Topological quantum computation is a computational paradigm based on topo-
logical phases of matter, which are governed by topological quantum field theories.
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 together and observing the result. 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 implemented with
exponential accuracy. Experimental realizations are pursued in systems such as
fractional quantum Hall liquids and topological insulators.
This book expands on the author’s CBMS lectures on knots and topological
quantum computing and is intended as a primer for mathematically inclined
graduate students. With an emphasis on introducing basic notions and current
research, this book gives the first coherent account of the field, covering a wide
range of topics: Temperley-Lieb-Jones theory, the quantum circuit model, ribbon
fusion category theory, topological quantum field theory, anyon theory, additive
approximation of the Jones polynomial, anyonic quantum computing models, and
mathematical models of topological phases of matter.
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