, Z.: Topological quantum computation with gapped boundaries. arXiv:1609.02037 (2016)
  13. 13.
    Cong I., Cheng M., Wang Z.: Universal quantum computation with gapped boundaries. Phys. Rev. Lett. 119, 170504 (2017) arXiv:1707.05490 ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Yoshida B.: Gapped boundaries, group cohomology and fault-tolerant logical gates. Ann. Phys. 377, 387 (2017) arXiv:1509.03626 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kesselring M.S., Pastawski F., Eisert J., Brown B.J.: The boundaries and twist defects of the color code and their applications to topological quantum computation. Quantum 2, 101 (2018) arXiv:1806.02820 CrossRefGoogle Scholar
  16. 16.
    Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Atiyah M.: Topological quantum field theories. Institut des Hautes Études Scientifiques. Publications Mathématiques 68, 175 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Baez J.C., Dolan J.: Higher-dimensional algebra and topological quantum field theory. J. Math. Phys. 36, 6073 (1995) arXiv:q-alg/9503002 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Turaev V., Virelizier A.: Monoidal Categories and Topological Field Theory. Progress in Mathematics, vol. 322, pp. xii+523. Birkhäuser, Cham (2017)CrossRefzbMATHGoogle Scholar
  20. 20.
    Chow J.M., Gambetta J.M., Magesan E., Abraham D.W., Cross A.W., Johnson B., Masluk N.A., Ryan C.A., Smolin J.A., Srinivasan S.J. et al.: Implementing a strand of a scalable fault-tolerant quantum computing fabric. Nat. Commun. 5, 4015 (2014) arXiv:1311.6330 ADSCrossRefGoogle Scholar
  21. 21.
    Gambetta J.M., Chow J.M., Steffen M.: Building logical qubits in a superconducting quantum computing system. NPJ Quantum Inf. 3, 2 (2017) arXiv:1510.04375 ADSCrossRefGoogle Scholar
  22. 22.
    Levin M., Wen X.-G.: String-net condensation: a physical mechanism for topological phases. Phys. Rev. B 71, 045110 (2005) arXiv:cond-mat/0404617 ADSCrossRefGoogle Scholar
  23. 23.
    Fuchs J., Runkel I., Schweigert C.: TFT construction of RCFT correlators I: partition functions. Nucl. Phys. B 646, 353 (2002) arXiv:hep-th/0204148 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fuchs J., Priel J., Schweigert C., Valentino A.: On the Brauer groups of symmetries of abelian Dijkgraaf–Witten theories. Commun. Math. Phys. 339, 385 (2015) arXiv:1404.6646 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kitaev A., Kong L.: Models for gapped boundaries and domain walls. Commun. Math. Phys. 313, 351 (2012) arXiv:1104.5047 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kong L.: Anyon condensation and tensor categories. Nucl. Phys. B 886, 436 (2014) arXiv:1307.8244 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Morrison S., Walker K.: Blob homology. Geom. Topol. 16, 1481 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Fröhlich J., Fuchs J., Runkel I., Schweigert C.: Correspondences of ribbon categories. Adv. Math. 199, 192 (2006) arXiv:math/0309465 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Selinger P.: New Structures for Physics. Lecture Notes in Physics, vol. 813, pp. 289–355. Springer, Heidelberg (2011)Google Scholar
  30. 30.
    Etingof, P., Nikshych, D., Ostrik, V.: Fusion categories and homotopy theory. Quantum Topol. 1, 209, with an appendix by Ehud Meir, arXiv:0909.3140 (2010)
  31. 31.
    Cui, S.X., Zini, M.S., Wang, Z.: On generalized symmetries and structure of modular categories. arXiv:1809.00245 (2018)
  32. 32.
    Barter, D., Bridgeman, J.C., Jones, C.: in preparation Google Scholar
  33. 33.
    Etingof P., Gelaki S., Nikshych D., Ostrik V.: Tensor Categories. Mathematical Surveys and Monographs, vol. 205, pp. xvi+343. American Mathematical Society, Providence (2015)zbMATHGoogle Scholar
  34. 34.
    There are multiple ways to take the opposite of a tensor category. The reader should consult Ref. [33] for the definitions of all the tensor category opposite constructions and how they are relatedGoogle Scholar
  35. 35.
    Douglas, C.L., Schommer-Pries C., Snyder, N.: The balanced tensor product of module categories. arXiv:1406.4204 (2014)
  36. 36.
    Bar-Natan D., Morrison S.: The Karoubi envelope and Lee’s degeneration of Khovanov homology. Algebr. Geom. Topol. 6, 1459 arXiv:math/0606542 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Schaumann, G.: Duals in tricategories and in the tricategory of bimodule categories. Ph.D. thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) (2013)Google Scholar
  38. 38.
    Barter, D.: Computing the minimal model for the quantum symmetric algebra. arXiv:1610.05204 (2016)
  39. 39.
    Lawson, T.: Computing an explicit homotopy inverse for \({B(*,H,*) \hookrightarrow B(*,G,G/H)}\) , MathOverflow. (version: 2017-12-12)
  40. 40.
    Bridgeman J.C., Doherty A.C., Bartlett S.D.: Tensor networks with a twist: anyon-permuting domain walls and defects in projected entangled pair states. Phys. Rev. B 96, 245122 (2017) arXiv:1708.08930 ADSCrossRefGoogle Scholar
  41. 41.
    Delfosse, N., Iyer, P., Poulin, D.: Generalized surface codes and packing of logical qubits. arXiv:1606.07116 (2016)
  42. 42.
    Bridgeman, J., Barter, D., Jones, C.: Fusing binary interface defects in topological phases: The \({\mathbb{Z}/p\mathbb{Z}}\) case. arXiv:1810.09469 (2018)
  43. 43.
    Bombin H., Martin-Delgado M.: Topological quantum distillation. Phys. Rev. Lett. 97, 180501 (2006) arXiv:quant-ph/0605138 ADSCrossRefGoogle Scholar
  44. 44.
    Barkeshli M., Jian C.-M., Qi X.-L.: Theory of defects in Abelian topological states. Phys. Rev. B 88, 235103 (2013) arXiv:1305.7203 ADSCrossRefGoogle Scholar
  45. 45.
    Barkeshli, M., Bonderson, P., Cheng, M., Wang, Z.: Symmetry, defects, and gauging of topological phases. arXiv:1410.4540 (2014)
  46. 46.
    Williamson, D.J., Bultinck, N., Verstraete, F.: Symmetry-enriched topological order in tensor networks: defects, gauging and anyon condensation. arXiv:1711.07982 (2017)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Daniel Barter
    • 1
  • Jacob C. Bridgeman
    • 2
    Email author
  • Corey Jones
    • 1
  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  2. 2.Centre for Engineered Quantum Systems, School of PhysicsThe University of SydneySydneyAustralia

Personalised recommendations