Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as Non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the ν = 5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the ν = 5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

No abstract

Abstract. The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like e −α , where is a length scale, and α is some positive constant. In contrast, the "presumptive" qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about 10 −4 ) before computation can be stabilized.Quantum computation is a catch-all for several models of computation based on a theoretical ability to manufacture, manipulate and measure quantum states. In this context, there are three areas where remarkable algorithms have been found: searching a data base [15], abelian groups (factoring and discrete logarithm) [19], [27], and simulating physical systems [5], [21]. To this list we may add a fourth class of algorithms which yield approximate, but rapid, evaluations of many quantum invariants of three dimensional manifolds, e.g., the absolute value of the Jones polynomial of a link L at certain roots of unity: |V L (e 2πi 5 )|. This seeming curiosity is actually the tip of an iceberg which links quantum computation both to low dimensional topology and the theory of anyons; the motion of anyons in a two dimensional system defines a braid in 2 + 1 dimension. This iceberg is a model of quantum computation based on topological, rather than local, degrees of freedom.The class of functions, BQP (functions computable with bounded error, given quantum resources, in polynomial time), has been defined in three distinct but equivalent ways: via quantum Turing machines [2], quantum circuits [3], [6], and modular functors [7], [8]. The last is the subject of this article. We may now propose a "thesis" in the spirit of Alonzo Church: all "reasonable" computational models which add the resources of quantum mechanics (or quantum field theory) to classical computation yield (efficiently) inter-simulable classes: there is one quantum theory of computation. (But alas, we are not so sure of our thesis at Planck scale energies. Who is to say that all the observables there must even be computable functions in the sense of Turing?)The case for quantum computation rests on three pillars: inevitability-Moore's law suggests we will soon be doing it whether we want to or not, desirability-the above mentioned algorithms, and finally feasibility-which in the past has been

The Pfaffian state is an attractive candidate for the observed quantized Hall plateau at a Landau-level filling fraction nu=5/2. This is particularly intriguing because this state has unusual topological properties, including quasiparticle excitations with non-Abelian braiding statistics. In order to determine the nature of the nu=5/2 state, one must measure the quasiparticle braiding statistics. Here, we propose an experiment which can simultaneously determine the braiding statistics of quasiparticle excitations and, if they prove to be non-Abelian, produce a topologically protected qubit on which a logical Not operation is performed by quasiparticle braiding. Using the measured excitation gap at nu=5/2, we estimate the error rate to be 10(-30) or lower.