== Topological quantum information processing and computation: current status ==

Topological quantum computation (TQC) is an approach to quantum information processing that eliminates decoherence at the hardware level by encoding quantum states and gates in global, delocalized properties of the hardware medium.

Most of the current quantum computing schemes assume nearly perfect shielding from the environment. Decoherence makes quantum computing prone to error and nonscalable, allowing only for very small `proof of principle' devices. Error correction software can in principle solve this problem, but progress along this path will take a long time. While much of the current research on other approaches to quantum computation is focused on improving control over well-understood physical systems, TQC research promises fundamental breakthroughs.

Delocalized, or topological degrees of freedom are intrinsically immune to all forms of noise which do not impact the entire medium at once and coherently. For media which exhibit an energy gap, kept at low enough temperatures, this is in fact all conceivable noise. If such materials can be constructed or found in nature, they will allow a much cleaner and faster realization of scalable quantum computation than other schemes.

TQC can be realized in effectively planar (2D) systems whose quasiparticles are anyons, that is they have nontrivial exchange behavior, different from that of bosons or fermions. If, in a system of three or more anyons, the result of sequential exchanges depends on the order in which they are performed, they are called non-Abelian anyons. Systems with non-abelian anyons allow for scalable quantum computation: many-anyon systems have an exponentially large set of topologically protected low-energy states which can be manipulated and distinguished from one another by experimental techniques, such as anyon interferometry recently realized in fractional quantum Hall systems.

A physical system which harbours anyons is said to be topologically ordered, or in a topological phase. One of the most important goals is to study such phases and their non-Abelian anyonic quasiparticles. The most advanced experiments in this direction are done in the context of the fractional quantum Hall effect (FQHE), where phases with fractionally charged Abelian anyons have already been seen and strong experimental evidence for the existence of non-Abelian anyons is emerging. In addition, very promising results have recently been obtained on engineered topologically ordered phases in Josephson junction arrays.

In addition to its natural fault-tolerance, topological quantum computation - though computationally equivalent to the conventional quantum circuit model - is a unique operational model of computation, which represents an original path to new quantum algorithms. New algorithms for approximation of certain hard #P hard computational problems have already been developed and this is opening up new areas of quantum algorithmic research.

### Objectives

The research objectives cover all aspects of topological quantum computation and include:

- Produce clear experimental evidence of topological phases suitable for TQC;

- Design, simulate and build devices for fully scalable topological memory and gates;

- Develop theoretical and algorithmic aspects of topological quantum computation as a new quantum computing paradigm;

- Characterize topological phases and topological phase transitions;

- Propose engineered experimental realizations of topological phases;

- Develop analytical and numerical computing skills for the FQHE and other topological systems.

### Key references

**a) Review articles**

[1] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, *Rev. Mod. Phys.* **80**, 1083 (2008).

[2] G.P. Collins, Computing with quantum knots, *Scientific American* **294**, Issue 4, 56 (2006).

**b) Theory**

[3] M.H. Freedman, M.J. Larsen, and Z. Wag, A modular functor which is universal for quantum computation, *Commun. Math. Phys.* **227**, 605 (2002).

[4] A. Yu. Kitaev, Fault-tolerant quantum computation by anyons, *Ann. Phys* **303**, 2 (2003).

[5] G. Kells, J. K. Slingerland, and J. Vala, Description of Kitaev's honeycomb model with toric-code stabilizers, *Phys. Rev. B* **80**, 125415 (2009).

[6] W. Bishara, P. Bonderson, C. Nayak, K. Shtengel, and J. K. Slingerland, Interferometric signature of non-Abelian anyons, *Phys. Rev. B* **80**, 155303 (2009).

**c) Experiment**

[7] M. Dolev, M. Heiblum, V. Umansky, Ady Stern, D. Mahalu, Observation of a quarter of an electron charge at the : nu = 5/2 quantum Hall state *Nature* **452**, 829 (2008).

[8] I. P. Radu, J. B. Miller, C. M. Marcus, M. A. Kastner, L. N. Pfeiffer, K. W. West, Quasiparticle Properties from Tunneling in the $\nu = 5/2$ Fractional Quantum Hall State, *Science* **320**, 899 (2008).

[9] S. Gladchenko, D. Olaya, E. Dupont-Ferrier, B. Doucot, L.B. Ioffe, and M.E. Gershenson, Superconducting nanocircuits for topologically protected qubits, *Nature Physics* **5**, 48 (2008).

[10] R. L. Willett, L. N. Pfeiffer, and K. W. West, Measurement of filling factor 5/2 quasiparticle interference with observation of charge e/4 and e/2 period oscillations, *Proc. Natl. Acad. Sci.* **106**, 8853 (2009).