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Universal quantum information processing requires the execution of single-qubit and two-qubit logic. Across all qubit realizations 1 , spin qubits in quantum dots have great promise to become the central building block for quantum computation 2 . Excellent quantum dot control can be achieved in gallium arsenide 3 – 5 , and high-fidelity qubit rotations and two-qubit logic have been demonstrated in silicon 6 – 9 , but universal quantum logic implemented with local control has yet to be demonstrated. Here we make this step by combining all of these desirable aspects using hole quantum dots in germanium. Good control over tunnel coupling and detuning is obtained by exploiting quantum wells with very low disorder, enabling operation at the charge symmetry point for increased qubit performance. Spin–orbit coupling obviates the need for microscopic elements close to each qubit and enables rapid qubit control with driving frequencies exceeding 100 MHz. We demonstrate a fast universal quantum gate set composed of single-qubit gates with a fidelity of 99.3 per cent and a gate time of 20 nanoseconds, and two-qubit logic operations executed within 75 nanoseconds. Planar germanium has thus matured within a year from a material that can host quantum dots to a platform enabling two-qubit logic, positioning itself as an excellent material for use in quantum information applications. Spin qubits based on hole states in strained germanium could offer the most scalable platform for quantum computation.

Certain insulators have exotic metallic states on their surfaces. These states are formed by topological effects that also render the electrons travelling on such surfaces insensitive to scattering by impurities. Such topological insulators may provide new routes to generating novel phases and particles, possibly finding uses in technological applications in spintronics and quantum computing.

Graphene and topological insulator two-dimensional electron systems are described by massless Dirac equations. Although the two systems have similar Hamiltonians, they are polar opposites in terms of spin–orbit coupling strength. The status of efforts to achieve long spin-relaxation times in weakly spin–orbit-coupled graphene, and large current-induced spin-polarizations in strongly spin–orbit-coupled topological insulator surface states are reviewed in this Progress Article. The two-dimensional electron systems in graphene and in topological insulators are described by massless Dirac equations. Although the two systems have similar Hamiltonians, they are polar opposites in terms of spin–orbit coupling strength. We briefly review the status of efforts to achieve long spin-relaxation times in graphene with its weak spin–orbit coupling, and to achieve large current-induced spin polarizations in topological-insulator surface states that have strong spin–orbit coupling. We also comment on differences between the magnetic responses and dilute-moment coupling properties of the two systems, and on the pseudospin analogue of giant magnetoresistance in bilayer graphene.

Exciting phenomena may emerge in non-centrosymmetric two-dimensional electronic systems when spin–orbit coupling (SOC) 1 interplays dynamically with Coulomb interactions 2 , 3 , band topology 4 , 5 and external modulating forces 6 – 8 . Here we report synergetic effects between SOC and the Stark effect in centrosymmetric few-layer black arsenic, which manifest as particle–hole asymmetric Rashba valley formation and exotic quantum Hall states that are reversibly controlled by electrostatic gating. The unusual findings are rooted in the puckering square lattice of black arsenic, in which heavy 4 p orbitals form a Brillouin zone-centred Γ valley with p z symmetry, coexisting with doubly degenerate D valleys of p x origin near the time-reversal-invariant momenta of the X points. When a perpendicular electric field breaks the structure inversion symmetry, strong Rashba SOC is activated for the p x bands, which produces spin–valley-flavoured D ± valleys paired by time-reversal symmetry, whereas Rashba splitting of the Γ valley is constrained by the p z symmetry. Intriguingly, the giant Stark effect shows the same p x -orbital selectiveness, collectively shifting the valence band maximum of the D ± Rashba valleys to exceed the Γ Rashba top. Such an orchestrating effect allows us to realize gate-tunable Rashba valley manipulations for two-dimensional hole gases, hallmarked by unconventional even-to-odd transitions in quantum Hall states due to the formation of a flavour-dependent Landau level spectrum. For two-dimensional electron gases, the quantization of the Γ Rashba valley is characterized by peculiar density-dependent transitions in the band topology from trivial parabolic pockets to helical Dirac fermions. Two-dimensional electronic systems in few-layer black arsenic show gate-tunable Rashba bands with unique spin–valley flavours and unconventional quantum Hall states due to synergetic spin–orbit coupling and the Stark effect.

The spin Hall effect is a relativistic spin–orbit coupling phenomenon, which can be used to electrically generate or detect spin currents in non-magnetic systems. This Review discusses the experiments that have established the basic physical understanding of the effect, and the role that several of the spin Hall devices have had in the demonstration of spintronic functionalities and physical phenomena. The spin Hall effect is a relativistic spin–orbit coupling phenomenon that can be used to electrically generate or detect spin currents in non-magnetic systems. Here we review the experimental results that, since the first experimental observation of the spin Hall effect less than 10 years ago, have established the basic physical understanding of the phenomenon, and the role that several of the spin Hall devices have had in the demonstration of spintronic functionalities and physical phenomena. We have attempted to organize the experiments in a chronological order, while simultaneously dividing the Review into sections on semiconductor or metal spin Hall devices, and on optical or electrical spin Hall experiments. The spin Hall device studies are placed in a broader context of the field of spin injection, manipulation, and detection in non-magnetic conductors.

The Hall effect usually occurs in conductors when the Lorentz force acts on a charge current in the presence of a perpendicular magnetic field. Neutral quasi-particles such as phonons and spins can, however, carry heat current and potentially exhibit the thermal Hall effect without resorting to the Lorentz force. We report experimental evidence for the anomalous thermal Hall effect caused by spin excitations (magnons) in an insulating ferromagnet with a pyrochlore lattice structure. Our theoretical analysis indicates that the propagation of the spin waves is influenced by the Dzyaloshinskii-Moriya spin-orbit interaction, which plays the role of the vector potential, much as in the intrinsic anomalous Hall effect in metallic ferromagnets.

Commonly, materials are classified as either electrical conductors or insulators. The theoretical discovery of topological insulators has fundamentally challenged this dichotomy. In a topological insulator, the spin–orbit interaction generates a non-trivial topology of the electronic band structure dictating that its bulk is perfectly insulating, whereas its surface is fully conducting. The first topological insulator candidate material put forward—graphene—is of limited practical use because its weak spin–orbit interactions produce a bandgap of ~ 0.01 K. Recent reexaminations of Bi 2 Se 3 and Bi 2 Te 3 , however, have firmly categorized these materials as strong three-dimensional topological insulators. We have synthesized the first bulk material belonging to an entirely different, weak, topological class, built from stacks of two-dimensional topological insulators: Bi 14 Rh 3 I 9 . Its Bi–Rh sheets are graphene analogues, but with a honeycomb net composed of RhBi 8 cubes rather than carbon atoms. The strong bismuth-related spin–orbit interaction renders each graphene-like layer a topological insulator with a 2,400 K bandgap. Experimental realizations of topological insulators are relatively rare at present. Now, a structurally complex bismuth rhodium iodide is synthesized and shown to have a honeycomb-layered structure akin to that of graphene, but made up of bismuth and rhodium sheets.

The best of both worlds Two of the hottest topics in fundamental condensed matter physics are relativistic Dirac particles and the quantum spin Hall phase. Materials that realize either one of these phenomena are scarce, but new work in bismuth-antimony crystals points to a novel state of quantum matter with both properties. Dirac particles have so far been discovered only in graphene, and the topological edge states central to the quantum spin Hall phase have yet to be directly observed. Hsieh et al . now show, through experimental observation of the simple crystal system Bi 1− x Sb x , that the three-dimensional generalization of both these exotic quantum phases coexist and are highly coupled. This 'topological metal' could be of interest for developing next-generation quantum computing devices. In the conventional quantum Hall effect, a two-dimensional electronic system in the presence of a magnetic field forms metallic conduction paths at the edge of the sample. This paper experimentally demonstrates a sought-after three-dimensional and spontaneous version of this effect; the bulk of a Bi0.9Sb0.1 crystal is shown to be insulating, while two-dimensional metallic conduction paths exist at the surface, without any applied magnetic field. When electrons are subject to a large external magnetic field, the conventional charge quantum Hall effect 1 , 2 dictates that an electronic excitation gap is generated in the sample bulk, but metallic conduction is permitted at the boundary. Recent theoretical models suggest that certain bulk insulators with large spin–orbit interactions may also naturally support conducting topological boundary states in the quantum limit 3 , 4 , 5 , which opens up the possibility for studying unusual quantum Hall-like phenomena in zero external magnetic fields 6 . Bulk Bi 1- x Sb x single crystals are predicted to be prime candidates 7 , 8 for one such unusual Hall phase of matter known as the topological insulator 9 , 10 , 11 . The hallmark of a topological insulator is the existence of metallic surface states that are higher-dimensional analogues of the edge states that characterize a quantum spin Hall insulator 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 . In addition to its interesting boundary states, the bulk of Bi 1- x Sb x is predicted to exhibit three-dimensional Dirac particles 14 , 15 , 16 , 17 , another topic of heightened current interest following the new findings in two-dimensional graphene 18 , 19 , 20 and charge quantum Hall fractionalization observed in pure bismuth 21 . However, despite numerous transport and magnetic measurements on the Bi 1- x Sb x family since the 1960s 17 , no direct evidence of either topological Hall states or bulk Dirac particles has been found. Here, using incident-photon-energy-modulated angle-resolved photoemission spectroscopy (IPEM-ARPES), we report the direct observation of massive Dirac particles in the bulk of Bi 0.9 Sb 0.1 , locate the Kramers points at the sample’s boundary and provide a comprehensive mapping of the Dirac insulator’s gapless surface electron bands. These findings taken together suggest that the observed surface state on the boundary of the bulk insulator is a realization of the ‘topological metal’ 9 , 10 , 11 . They also suggest that this material has potential application in developing next-generation quantum computing devices that may incorporate ‘light-like’ bulk carriers and spin-textured surface currents.

Towards the nanowire qubit A promising new form of quantum bit (qubit), the spin–orbit qubit, may be an improvement over both charge and spin qubits. In quantum physics, the motion of electrons can influence their spins through a fundamental effect called the spin–orbit interaction. Nadj-Perge et al . implement a spin–orbit quantum bit in an indium arsenide nanowire. The spin–orbit qubit is electrically controllable, and information can be stored in the spin. Nanowires are particularly suited to quantum computing as they can serve as one-dimensional templates for scalable qubit registers and can function in both electronic and optical devices. Motion of electrons can influence their spins through a fundamental effect called the spin–orbit interaction. Here, a spin–orbit quantum bit (qubit) is implemented in an indium arsenide nanowire, which should offer significant advantages for quantum computing. The spin–orbit qubit is electrically controllable, and information can be stored in the spin. Moreover, nanowires can serve as one dimensional templates for scalable qubit registers, and are suited for both electronic and optical devices. Motion of electrons can influence their spins through a fundamental effect called spin–orbit interaction. This interaction provides a way to control spins electrically and thus lies at the foundation of spintronics 1 . Even at the level of single electrons, the spin–orbit interaction has proven promising for coherent spin rotations 2 . Here we implement a spin–orbit quantum bit (qubit) in an indium arsenide nanowire, where the spin–orbit interaction is so strong that spin and motion can no longer be separated 3 , 4 . In this regime, we realize fast qubit rotations and universal single-qubit control using only electric fields; the qubits are hosted in single-electron quantum dots that are individually addressable. We enhance coherence by dynamically decoupling the qubits from the environment. Nanowires offer various advantages for quantum computing: they can serve as one-dimensional templates for scalable qubit registers, and it is possible to vary the material even during wire growth 5 . Such flexibility can be used to design wires with suppressed decoherence and to push semiconductor qubit fidelities towards error correction levels. Furthermore, electrical dots can be integrated with optical dots in p–n junction nanowires 6 . The coherence times achieved here are sufficient for the conversion of an electronic qubit into a photon, which can serve as a flying qubit for long-distance quantum communication.

Ferromagnetic surprise The dynamics of spin ordering in magnetic materials is of interest for both fundamental understanding and progress in information-processing and recording technology. Radu et al . study spin dynamics in a ferrimagnetic gadolinium–iron–cobalt (GdFeCo) alloy that is optically excited at a timescale shorter than the characteristic magnetic exchange interaction between the Gd and Fe spins. Using element-specific X-ray magnetic circular dichroism spectroscopy, they show that the Gd and Fe spins switch directions at very different timescales. As a consequence, an unexpected transient ferromagnetic state emerges. These surprising observations, supported by simulations, provide a possible new concept of manipulating magnetic order on a timescale of the exchange interaction. Ferromagnetic or antiferromagnetic spin ordering is governed by the exchange interaction, the strongest force in magnetism 1 , 2 , 3 , 4 . Understanding spin dynamics in magnetic materials is an issue of crucial importance for progress in information processing and recording technology. Usually the dynamics are studied by observing the collective response of exchange-coupled spins, that is, spin resonances, after an external perturbation by a pulse of magnetic field, current or light. The periods of the corresponding resonances range from one nanosecond for ferromagnets down to one picosecond for antiferromagnets. However, virtually nothing is known about the behaviour of spins in a magnetic material after being excited on a timescale faster than that corresponding to the exchange interaction (10–100 fs), that is, in a non-adiabatic way. Here we use the element-specific technique X-ray magnetic circular dichroism to study spin reversal in GdFeCo that is optically excited on a timescale pertinent to the characteristic time of the exchange interaction between Gd and Fe spins. We unexpectedly find that the ultrafast spin reversal in this material, where spins are coupled antiferromagnetically, occurs by way of a transient ferromagnetic-like state. Following the optical excitation, the net magnetizations of the Gd and Fe sublattices rapidly collapse, switch their direction and rebuild their net magnetic moments at substantially different timescales; the net magnetic moment of the Gd sublattice is found to reverse within 1.5 picoseconds, which is substantially slower than the Fe reversal time of 300 femtoseconds. Consequently, a transient state characterized by a temporary parallel alignment of the net Gd and Fe moments emerges, despite their ground-state antiferromagnetic coupling. These surprising observations, supported by atomistic simulations, provide a concept for the possibility of manipulating magnetic order on the timescale of the exchange interaction.