Topological phases of matter and quantum computation : AMS Special Session on Topological Phases of Matter and Quantum Computation, September 24-25, 2016, Bowdoin College, Brunswick, Maine

This volume contains the proceedings of the AMS Special Session on Topological Phases of Matter and Quantum Computation, held from September 24–25, 2016, at Bowdoin College, Brunswick, Maine. Topological quantum computing has exploded in popularity in recent years. Sitting at the triple point between mathematics, physics, and computer science, it has the potential to revolutionize sub-disciplines in these fields. The academic importance of this field has been recognized in physics through the 2016 Nobel Prize. In mathematics, some of the 1990 Fields Medals were awarded for developments in topics that nowadays are fundamental tools for the study of topological quantum computation. Moreover, the practical importance of this discipline has been underscored by recent industry investments. The relative youth of this field combined with a high degree of interest in it makes now an excellent time to get involved. Furthermore, the cross-disciplinary nature of topological quantum computing provides an unprecedented number of opportunities for cross-pollination of mathematics, physics, and computer science. This can be seen in the variety of works contained in this volume. With articles coming from mathematics, physics, and computer science, this volume aims to provide a taste of different sub-disciplines for novices and a wealth of new perspectives for veteran researchers. Regardless of your point of entry into topological quantum computing or your experience level, this volume has something for you. Cover Title page Contents Preface Bibliography Lie theory for fusion categories: A research primer 1. Lie algebras 2. Representation theory of semisimple Lie algebras 3. Representation theory of quantized enveloping algebras 4. The categories C(g,l,q) 5. Fusion 6. Modular data 7. Symmetries and quantum subgroups References Entanglement and the Temperley-Lieb category 1. Introduction 2. Preliminaries 3. Entanglement analysis 4. O_{N}+-equivariant quantum channels and minimum output entropy estimates 5. The Choi map and Planar Isotopy 6. Future work and open problems References Lifting shadings on symmetrically self-dual subfactor planar algebras References Q-systems and compact W*-algebra objects 1. Introduction 2. Background 3. From Q-systems to W*-algebra objects 4. From W*-algebra objects to Q-systems 5. Equivalence of Q-systems and W*-algebra objects References Dimension as a quantum statistic and the classification of metaplectic categories 1. Introduction 2. Preliminaries and Dimension Functions 3. Properties Determined by Dimension 4. Metaplectic modular categories 5. Modular categories of dimension 16m, with m odd square-free integer References The rank of G-crossed braided extensions of modular tensor categories 1. Introduction 2. The rank of module categories and G-crossed braided extensions References Symmetry defects and their application to topological quantum computing 1. Introduction 2. Algebraic theory of symmetry defects 3. Topological quantum computing with symmetry defects 4. Bilayer Ising with \mathds{Z}2 layer-exchange symmetry 5. Discussion Acknowledgements Appendix A. Data Appendix B. Graphical calculus Appendix C. Calculation of matrix entries of the T-gate protocol References Topological quantum computation with gapped boundaries and boundary defects 1. Introduction 2. Summary 3. Open questions References Classification of gapped quantum liquid phases of matter 1. Introduction 2. Classification of gapped quantum liquids 3. Summary References Schur-type invariants of branched G-covers of surfaces 1. Introduction Acknowledgments 2. C-branched G-covers 3. Stable orbits 4. Outlook References Quantum error-correcting codes over finite Frobenius rings 1. Introduction 2. Discretization and Nice Rings 3. Quantum Error-Correcting Codes 4. Quadratic Residue Codes 5. Conclusions References A short history of frames and quantum designs 1. Introduction 2. From quantum state tomography to quantum designs 3. Known quantum 2-designs References Back Cover