PREFACE xi few open problems. Chapters 1,2,3 give a self-contained treatment of the additive approximation algorithm. Moreover, universal topological quantum computation models can be built from some even-half theories of Jones algebroids such as the Fibonacci theory. Combining the results together, we obtain an equivalence of the topological quantum computational model with the quantum circuit model. Chap- ters 1,2,3, based on graphical calculus of ribbon fusion categories, are accessible to entry-level graduate students in mathematics, physics, or computer science. A rib- bon fusion category, defined with 6j symbols, is just some point up to equivalence on an algebraic variety of polynomial equations. Therefore the algebraic theory of anyons is elementary, given basic knowledge of surfaces and their mapping class groups of invertible self-transformations up to deformation. Some useful books on related topics are: for mathematics, Bakalov-Kirillov [BK], Kassel [Kas], Kauffman-Lins [KL], and Turaev [Tu] for quantum computation, Kitaev-Shen-Vyalyi [KSV] and Nielsen-Chuang [NC] and for physics, Altland- Simons [AS], Di Francesco–Mathieu-Senechal [DMS], and Wen [Wen7]. Topological quantum computation sits at the triple juncture of quantum topol- ogy, quantum physics, and quantum computation: TQC QP QT QC The existence of topological phases of matter (TPM) with non-abelian anyons would lead us to topological quantum computation (TQC) via unitary modular tensor categories (UMTC): TPM UMTC TQC Thus the practical aspect of topological quantum computation hinges on the exis- tence of non-abelian topological states. Will we succeed in building a large scale quantum computer? Only time will tell. To build a useful quantum computer requires unprecedented precise control of quantum systems, and complicated dialogues between the classical and quantum worlds. Though Nature seems to favor simplicity, she is also fond of complexity as evidenced by our own existence. Therefore, there is no reason to believe that she would not want to claim quantum computers as her own.

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