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13
Faulttolerant quantum computation by anyons
, 2003
"... A twodimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation ..."
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Cited by 94 (3 self)
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A twodimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is faulttolerant by its physical nature.
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 27 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
The universal chiral partition function for exclusion statistics
, 1998
"... We demonstrate the equality between the universal chiral partition function, which was first found in the context of conformal field theory and RogersRamanujan identities, and the exclusion statistics introduced by Haldane in the study of the fractional quantum Hall effect. The phenomena of multipl ..."
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Cited by 18 (0 self)
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We demonstrate the equality between the universal chiral partition function, which was first found in the context of conformal field theory and RogersRamanujan identities, and the exclusion statistics introduced by Haldane in the study of the fractional quantum Hall effect. The phenomena of multiple representations of the same conformal field theory by different sets of exclusion statistics is discussed in the context of the û(1) theory of a compactified boson of radius R.
Nonabelian anyons and topological quantum computation
 Reviews of Modern Physics
"... Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are partic ..."
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Cited by 10 (0 self)
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Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant 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 NonAbelian anyons, meaning that they obey nonAbelian braiding statistics. Quantum information is stored in states with multiple quasiparticles,
NonAbelian Anyons and Topological Quantum Computation. arxiv: condmat.strel/0707.1889
"... Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are partic ..."
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Cited by 7 (1 self)
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Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant 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 NonAbelian anyons, meaning that they obey nonAbelian braiding statistics. Quantum information is stored in states with multiple quasiparticles,
From new states of matter to a unification of light and electrons
, 2006
"... For a long time, people believe that all possible states of matter are described by Landau symmetrybreaking theory. Recently we find that stringnet condensation provide a mechanism to produce states of matter beyond the symmetrybreaking description. The collective excitations of the stringnet co ..."
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Cited by 3 (0 self)
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For a long time, people believe that all possible states of matter are described by Landau symmetrybreaking theory. Recently we find that stringnet condensation provide a mechanism to produce states of matter beyond the symmetrybreaking description. The collective excitations of the stringnet condensed states turn out to be our old friends, photons and electrons (and other gauge bosons and fermions). This suggests that our vacuum is a stringnet condensed state. Light and electrons in our vacuum have a unified origin – stringnet condensation. 1
Direct observation of fractional statistics in two dimensions
"... In two dimensions, the laws of physics permit existence of anyons, particles with fractional statistics which is neither Fermi nor Bose. That is, upon exchange of two such particles, the quantum state of a system acquires a phase which is neither 0 nor π, but can be any value. The elementary excitat ..."
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Cited by 1 (0 self)
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In two dimensions, the laws of physics permit existence of anyons, particles with fractional statistics which is neither Fermi nor Bose. That is, upon exchange of two such particles, the quantum state of a system acquires a phase which is neither 0 nor π, but can be any value. The elementary excitations (Laughlin quasiparticles) of a fractional quantum Hall fluid have fractional electric charge and are expected to obey fractional statistics. Here we report experimental realization of a novel Laughlin quasiparticle interferometer, where quasiparticles of the 1/3 fluid execute a closed path around an island of the 2/5 fluid and thus acquire statistical phase. Interference fringes are observed as conductance oscillations as a function of magnetic flux, similar to the AharonovBohm effect. We observe the interference shift by one fringe upon introduction of five magnetic flux quanta (5h/e) into the island. The corresponding 2e charge period is confirmed directly in calibrated gate experiments. These results constitute direct observation of fractional statistics of Laughlin quasiparticles.
Anyons and Lowest Landau Level Anyons
 SÉMINAIRE POINCARÉ XI
, 2007
"... Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quant ..."
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Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum manybody states with a multivalued phase factor, which encodes the anyonic nature of particle windings. Bosons and fermions appear as two limiting cases. Gauging away the phase leads to the socalled anyon model, where the charge of each particle interacts ”à la AharonovBohm” with the fluxes carried by the other particles. The multivaluedness of the wave function has been traded off for topological interactions between ordinary particles. An alternative Lagrangian formulation uses a topological ChernSimons term in 2+1 dimensions. Taking into account the short distance repulsion between particles leads to an Hamiltonian well defined in perturbation theory, where all perturbative divergences have disappeared. Together with numerical and semiclassical studies, perturbation theory is a basic analytical tool at disposal to study the model, since
Topological Investigation of the Fractionally Quantized Hall Conductivity
, 1995
"... Abstract. Using the fiber bundle concept developed in geometry and topology, the fractionally quantized Hall conductivity is discussed in the relevant many–particle configuration space. Electronmagnetic field and electronelectron interactions under FQHE conditions are treated as functional connect ..."
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Abstract. Using the fiber bundle concept developed in geometry and topology, the fractionally quantized Hall conductivity is discussed in the relevant many–particle configuration space. Electronmagnetic field and electronelectron interactions under FQHE conditions are treated as functional connections over the torus, the torus being the underlying twodimensional manifold. Relations to the (2+1)–dimensional Chern–Simons theory are indicated. The conductivity being a topological invariant is given as e2 times a linking number which is the quotient of the winding numbers h of the selfconsistent field and the magnetic field, respectively. Odd denominators are explained by the two spin structures which have been considered for the FQHE correlated electron system.
Statistical Phases and Momentum Spacings for One Dimensional Anyons
, 707
"... Anyons and fractional statistics1, 2 are by now well established in twodimensional systems. In one dimension, fractional statistics has been established so far only through Haldane’s fractional exclusion principle3, but not via a fractional phase the wave function acquires as particles are intercha ..."
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Anyons and fractional statistics1, 2 are by now well established in twodimensional systems. In one dimension, fractional statistics has been established so far only through Haldane’s fractional exclusion principle3, but not via a fractional phase the wave function acquires as particles are interchanged. At first sight, the topology of the configuration space appears to preclude such phases in one dimension. Here we argue that the crossings of onedimensional anyons are always unidirectional, which makes it possible to assign phases consistently and hence to introduce a statistical parameter θ. The fractional statistics then manifests itself in fractional spacings of the singleparticle momenta of the anyons when periodic boundary conditions are imposed. These spacings are given by ∆p = 2π�/L (θ/π + nonnegative integer) for a system of length L. This condition is the analogue of the quantisation of relative angular momenta according to lz = �(−θ/π + 2 · integer) for twodimensional anyons. The concept of fractional statistics, as introduced by Leinaas and Myrheim4 and Wilczek5, has generically been associated with identical particles in two space dimensions. It is intimately related to the topology of the configuration space, or the existence of fractional relative angular momentum. Angular momentum does not exist in one dimension (1D), and is quantised in units