Topological Quantum Computation
About this Title
Zhenghan Wang, Microsoft, Santa Barbara, CA
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 2010: Volume 112
ISBNs: 978-0-8218-4930-9 (print); 978-1-4704-1570-9 (online)
MathSciNet review: MR2640343
MSC: Primary 81P68; Secondary 18D10, 57M27, 57R56, 68Q12, 81T45
Topological quantum computation is a computational paradigm based on topological 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.
Graduate students and research mathematicians interested in quantum computers, topological quantum field theory.
Table of Contents
- Chapter 1. Temperley-Lieb-Jones theories
- Chapter 2. Quantum circuit model
- Chapter 3. Approximation of the Jones polynomial
- Chapter 4. Ribbon fusion categories
- Chapter 5. (2+1)-TQFTs
- Chapter 6. TQFTs in nature
- Chapter 7. Topological quantum computers
- Chapter 8. Topological phases of matter
- Chapter 9. Outlook and open problems