Quantum Computation with Topological Phases of Matter (08w5103)

Objectives

Although the idea of topological quantum computation as a mathematical
concept is now well-established [Kitaev (1997); Freedman (2001);
Freedman, Kitaev, Larsen and Wang (2001); Freedman, Larsen and Wang
(2002)], relatively little work has gone into its physical
realization. In fact, very few basic principles underlying topological
quantum computation using anyons have been worked out, and the
existence of suitable condensed matter systems (e.g. quantum Hall
systems, quantum magnets, quantum dot and Josephson junction arrays)
is still in doubt. In addition, currently there are no firmly
established practical schemes for carrying out the final state
read-out in a topological quantum computer (even if a suitable system,
e.g. some exotic fractional quantized Hall state, can be
identified). On the other hand, the conceptual proposal of
fault-tolerant topological quantum computation has had a large impact
because of the deep connections it establishes between topology,
condensed matter physics, and quantum computation.

The goal of this workshop is to both study the physical nature of
topological phases as well as to address the most important
theoretical issues connected with any attempt to practically realize a
topological quantum computer. One of the main objectives of the
proposed workshop is to bring together experts from different fields
including condensed matter physics, quantum optics, mathematics, and
quantum information. The exchange of ideas between researchers working
on these subjects will hopefully result in making new inroads into
this broad, inter-disciplinary field. Specifically, we hope to focus
on the following topics.

1. Physical systems with topological order. So far, of experimentally
observed systems, only Fractional Quantum Hall systems are believed to
possess topological order. Numerous other possibilities have been
discussed recently, including atomic systems in optical lattices,
Josephson junction arrays, frustrated magnetic systems and
unconventional superconductors. How to engineer systems supporting a
topological phase remains an open question. No one has yet provided
clear experimental evidence of any exotic statistics in any system.
Experimental detection of topological order, or even detection of its
slightly less exotic cousin -- fractional statistics -- remains an
open problem. How to control topological excitations, a crucial part
of building a topological quantum computer, also remains to be
explored.

2. Model Hamiltonians and mathematical structures. Recently,
advances have been made towards understanding which specific types of
materials and/or Hamiltonians could support a topological phase. Yet,
much remains unknown. Most disturbingly, a general scheme for
classifying topological orders (akin to the Landau theory of symmetry
breaking for conventional orders) is lacking. Part of the workshop
proposed here is concerned with understanding which model Hamiltonians
might have topological phases, how to classify these in full
generality, and which could support universal quantum computation.

3. Topological Algorithms and Computational Architectures. Given that
such topological phases do exist, and are capable of performing
universal quantum computation, one must ask how to build efficient
architecture and how one would actually go about performing such
computations. The structure of topological phases is extremely
complex and imposing the traditional qubit architecture may not be the
most efficient scheme.