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

Anyons in an exactly solved model and beyond

The invention concerns a quantum cascade laser comprising in particular a gain region (14) consisting of several layers (20) each including: alternating strata of a first type (28) defining each an AllnAs quantum barrier and strata of a second type (28) defining each an InGaAs quantum barrier, and injection barriers (22), interposed between two of the layers (20). The layers of the gain region (14) form each an active zone extending from one to the other of the injection barriers (22) adjacent thereto. The strata (26, 28) are dimensioned such that: each of the wells comprises, in the presence of an electric field, at least a first upper subband, a second median subband, and a third lower subband, and the probability of an electron being present in the first subband is highest in the proximity of one of the adjacent injection barriers, in the second subband in the median part of the zone and in the third subband in the proximity of the other adjacent barriers. The laser is formed by a succession of active zones and injection barriers, without interposition of a relaxation zone.

https://science.sciencemag.org/content/sci/264/5158/553.full.pdf

Quantum cascade laser

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.

The canonical example of a quantum-mechanical two-level system is spin. The simplest picture of spin is a magnetic moment pointing up or down. The full quantum properties of spin become apparent in phenomena such as superpositions of spin states, entanglement among spins, and quantum measurements. Many of these phenomena have been observed in experiments performed on ensembles of particles with spin. Only in recent years have systems been realized in which individual electrons can be trapped and their quantum properties can be studied, thus avoiding unnecessary ensemble averaging. This review describes experiments performed with quantum dots, which are nanometer-scale boxes defined in a semiconductor host material. Quantum dots can hold a precise but tunable number of electron spins starting with 0, 1, 2, etc. Electrical contacts can be made for charge transport measurements and electrostatic gates can be used for controlling the dot potential. This system provides virtually full control over individual electrons. This new, enabling technology is stimulating research on individual spins. This review describes the physics of spins in quantum dots containing one or two electrons, from an experimentalist’s viewpoint. Various methods for extracting spin properties from experiment are presented, restricted exclusively to electrical measurements. Furthermore, experimental techniques are discussed that allow for 1 the rotation of an electron spin into a superposition of up and down, 2 the measurement of the quantum state of an individual spin, and 3 the control of the interaction between two neighboring spins by the Heisenberg exchange interaction. Finally, the physics of the relevant relaxation and dephasing mechanisms is reviewed and experimental results are compared with theories for spin-orbit and hyperfine interactions. All these subjects are directly relevant for the fields of quantum information processing and spintronics with single spins i.e., single spintronics.

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Spins in few-electron quantum dots

A simple quantum system, the semiconductor-based double-barrier resonant-tunneling structure, exhibits intrinsic bistability which we attribute to the feedback dependence of the energy of the electronic states in the well on the tunneling current density.

Observation of intrinsic bistability in resonant tunneling structures.

In experiments on resonant tunneling through a "quantum antidot" (a potential hill) in the quantum Hall (QH) regime, periodic conductance peaks were observed as a function of both magnetic field and back gate voltage. A combination of the two periods constitutes a measurement of the charge of the tunneling particles and implies that charge deficiency on the antidot is quantized in units of the charge of quasi-particles of the surrounding QH condensate. The experimentally determined value of the electron charge e is 1.57 x 10–19 coulomb = (0.98 ± 0.03) e for the states v = 1 and v = 2 of the integer QH effect, and the quasi-particle charge is 5.20 x 10–20 coulomb = (0.325 ± 0.01)e for the state v = ⅓ of the fractional QH effect.

https://science.sciencemag.org/content/sci/267/5200/1010.full.pdf

Resonant tunneling in the quantum Hall regime: measurement of fractional charge.

We report observations of an electric-field threshold conduction and of related ac voltage (broad-band noise) generation in low-disorder two-dimensioanl electron systems in the extreme magnetic quantum limit. We interpret these phenomena as definitive evidence for formation of a pinned quantum Wigner crystal and determine its melting phase diagram from the disappearance of threshold and noise behavior at higher temperatures.

V. J. Goldman

Papers

Observation of intrinsic bistability in resonant tunneling structures.

Resonant tunneling in the quantum Hall regime: measurement of fractional charge.

Evidence for two-dimentional quantum Wigner crystal.

Evidence for LO-phonon-emission-assisted tunneling in double-barrier heterostructures.

Detection of composite fermions by magnetic focusing.