Departmental Bulletin Paper TQCにおけるモジュラーS行列をめぐって
Modular S-matrix in TQC theory

藤井, 淳一

64 ( 2 )  , pp.15 - 33 , 2016-02-29 , 大阪教育大学
The term `TQC' means the topological quantum computation, which is one of remarkable theories to realize quantum computers. This theory is related to various fields; theory of quantum groups, Lie algebras, Hopf algebras and conformal field or string theory. But there are few textbooks which readers of other fields can read easily. We wrote a survay[10]easy to observe from the basics of TQC theory to basic quantum gates for quantum computers. But we cannot explain modular S-matrices that is one of the keys in TQC in the preceding survey. So, in this paper, we try to express total basic issue including theories around modular S-matrix. In the many papers for TQC, there are some subtle mistakes and we correct them in this paper. At first we introduce formulae for modular S-matrix in the axiomatic (categorical) approach. Secondly we give an important model for TQC; spin network model based on the Temperly-Lieb-Jones algebra. Finally we mention another important model suk(2). In each levels, Fibannaci anyon is a basic example to explain them.

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