Topological Quantum Computing

A Symposium at ETH Zurich 29th-31st August 2009

This is the 8th in the series of Topological Symposia. The preceeding ones being in Paris, Dublin, Leeds and Munich.

Topological Quantum Computation is an exciting young field which draws on many areas of Theoretical Physics. Its focus is the exploitation of topologically ordered systems for quantum information processing. The aim of the symposium is to bring together an international set of experts and encourage them to share their latest ideas and start new collaborations. In particular, we hope to bridge the gap between the (already numerous) different communities (solid state, numerics, mathematical physics, quantum information, etc.) working in this area.

The 8th symposium will be held at ETH Zurich from the 29th to the 31st August 2009 and involves the groups of Atac Imamoglu, Renato Renner and Matthias Troyer. The main scientific organizers are Roger Colbeck and Cyril Stark.

Registration

If you wish to participate, please email Roger Colbeck: with your dates of attendance. If you are interested in giving a talk, please send us a title and short abstract. Note that we may not be able to accept all talks. Please also let us know whether you wish to attend the conference dinner on 30th August (deadline 15th August if you wish to attend the dinner). This will take place at the magnificent UTO Kulm restaurant which overlooks Zurich. If you are attending the dinner, please also let us know if you are vegetarian.

There will be no conference fee, but participants are expected to take care of their own expenses.

Location

The meeting will take place in the Computer Science Department (building RZ, room F 21) of ETH. This is a short walk from Zurich Hauptbahnhof (Zurich main station), a 10 minute train ride from Zurich airport.

If you are travelling by air, we recommend flying to Zurich.  However, both Basel and Bern are reachable by train in roughly 1hr30, and Geneva in about 4hrs.  The connection can be checked on the SBB website.

For local directions, please consult the site information page. We have also made a map showing the venue as well as the hotels and a few local restaurants.

To access the building, please use the entrance at Clausiusstrasse 59 (see map above).

Participants

• Eddy Ardonne (Nordita)
• Dave Bacon (University of Washington)
• Charles Bardyn (ETH Zurich)
• Gavin Brennen (Macquarie University)
• Oliver Buerschaper (Max Planck Institute of Quantum Optics)
• Stefano Chesi (University of Basel)
• Matthias Christandl (LMU)
• Roger Colbeck (ETH Zurich)
• Jürg Fröhlich (ETH Zurich)
• Matthias Gaberdiel (ETH Zurich)
• Viktor Galliard (ETH Zurich)
• Esther Hänggi (ETH Zurich)
• Atac Imamoglu (ETH Zurich)
• Graham Kells (National University of Ireland, Maynooth)
• Ville Lahtinen (University of Leeds)
• Rudolf Morf (Paul Scherrer Institute)
• Ozgur Mustecaplioglu (Koc University)
• Jiannis Pachos (University of Leeds)
• Fabio Pedrocchi (University of Basel)
• Joe Renes (Technische Universität Darmstadt)
• Renato Renner (ETH Zurich)
• Ivan Rodriguez (National University of Ireland, Maynooth)
• Beat Röthlisberger (University of Basel)
• Abbas Al Shimary (University of Leeds)
• Joost Slingerland (Dublin IAS/NUI Maynooth)
• Cyril Stark (ETH Zurich)
• Ady Stern (Weizmann Institute)
• Simon Trebst (Microsoft Research/UC Santa Barbara)
• Matthias Troyer (ETH Zurich)

Programme and Slides of Talks

• The physics of interacting anyonic chains
  Eddy Ardonne, Nordita - Slides
  The study of anyonic quantum spin chains has revealed that these chains have a remarkable rich structure, which is governed by a so-called topological symmetry. The role of this topological symmetry on the various gapped and critical phases will be discussed, as well as the nucleation of novel topological phases.
• Adiabatic Gate Teleportation On Topological Quantum Systems
  Dave Bacon, Washington - Slides
  Adiabatic gate teleportation is a simple protocol for performing universal quantum computing by adiabatically deforming between Hamiltonians whose energy eigenstates are simple quantum error correcting codewords. Here I will discuss this protocol as well as its extension to simple topological models. This later extension will allow for rigorous exploration of the robustness of topological quantum computing models during computation, a subject of some recent controversy.
• Quantum Walks with Anyons: Decoherence by Knots
  Gavin Brennen, Macquarie University - Slides
  Quantum walks describe coherent dynamics which are distinct from classical random walks (e.g. quadratic vs. linear dispersion) but reducing to classical walks in the presence of decoherence. I will describe quantum walks with anyonic particles on a disk. The fusion degrees of freedom of non-Abelian anyons act as an environment which ``entangles'' the walkers trajectory with a knot in the world lines of the anyons and the degree of decoherence is quantified by evaluating link invariants for the quantum group representation of the anyons. Limits toward classical and quantum walk behavior will be discussed.
• Quantum double lattice models from Hopf algebras, and tensor networks
  Oliver Buerschaper, Max Planck Institute of Quantum Optics
  We show how to extend quantum double lattice models to certain non-trivial Hopf algebras and derive tensor network representations for their ground states. We analyze the structure of these tensor networks and relate them to other examples with topological order obtained from string-net models.
• An Introduction to the Theory of Incompressible Quantum Hall Fluids (tutorial)
  Jürg Fröhlich, ETH Zurich
  As suggested by several people, incompressible Hall fluids exhibiting quasi-particles with non-abelain braid statistics - assuming they can be realized - may be systems that can be used for topological quantum computing, although they will not take the form of "laptops", any time soon. In this lecture, I will present a short summary of a theory of incompressible Hall fluids that I have been involved in developing. I will recall what incompressible Hall fluids are and how to characterize their main physical properties. Subsequently, I will outline a general classification of such fluids. The special example of Hall fluids with a filling factor of 5/2 will be considered in more detail. Some remarks about quantum Hall interferometry will be made. Applications to topological quantum computing will be left to other speakers.
• Some basics of CFT (tutorial)
  Matthias Gaberdiel, ETH Zurich
  I will give a basic introduction to 2-dimensional conformal field theory with a particular emphasis on explaining the structure of WZW models.
• A stabilized solution to Kitaev's Honeycomb model
  Graham Kells, National University of Ireland, Maynooth
  In this talk I will discuss a new solution to Kitaev's honeycomb lattice model , see arXiv:0903.5211. The method of solution is by fermionization through a Jordan-Wigner type transform that has been adapted from perturbative methods used to analyse the Abelian phase. Specifically I will show that the ground-state of the system, valid in both Abelian and non-Abelian phases, is a BCS condensate of fermion pairs over a toric code vacuum state with the same vorticity. This constitutes a complete closed form expression for the system ground-state and clearly illustrates the relationship between Abelian and non-Abelian phases in the system. A brief discussion of the ground-state degeneracy on a torus will also be given and I will discuss the mechanism by which the non-Abelian phase becomes 3-fold degenerate.
• Interaction driven phase transitions in an exactly solvable model
  Ville Lahtinen, University of Leeds
• Non-abelian states in the fractional quantum Hall effect: present status
  Rudolf Morf, Paul Scherrer Institute
  We review some of the theoretical proposals for non-abelian fractional quantum Hall (FQH) states. We discuss which ones may be realizable in experimental systems and discuss recent experimental and theoretical results which make us hopeful in the case of the \nu=5/2 FQH state. We also review recent numerical results for a possible non-abelian state at \nu=12/5.
• Why should anyone care about computing with anyons?
  Jiannis Pachos, University of Leeds - Slides
  Two dimensional topological models appear in strongly correlated quantum systems that support anyons, vortex-like quasiparticles with classical and quantum properties. They are coupled to each other by effective gauge theories that give rise to their exotic statistics. After a brief overview of anyonic systems we will look in detail how such vortices can emerge in an exactly solvable lattice model.
• Hierarchical Hall states for the second Landau level
  Joost Slingerland, Dublin IAS/NUI Maynooth
• Fractional topological insulators
  Ady Stern, Weizmann Institute - Slides
  We analyze generalizations of two dimensional topological insulators which can be realized in interacting, time reversal invariant electron systems. These states, which we call fractional topological insulators, contain excitations with fractional charge and statistics in addition to protected edge modes. In the case of s^z conserving toy models, we show that a system is a fractional topological insulator if and only if \sigma_{sH}/e^* is odd, where \sigma_{sH} is the spin-Hall conductance in units of e/2\pi, and e^* is the elementary charge in units of e. We find that systems with 1/e^* even cannot support fractional topological insulators.
• Collective states of interacting anyons, edge states, and the nucleation of topological liquids
  Simon Trebst, Microsoft Station Q - Slides
  Interactions mediated by quasiparticle tunneling split the degenerate space of states formed by a set of localized, non-Abelian anyons in two spatial dimensions. Here we show that this splitting selects a unique collective state as new ground states and results in the nucleation of a novel gapped quantum liquid inside the original parent liquid (of which the anyons are excitations). The nucleated liquid is separated from the parent liquid by a neutral, chiral edge state which we characterize. This physics is at play for non-Abelian quantum Hall states o the center of the plateau.

Sponsorship

We are pleased to receive support from the Center For Theoretical Studies, which is funded by the Swiss National Science Foundation.