Preface The factors of any integer can be found quickly by a quantum computer. Since P. Shor discovered this eﬃcient quantum factoring algorithm in 1994 [S], people have started to work on building these new machines. As one of those people, I joined Microsoft Station Q in Santa Barbara to pursue a topological approach in 2005. My dream is to braid non-abelian anyons. So long hours are spent on picturing quasiparticles in fractional quantum Hall liquids. From my oﬃce on UCSB campus, I often see small sailboats sailing in the Pacific Ocean. Many times I am lost in thought imagining that the small sailboats are anyons and the ocean is an electron liquid. Then to carry out a topological quantum computation is as much fun as jumping into such small sailboats and steering them around each other. Will we benefit from such man-made quantum systems besides knowing factors of large integers? A compelling reason for a yes comes from the original idea of R. Feynman: a quantum computer is an eﬃcient universal simulator of quantum mechanics. This was suggested in his original paper [Fe82]. Later, an eﬃcient sim- ulation of topological quantum field theories was given by M. Freedman, A. Kitaev, and the author [FKW]. These results provide support for the idea that quantum computers can eﬃciently simulate quantum field theories, although rigorous results depend on mathematical formulations of quantum field theories. So quantum com- puting literally promises us a new world. More speculatively, while the telescope and microscope have greatly extended the reach of our eyes, quantum computers would enhance the power of our brains to perceive the quantum world. Would it then be too bold to speculate that useful quantum computers, if built, would play an essential role in the ontology of quantum reality? Topological quantum computation is a paradigm to build a large-scale quantum computer based on topological phases of matter. In this approach, information is stored in the lowest energy states of many-anyon systems, and processed by braiding non-abelian anyons. The computational answer is accessed by bringing anyons to- gether and observing the result. Topological quantum computation stands uniquely at the interface of quantum topology, quantum physics, and quantum computing, enriching all three subjects with new problems. The inspiration comes from two seemingly independent themes which appeared around 1997. One was Kitaev’s idea of fault-tolerant quantum computation by anyons [Ki1] the other was Freedman’s program to understand the computational power of topological quantum field the- ories [Fr1]. It turns out the two ideas are two sides of the same coin: the algebraic theory of anyons and the algebraic data of a topological quantum field theory are both modular tensor categories. The synthesis of the two ideas ushered in topo- logical quantum computation. The topological quantum computational model is ix

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