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Witness algebra and anyon braiding

Published online by Cambridge University Press:  16 March 2020

Andreas Blass  and
Yuri Gurevich Open the ORCID record for Yuri Gurevich [Opens in a new window]
Andreas Blass
Affiliation:
Mathematics Department, University of Michigan, Ann Arbor, MI48109-1043, USA
Yuri Gurevich*
Affiliation:
Computer Science and Engineering, University of Michigan, Ann Arbor, MI48109-2121, USA
*
*Corresponding author. Email: [email protected]
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Abstract

Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.

Keywords

Anyonsbraidingmodular tensor categoriestopological quantum computingwitness algebra
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 3 , March 2020 , pp. 234 - 270
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Artemov, S. N. (1994). Logic of proofs. Annals of Pure and Applied Logic 67 29–59.CrossRefGoogle Scholar
Blass, A. and Gurevich, Y. (2016). On quantum computation, anyons, and categories. In: Amodeo, E. and Policriti, A. (eds.) Martin Davis on Computability, Computational Logic, and Mathematical Foundations, Outstanding Contributions to Logic, vol. 10, Springer International Publishing Switzerland, Springer-Verlag, 209–241.CrossRefGoogle Scholar
Blass, A. and Gurevich, Y. (2020). Braided distributivity. Theoretical Computer Science 807 73–94. Prepublication at arXiv:1807.11403.CrossRefGoogle Scholar
Bonderson, P. (2007). Non-Abelian Anyons and Interferometry. Phd thesis, Caltech.Google Scholar
Freedman, M. H., Kitaev, A., Larsen, M. J. and Wang, Z. (2003). Topological quantum computation. Bulletin of the American Mathematical Society 40 (1) 31–38.CrossRefGoogle Scholar
Freyd, P. (1964). Abelian Categories, Harper & Row. Republished with new foreword in Reprints in Theory and Applications of Categories 3 (2003) (–25)–164; see http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html.Google Scholar
Gödel, K. (1995). Vortrag bei Zilsel (with introductory note by W. Sieg and C. Parsons, and with English translation). In: Feferman, S., Dawson, J. W., Goldfarb, W., Parsons, C. and Solovay, R. M. (eds.) Kurt Gödel Collected Works, Volume III, Unpublished Essays and Lectures, Oxford University Press, 62–113.Google Scholar
Joyal, A. and Street, R. (1986). Braided Monoidal Categories, Macquarie Mathematics Reports 860081.Google Scholar
Joyal, A. and Street, R. (1993). Braided tensor categories. Advances in Mathematics 102 20–78.CrossRefGoogle Scholar
Kelly, G. M. (1964). On MacLane’s conditions for coherence of natural associativities, commutativities, etc. Journal of Algebra 1 397–402.Google Scholar
Kitaev, A. (1997). Fault-tolerant quantum computation by anyons. arXiv:quant-ph/9707021, 9 July 1997.Google Scholar
Laplaza, M. (1972). Coherence for distributivity. In: Mac Lane, S. (ed.) Coherence in Categories, Lecture Notes in Mathematics, vol. 281, Berlin, Heidelberg, New York, Springer-Verlag, 29–65.CrossRefGoogle Scholar
Mac Lane, S. (1963). Natural associativity and commutativity. Rice University Studies 49 (4) 28–46.Google Scholar
Panangaden, P. and Paquette, É. O. (2011). A categorical presentation of quantum computation with anyons. In: Coecke, B. (ed.) New Structures For Physics, Lecture Notes in Physics, vol. 813, Springer-Verlag, 983–1025.Google Scholar
Trebst, S., Troyer, M., Wang, Z. and Ludwig, A. W. W. (2008). A short introduction to Fibonacci anyon models. Progress of Theoretical Physics Supplement 176 384–407.CrossRefGoogle Scholar
Wang, Z. (2010). Topological Quantum Computation, CBMS Regional Conference Series in Mathematics, vol. 112, American Mathematical Society.Google Scholar