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

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.

Oscillatory effects in a strong magnetic field B and magnetic susceptibility are investigated, as applied to 2D systems, in which the twofold spin degeneracy is lifted by the spin-orbit-interaction Hamiltonian HSO= alpha ( sigma *k). nu . The term HSO is shown to change greatly the usual patterns of B-1-periodic oscillations; some oscillations are strongly suppressed due to the diminishing of the gaps between adjacent levels, and new oscillations appear due to intersections of levels.

https://iopscience.iop.org/article/10.1088/0022-3719/17/33/015/pdf

Oscillatory effects and the magnetic susceptibility of carriers in inversion layers

Random-matrix theories in quantum physics : common concepts

Impurity centres in diamond have recently attracted attention in the context of quantum information processing. Now their use as magnetic-field sensors is explored, promising a fresh approach to single-spin detection and magnetic-field imaging at the nanoscale.

http://yacoby.physics.harvard.edu/Publications/High-sensitivity%20diamond%20magnetometer%20with%20nanoscale%20resolution%20_2008.pdf

High-sensitivity diamond magnetometer with nanoscale resolution

We describe the implementation of a scanning Hall probe microscope of outstanding magnetic field sensitivity (∼0.1 G) and unprecedented spatial resolution (∼0.35 μm) to detect surface magnetic fields at close proximity to a sample. Our microscope combines the advantages of a submicron Hall probe fabricated on a GaAs/Al0.3Ga0.7As heterostructure chip and the scanning tunneling microscopy technique for precise positioning. We demonstrate its usefulness by imaging individual vortices in high Tc La1.85Sr0.15CuO4 films and superconducting networks, and magnetic bubble domains.

Scanning Hall probe microscopy

The edges of quantum Hall fluids behave as one-dimensional conductors. This article reviews electron transport into these edge states, covering both the theory based on the chiral Luttinger liquid and the experimental findings using electron tunneling as the probe. The first part of the review presents a basic description of this theory, including a derivation of the density of states, to provide a framework and language for discussing the experimental observations. The signature of the chiral Luttinger liquid is a power-law behavior for the density of states and the tunneling conductances. Experimentally, two techniques have been applied to study the tunneling conductance, using a gated point contact between two quantum Hall edges, or using a cleaved-edge barrier between an edge and a normal conductor. The point-contact method exhibits resonant tunneling, which appears to show some aspects of the Luttinger liquid, and the cleaved-edge method has yielded clear power-law dependences in the off-resonance conductances. Power-law behavior over many orders of magnitude is observed, confirming the Luttinger-liquid character of the edge states. However, the power-law exponents, while in agreement with finite-size numerical calculations, can differ from the universal values predicted by the Chern-Simon field theory. This disagreement is still not well understood. The review concludes with a brief survey of other one-dimensional conductors that have been studied to look for characteristics of the nonchiral Tomonaga-Luttinger liquid.

Chiral Luttinger liquids at the fractional quantum Hall edge

Combined magnetotransport and cyclotron-resonance experiments in a two-dimensional hole system at a modulation-doped GaAs-(AlGa)As heterojunction show that the Kramers degeneracy of the lowest subband is lifted for finite $k$ giving rise to two cyclotron masses ${{m}_{1}}^{*}=0.38{m}_{0}$ and ${{m}_{2}}^{*}=0.60{m}_{0}$ at ${E}_{\mathrm{F}}=2.4$ meV. The observation of plateaus in ${\ensuremath{\rho}}_{xy}$ shows that the quantized Hall effect is independent of the details of the host band structure.

Energy structure and quantized Hall effect of two-dimensional holes

Double quantum dots provide an ideal model system for studying interactions between localized impurity spins. We report on the transport properties of a series-coupled double quantum dot as electrons are added one by one onto the dots. When the many-body molecular states are formed, we observe a splitting of the Kondo resonance peak in the differential conductance. This splitting reflects the energy difference between the bonding and antibonding states formed by the coherent superposition of the Kondo states of each dot. The occurrence of the Kondo resonance and its magnetic field dependence agree with a simple interpretation of the spin status of a double quantum dot.

https://science.sciencemag.org/content/sci/293/5538/2221.full.pdf

The Kondo effect in an artificial quantum dot molecule.

We measure the tunneling conductance ( $G$) and current-voltage ( $I\ensuremath{-}V$) characteristics for electron tunneling from a bulk doped-GaAs normal metal into the abrupt edge of a fractional quantum Hall effect. We observe clear power law behavior for both the $I\ensuremath{-}V$ relationship and the conductance versus temperature. For tunneling into the $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3$ edge the power law persists for one decade in $V$ ( $T$), and 2.7 (1.7) decades in $I$ ( $G$). This strongly indicates the 1/3 edge behaves like a chiral Luttinger liquid. In contrast, the $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ edge is essentially linear, while the 2/3 edge is slightly nonlinear.

Albert M. Chang

Papers

Scanning Hall probe microscopy

Chiral Luttinger liquids at the fractional quantum Hall edge

Energy structure and quantized Hall effect of two-dimensional holes

The Kondo effect in an artificial quantum dot molecule.

Observation of Chiral Luttinger Behavior in Electron Tunneling into Fractional Quantum Hall Edges.