We are a Microsoft research group working on topological quantum computing.

Quantum computers should be capable of performing tasks that would be very difficult, if not impossible, with digital computers, such as finding the prime factors of large numbers, searching large databases, and simulating quantum systems. However, enormous scientific and engineering challenges must be overcome for scalable quantum computers to be realized. A basic problem is that much of the information contained in a quantum system is encoded in phase relations which one might expect to be easily destroyed by its interactions with the outside world. Therefore, error-correction is particularly important for quantum computation. However, error correction can only succeed if the basic error rate is very small. Hence, it is extremely important to build a quantum computer around a quantum system with a low error rate.

Topological quantum computation is a proposal of a particular class of quantum systems. A system with many microscopic degrees of freedom can have ground states whose degeneracy is determined by the topology of the system. The quasiparticle excitations of such a system have exotic braiding statistics, which is a topological effective interaction between them. It is then said to be in a topological phase. This is ideal for quantum information processing, since quantum states can be encoded in nonlocal degrees of freedom which are intrinsically resistant to the debilitating effects of local noise. Errors occur only when the environment supplies enough energy to create quasiparticle excitations which can migrate across the system, thereby affecting its topological degrees of freedom. When the temperature is low compared to the energy gap of the system, such events are exponentially rare.

If the quasiparticle excitations of our topological phase system obey non-Abelian braiding statistics, then braiding operations alone - which are also topologically protected - are sufficient for universal computation. However, very little is known about the conditions under which such states occur. Topological phases occur in the fractional quantum Hall regime (high magnetic fields and low temperatures in two-dimensional electron gases in semiconductor heterostructures and quantum wells). Some of these may be non-Abelian. It has been conjectured that such phases also occur in frustrated magnets, Josephson junction arrays, superconductors of p+ip pairing symmetry, and ultra-cold atoms in optical traps.

In our group, we are (1) investigating the possible occurrence of non-Abelian topological phases in the quantum Hall regime, especially at ν=5/2; (2) determining the conditions under which non-Abelian topological phases could occur in other systems; (3) developing ideas for how topological quantum computation operations could be performed using quantum Hall and other systems; and (4) exploring the mathematical properties of non-Abelian topological phases including the classification of (2+1)-dimensional topological quantum field theories and their higher dimensional analogs.

In order to pursue these aims, our interdisciplinary group brings together expertise in a variety techniques from mathematics, physics, and computer science, including topology, conformal and topological quantum field theory, solid-state physics, statistical mechanics, and numerical simulation methods such as (quantum) Monte Carlo simulations, exact diagonalization and the density-matrix renormalization group.