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arXiv:0904.2771

Topological phases and quantum computation

Alexei Kitaev, Chris R. Laumann

Published Apr 20, 2009• 76 citations
arXivHTMLPDF

Abstract

This is a collection of lecture notes from three lectures given by Alexei Kitaev at the 2008 Les Houches summer school "Exact methods in low-dimensional physics and quantum computing." They provide a pedagogical introduction to topological phenomena in 1-D superconductors and in the 2-D topological phases of the toric code and honeycomb model.

DOI
10.48550/arxiv.0904.2771

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This is a collection of lecture notes from three lectures given by Alexei Kitaev at the 2008 Les Houches summer school "Exact methods in low-dimensional physics and quantum computing." They provide a pedagogical introduction to topological phenomena in 1-D superconductors and in the 2-D topological phases of the toric code and honeycomb model.
Abstract Topological condensed matter physics is a recent arrival among the disciplines of modern physics of a distinctive and substantive nature. Its roots reach far back, but much of its current importance derives from exciting developments in the last half-century. The field is advancing rapidly, growing explosively, and diversifying greatly. There is now a zoo of topological phenomena–the quantum spin Hall effect, topological insulators, Coulomb spin liquids, non-Abelian anyonic statistics and their potential application in topological quantum computing, to name but a few–as well as an increasingly sophisticated set of concepts and methods underpinning their understanding. The aim of this Les Houches Summer School was to present an overview of this field, along with a sense of its origins and its place on the map of advances in fundamental physics. The school comprised a set of basic lectures (Part I) aimed at a pedagogical introduction to the fundamental concepts, which was accompanied by more advanced lectures (Part II) covering individual topics at the forefront of today’s research in condensed matter physics.
Abstract A pedagogical introduction to topologically ordered states of matter is given, with the aim of familiarizing the reader with their axiomatic topological quantum field theory description. Basic non-interacting topological phases of matter protected by symmetries are introduced, including the Su–Schrieffer–Heeger model and the one-dimensional p-wave superconductor. The defining properties of topologically ordered states are illustrated explicitly using the toric code and—on a more abstract level—Kitaev’s 16-fold classification of two-dimensional topological superconductors. Subsequently, a short review of category theory as an axiomatic description of topological order in two-dimensions is presented. With this structure available, Kitaev’s 16-fold way is revisited.
LECTURES 1. Quantum impurity problems in condensed matter physics 2. Conformal field theory and statistical mechanics 3. Quantum Hall effect 4. Topological quantum phases and quantum computation 5. Four lectures on computational statistical physics 6. Loop models 7. Lectures on the integrability of the 6-vertex model 8. Mathematical aspects of 2D phase transitions 9. Numerical simulations of quantum statistical mechanics models 10. Rapidly rotating atomic Bose gases 11. Quantum Hall effect 12. The dimer model 13. Boundary loop models and 2D quantum gravity 14. Real-space condensation in stochastic mass transport models 15. Quantum spin liquids 16. Super spin chains and super sigma models: a short introduction 17. Integrability and combinatorics: selected topics SEMINARS 18. A rigorous perspective on Liouville quantum gravity and KPZ 19. Topologically protected qubits based on Josepshon junction arrays 20. On some quantum Hall states with negative flux 21. Supersolidity and what soluble models can tell us about it