Witness algebra and anyon braiding. (English) Zbl 1495.81024
Summary: 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.
MSC:
| 81P68 | Quantum computation |
| 18M20 | Fusion categories, modular tensor categories, modular functors |
| 68Q12 | Quantum algorithms and complexity in the theory of computing |
Keywords:
anyons; braiding; modular tensor categories; topological quantum computing; witness algebraReferences:
| [1] | Artemov, S. N. (1994). Logic of proofs. Annals of Pure and Applied Logic6729-59. · Zbl 0796.03029 |
| [2] | 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. · Zbl 1439.81028 |
| [3] | Blass, A. and Gurevich, Y. (2020). Braided distributivity. Theoretical Computer Science80773-94. Prepublication at arXiv:1807.11403. |
| [4] | Bonderson, P. (2007). Non-Abelian Anyons and Interferometry. Phd thesis, Caltech. · Zbl 1151.81004 |
| [5] | Freedman, M. H., Kitaev, A., Larsen, M. J. and Wang, Z. (2003). Topological quantum computation. Bulletin of the American Mathematical Society40 (1) 31-38. · Zbl 1019.81008 |
| [6] | 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. |
| [7] | 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 Kurt Gödel Collected Works, Volume III, Unpublished Essays and Lectures, Oxford University Press, 62-113. |
| [8] | Joyal, A. and Street, R. (1986). Braided Monoidal Categories, Macquarie Mathematics Reports 860081. · Zbl 0817.18007 |
| [9] | Joyal, A. and Street, R. (1993). Braided tensor categories. Advances in Mathematics10220-78. · Zbl 0817.18007 |
| [10] | Kelly, G. M. (1964). On MacLane’s conditions for coherence of natural associativities, commutativities, etc. Journal of Algebra1397-402. · Zbl 0246.18008 |
| [11] | Kitaev, A. (1997). Fault-tolerant quantum computation by anyons. arXiv:quant-ph/9707021, 9 July 1997. · Zbl 1012.81006 |
| [12] | 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. · Zbl 0244.18010 |
| [13] | Mac Lane, S. (1963). Natural associativity and commutativity. Rice University Studies49 (4) 28-46. · Zbl 0244.18008 |
| [14] | 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. · Zbl 1218.81036 |
| [15] | Trebst, S., Troyer, M., Wang, Z. and Ludwig, A. W. W. (2008). A short introduction to Fibonacci anyon models. Progress of Theoretical Physics Supplement176384-407. · Zbl 1173.81362 |
| [16] | Wang, Z. (2010). Topological Quantum Computation, CBMS Regional Conference Series in Mathematics, vol. 112, American Mathematical Society. · Zbl 1239.81005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
