x PREFACE efficiently equivalent to other models of quantum computation, such as the quan- tum circuit model, in the sense that all models solve the same class of problems in polynomial time [FKW, FLW1, FKLW]. Besides its theoretical esthetic appeal, the practical merit of the topological approach lies in its error-minimizing hypothetical hardware: topological phases of matter are fault-avoiding or deaf to most local noises, and unitary gates are imple- mented with exponential accuracy. There exist semi-realistic local model Hamil- tonians whose ground states are proven to be error-correcting codes such as the celebrated toric code. It is an interesting question to understand whether fault- avoidance will survive in more realistic situations, such as at finite temperatures or with thermal fluctuations. Perhaps no amount of modeling can be adequate for us to completely understand Mother Nature, who has repeatedly surprised us with her magic. We do not have any topological qubits yet. Since scalability is not really an issue in topological quantum computation—rather, the issue is controlling more anyons in the system—it follows that demonstrating a single topological qubit is very close to building a topological quantum computer. The most advanced experimental effort to build a topological quantum computer at this writing is fractional quantum Hall quantum computation. There is evidence both experimentally and numerically that non-abelian anyons exist in certain 2-dimensional electron systems that exhibit the fractional quantum Hall effect. Other experimental realizations are conceived in systems such as rotating bosons, Josephson junction arrays, and topological insulators. This book expands the plan of the author’s 2008 NSF-CBMS lectures on knots and topological quantum computing, and is intended as a primer for mathematically inclined graduate students. With an emphasis on introduction to basic notions and current research, the book is almost entirely about the mathematics of topological quantum computation. For readers interested in the physics of topological quantum computation with an emphasis on fractional quantum Hall quantum computing, we recommend the survey article [NSSFD]. The online notes of J. Preskill [P] and A. Kitaev’s two seminal papers [Ki1] [Ki2] are good references for physically inclined readers. The book of F. Wilczek [Wi2] is a standard reference for the physical theory of anyons, and contains a collection of reprints of classic papers on the subject. The CBMS conference gave me an opportunity to select a few topics for a coherent account of the field. No efforts have been made to be exhaustive. The selection of topics is personal, based on my competence. I have tried to cite the original reference for each theorem along with references which naturally extend the exposition. However, the wide-ranging and expository nature of this monograph makes this task very difficult if not impossible. I apologize for any omission in the references. The contents of the book are as follows: Chapters 1,2,4,5,6 are expositions, in some detail, of Temperley-Lieb-Jones theory, the quantum circuit model, ribbon fusion category theory, topological quantum field theory, and anyon theory, while Chapters 3,7,8 are sketches of the main results on the selected topics. Chapter 3 is on the additive approximation of the Jones polynomial, Chapter 7 is on the universality of certain anyonic quantum computing models, and Chapter 8 is on the mathematical models of topological phases of matter. Finally, Chapter 9 lists a
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