Topological quantum computation has 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 that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, 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 $\ensuremath{\nu}=5∕2$ state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, 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. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Non-Abelian Anyons and Topological Quantum Computation

The so-called Klein paradox—unimpeded penetration of relativistic particles through high and wide potential barriers—is one of the most exotic and counterintuitive consequences of quantum electrodynamics. The phenomenon is discussed in many contexts in particle, nuclear and astro-physics but direct tests of the Klein paradox using elementary particles have so far proved impossible. Here we show that the effect can be tested in a conceptually simple condensed-matter experiment using electrostatic barriers in single- and bi-layer graphene. Owing to the chiral nature of their quasiparticles, quantum tunnelling in these materials becomes highly anisotropic, qualitatively different from the case of normal, non-relativistic electrons. Massless Dirac fermions in graphene allow a close realization of Klein’s gedanken experiment, whereas massive chiral fermions in bilayer graphene offer an interesting complementary system that elucidates the basic physics involved.

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Chiral tunnelling and the Klein paradox in graphene

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 {it Non-Abelian anyons}, meaning that they obey {it 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 nu=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 nu=5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

We provide a broad review of fundamental electronic properties of two-dimensional graphene with the emphasis on density and temperature dependent carrier transport in doped or gated graphene structures. A salient feature of our review is a critical comparison between carrier transport in graphene and in two-dimensional semiconductor systems (e.g. heterostructures, quantum wells, inversion layers) so that the unique features of graphene electronic properties arising from its gap- less, massless, chiral Dirac spectrum are highlighted. Experiment and theory as well as quantum and semi-classical transport are discussed in a synergistic manner in order to provide a unified and comprehensive perspective. Although the emphasis of the review is on those aspects of graphene transport where reasonable consensus exists in the literature, open questions are discussed as well. Various physical mechanisms controlling transport are described in depth including long- range charged impurity scattering, screening, short-range defect scattering, phonon scattering, many-body effects, Klein tunneling, minimum conductivity at the Dirac point, electron-hole puddle formation, p-n junctions, localization, percolation, quantum-classical crossover, midgap states, quantum Hall effects, and other phenomena.

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Electronic transport in two-dimensional graphene

The term photonic crystals appears because of the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures. During the recent years the investigation of one-, two-and three-dimensional periodic structures has attracted a widespread attention of the world optics community because of great potentiality of such structures in advanced applied optical fields. The interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.

http://ieeexplore.ieee.org/iel5/6354/16969/00781867.pdf

Photonic crystals

A modulation‐doped Al0.35Ga0.65As/GaAs single interface structure with a 700 A undoped setback grown by solid‐source molecular beam epitaxy (MBE) shows a Hall mobility of 11.7×106 cm2/V s at a carrier density of 2.4×1011 electrons/cm2 measured in van der Pauw geometry after exposure to light at 0.35 K. This is the highest carrier mobility ever measured in a semiconductor. Similar Al0.32Ga0.68As/GaAs structures with 1000–2000 A setbacks show Hall mobilities in the dark at 0.35 K as high as 4.9×106 cm2 /V s for carrier densities of 5.4×1010 electrons/cm2 and lower.

Electron mobilities exceeding 107 cm2/V s in modulation‐doped GaAs

In the high-magnetic-field low-disorder limit the ground state at v=1/5 Landau-level filling is an incompressible quantum limit. This is determined by observing vanishing resistivity p xx as the temperature T→0. At filling factors below v=1/5 as well as in a narrow region above v=1/5, p xx diverges exponentially as T→0. This contrasts the T dependence at any higher v. The quantum limit at v=1/5 is surrounded by a different phase. In as much as the exponential divergencies are indicative of an electron solid this solid phase is reentrant in a narrow region above v=1/5

Quantum liquid versus electron solid around nu =1/5 Landau-level filling.

We introduce an electrostatic lens for ballistic electrons in two‐dimensional (2D) systems and demonstrate its focusing action in very high mobility GaAs‐(AlGa)As heterostructures. This is the first refractive element for the control of 2D electrons. It exemplifies the close analogy between ballistic propagation in 2D electron systems and traditional geometrical optics.

Electron focusing in two‐dimensional systems by means of an electrostatic lens

We report experimental transport results in 2D electron systems exposed to dipole radiation up to 20 GHz. Magnetoresistance oscillations occur as seen with higher frequency radiation; however, minima here can be seen to extend to negative biases, and zeros are not observed persistently around sample perimeters. Under radiation, voltages are generated from internal to external contacts in the absence of applied driving currents. These findings may be consistent with theoretical pictures of current instabilities due to local negative resistivities.

Evidence for current-flow anomalies in the irradiated 2D electron system at small magnetic fields.

We report on the transport properties of a two-dimensional electron gas (2DEG) confined in an AlGaN∕GaN heterostructure grown by plasma-assisted molecular-beam epitaxy on a semi-insulating GaN template prepared by hydride vapor phase epitaxy with a threading dislocation density of ∼5×107cm−2. Using a gated Hall bar structure, the electron density (ne) is varied from 4.1to9.1×1011cm−2. At T=300mK, the 2DEG displays a maximum mobility of 167000cm2∕Vs at a sheet density of 9.1×1011cm−2, corresponding to a mean-free-path of ∼3μm. Shubnikov–de Haas oscillations, typically not observed at magnetic fields below 2T in GaN, commence at B=0.6T.

K. W. West

Papers

Electron mobilities exceeding 107 cm2/V s in modulation‐doped GaAs

Quantum liquid versus electron solid around nu =1/5 Landau-level filling.

Electron focusing in two‐dimensional systems by means of an electrostatic lens

Evidence for current-flow anomalies in the irradiated 2D electron system at small magnetic fields.

Electron mobility exceeding 160000cm2∕Vs in AlGaN∕GaN heterostructures grown by molecular-beam epitaxy