Simultaneous single-qubit driving of semiconductor spin qubits at the fault-tolerant threshold.

Practical Quantum computing hinges on the ability to control large numbers of qubits with high fidelity. Quantum dots define a promising platform due to their compatibility with semiconductor manufacturing. Moreover, high-fidelity operations above 99.9% have been realized with individual qubits, though their performance has been limited to 98.67% when driving two qubits simultaneously. Here we present single-qubit randomized benchmarking in a two-dimensional array of spin qubits, finding native gate fidelities as high as 99.992(1)%. Furthermore, we benchmark single qubit gate performance while simultaneously driving two and four qubits, utilizing a novel benchmarking technique called N-copy randomized benchmarking, designed for simple experimental implementation and accurate simultaneous gate fidelity estimation. We find two- and four-copy randomized benchmarking fidelities of 99.905(8)% and 99.34(4)% respectively, and that next-nearest neighbor pairs are highly robust to cross-talk errors. These characterizations of single-qubit gate quality are crucial for scaling up quantum information technology.

Valley Splitting in Silicon from the Interference Pattern of Quantum Oscillations.

We determine the energy splitting of the conduction-band valleys in two-dimensional electrons confined in silicon metal oxide semiconductor Hall-bar transistors. These silicon metal oxide semiconductor Hall bars are made by advanced semiconductor manufacturing on 300 mm silicon wafers and support a two-dimensional electron gas of high quality with a maximum mobility of 17.6×10^{3} cm^{2}/Vs and minimum percolation density of 3.45×10^{10} cm^{-2}. Because of the low disorder, we observe beatings in the Shubnikov-de Haas oscillations that arise from the energy splitting of the two low-lying conduction band valleys. From the analysis of the oscillations beating patterns up to T=1.7 K, we estimate a maximum valley splitting of ΔE_{VS}=8.2 meV at a density of 6.8×10^{12} cm^{-2}. Furthermore, the valley splitting increases with density at a rate consistent with theoretical predictions for a near-ideal semiconductor-oxide interface.

Spin Relaxation Benchmarks and Individual Qubit Addressability for Holes in Quantum Dots.

We investigate hole spin relaxation in the single- and multihole regime in a 2 × 2 germanium quantum dot array. We find spin relaxation times T1 as high as 32 and 1.2 ms for quantum dots with single- and five-hole occupations, respectively, setting benchmarks for spin relaxation times for hole quantum dots. Furthermore, we investigate qubit addressability and electric field sensitivity by measuring resonance frequency dependence of each qubit on gate voltages. We can tune the resonance frequency over a large range for both single and multihole qubits, while simultaneously finding that the resonance frequencies are only weakly dependent on neighboring gates. In particular, the five-hole qubit resonance frequency is more than 20 times as sensitive to its corresponding plunger gate. Excellent individual qubit tunability and long spin relaxation times make holes in germanium promising for addressable and high-fidelity spin qubits in dense two-dimensional quantum dot arrays for large-scale quantum information.

A single-hole spin qubit.

Qubits based on quantum dots have excellent prospects for scalable quantum technology due to their compatibility with standard semiconductor manufacturing. While early research focused on the simpler electron system, recent demonstrations using multi-hole quantum dots illustrated the favourable properties holes can offer for fast and scalable quantum control. Here, we establish a single-hole spin qubit in germanium and demonstrate the integration of single-shot readout and quantum control. We deplete a planar germanium double quantum dot to the last hole, confirmed by radio-frequency reflectrometry charge sensing. To demonstrate the integration of single-shot readout and qubit operation, we show Rabi driving on both qubits. We find remarkable electric control over the qubit resonance frequencies, providing great qubit addressability. Finally, we analyse the spin relaxation time, which we find to exceed one millisecond, setting the benchmark for hole quantum dot qubits. The ability to coherently manipulate a single hole spin underpins the quality of strained germanium and defines an excellent starting point for the construction of quantum hardware.

Fast two-qubit logic with holes in germanium.

Universal quantum information processing requires the execution of single-qubit and two-qubit logic. Across all qubit realizations1, spin qubits in quantum dots have great promise to become the central building block for quantum computation2. Excellent quantum dot control can be achieved in gallium arsenide3-5, and high-fidelity qubit rotations and two-qubit logic have been demonstrated in silicon6-9, 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.

Gate-controlled quantum dots and superconductivity in planar germanium.

Superconductors and semiconductors are crucial platforms in the field of quantum computing. They can be combined to hybrids, bringing together physical properties that enable the discovery of new emergent phenomena and provide novel strategies for quantum control. The involved semiconductor materials, however, suffer from disorder, hyperfine interactions or lack of planar technology. Here we realise an approach that overcomes these issues altogether and integrate gate-defined quantum dots and superconductivity into germanium heterostructures. In our system, heavy holes with mobilities exceeding 500,000 cm2 (Vs)-1 are confined in shallow quantum wells that are directly contacted by annealed aluminium leads. We observe proximity-induced superconductivity in the quantum well and demonstrate electric gate-control of the supercurrent. Germanium therefore has great promise for fast and coherent quantum hardware and, being compatible with standard manufacturing, could become a leading material for quantum information processing.

Strong spin-photon coupling in silicon.

Long coherence times of single spins in silicon quantum dots make these systems highly attractive for quantum computation, but how to scale up spin qubit systems remains an open question. As a first step to address this issue, we demonstrate the strong coupling of a single electron spin and a single microwave photon. The electron spin is trapped in a silicon double quantum dot, and the microwave photon is stored in an on-chip high-impedance superconducting resonator. The electric field component of the cavity photon couples directly to the charge dipole of the electron in the double dot, and indirectly to the electron spin, through a strong local magnetic field gradient from a nearby micromagnet. Our results provide a route to realizing large networks of quantum dot-based spin qubit registers.

Dephasing rates for weak localization and universal conductance fluctuations in two dimensional Si:P and Ge:P δ-layers.

We report quantum transport measurements on two dimensional (2D) Si:P and Ge:P δ-layers and compare the inelastic scattering rates relevant for weak localization (WL) and universal conductance fluctuations (UCF) for devices of various doping densities (0.3-2.5 × 1018 m-2) at low temperatures (0.3-4.2 K). The phase breaking rate extracted experimentally from measurements of WL correction to conductivity and UCF agree well with each other within the entire temperature range. This establishes that WL and UCF, being the outcome of quantum interference phenomena, are governed by the same dephasing rate.

Spontaneous breaking of time-reversal symmetry in strongly interacting two-dimensional electron layers in silicon and germanium.

We report experimental evidence of a remarkable spontaneous time-reversal symmetry breaking in two-dimensional electron systems formed by atomically confined doping of phosphorus (P) atoms inside bulk crystalline silicon (Si) and germanium (Ge). Weak localization corrections to the conductivity and the universal conductance fluctuations were both found to decrease rapidly with decreasing doping in the Si:P and Ge:P delta layers, suggesting an effect driven by Coulomb interactions. In-plane magnetotransport measurements indicate the presence of intrinsic local spin fluctuations at low doping, providing a microscopic mechanism for spontaneous lifting of the time-reversal symmetry. Our experiments suggest the emergence of a new many-body quantum state when two-dimensional electrons are confined to narrow half-filled impurity bands.

Lithography and doping in strained Si towards atomically precise device fabrication.

We investigate the ability to introduce strain into atomic-scale silicon device fabrication by performing hydrogen lithography and creating electrically active phosphorus δ-doped silicon on strained silicon-on-insulator (sSOI) substrates. Lithographic patterns were obtained by selectively desorbing hydrogen atoms from a H resist layer adsorbed on a clean, atomically flat sSOI(001) surface with a scanning tunnelling microscope tip operating in ultra-high vacuum. The influence of the tip-to-sample bias on the lithographic process was investigated allowing us to pattern feature-sizes from several microns down to 1.3 nm. In parallel we have investigated the impact of strain on the electrical properties of P:Si δ-doped layers. Despite the presence of strain inducing surface variations in the silicon substrate we still achieve high carrier densities (>1.0 × 10(14) cm(-2)) with mobilities of ∼100 cm(2) V(-1) s(-1). These results open up the possibility of a scanning-probe lithography approach to the fabrication of strained atomic-scale devices in silicon.

Epitaxial top-gated atomic-scale silicon wire in a three-dimensional architecture.

Three-dimensional (3D) control of dopant profiles in silicon is a critical requirement for fabricating atomically precise transistors. We demonstrate conductance modulation through an atomic scale 3 nm wide δ-doped silicon-phosphorus wire using a vertically separated epitaxial doped Si:P top-gate. We show that intrinsic crystalline silicon grown at low temperatures (∼250 °C) serves as an effective gate dielectric permitting us to achieve large gate ranges (∼2.6 V) with leakage currents below 1 pA. Combining scanning tunneling lithography for precise lateral confinement, with monolayer doping and low temperature epitaxial overgrowth for precise vertical confinement, we can realize multiple layers of nano-patterned dopants in a single crystal material. These results demonstrate the viability of highly doped, vertically separated epitaxial gates in an all-crystalline architecture with long-term implications for monolithic 3D silicon circuits and for the realization of atomically precise donor architectures for quantum computing.

Stacking of 2D electron gases in Ge probed at the atomic level and its correlation to low-temperature magnetotransport.

Stacking of two-dimensional electron gases (2DEGs) obtained by δ-doping of Ge and patterned by scanning probe lithography is a promising approach to realize ultrascaled 3D epitaxial circuits, where multiple layers of active electronic components are integrated both vertically and horizontally. We use atom probe tomography and magnetotransport to correlate the real space 3D atomic distribution of dopants in the crystal with the quantum correction to the conductivity observed at low temperatures, probing if closely stacked δ-layers in Ge behave as independent 2DEGs. We find that at a separation of 9 nm the stacked-2DEGs, while interacting, still maintain their individuality in terms of electron transport and show long phase coherence lengths (∼220 nm). Strong vertical electron confinement is crucial to this finding, resulting in an interlayer scattering time much longer (∼1000 × ) than the scattering time within the dopant plane.

n-type doping of germanium from phosphine: early stages resolved at the atomic level.

To understand the atomistic doping process of phosphorus in germanium, we present a combined scanning tunneling microscopy, temperature programed desorption, and density functional theory study of the reactions of phosphine with the Ge(001) surface. Combining experimental and theoretical results, we demonstrate that PH(2) + H with a footprint of one Ge dimer is the only product of room temperature chemisorption. Further dissociation requires thermal activation. At saturation coverage, PH(2) + H species self-assemble into ordered patterns leading to phosphorus coverages of up to 0.5 monolayers.

Phosphorus atomic layer doping of germanium by the stacking of multiple δ layers.

In this paper we demonstrate the fabrication of multiple, narrow, and closely spaced δ-doped P layers in Ge. The P profiles are obtained by repeated phosphine adsorption onto atomically flat Ge(001) surfaces and subsequent thermal incorporation of P into the lattice. A dual-temperature epitaxial Ge overgrowth separates the layers, minimizing dopant redistribution and guaranteeing an atomically flat starting surface for each doping cycle. This technique allows P atomic layer doping in Ge and can be scaled up to an arbitrary number of doped layers maintaining atomic level control of the interface. Low sheet resistivities (280 Ω/ [symbol see text ) and high carrier densities (2 × 10(14) cm( - 2), corresponding to 7.4 × 10(19) cm( - 3)) are demonstrated at 4.2 K.

A complete fabrication route for atomic-scale, donor-based devices in single-crystal germanium.

Despite the rapidly growing interest in Ge for ultrascaled classical transistors and innovative quantum devices, the field of Ge nanoelectronics is still in its infancy. One major hurdle has been electron confinement since fast dopant diffusion occurs when traditional Si CMOS fabrication processes are applied to Ge. We demonstrate a complete fabrication route for atomic-scale, donor-based devices in single-crystal Ge using a combination of scanning tunneling microscope lithography and high-quality crystal growth. The cornerstone of this fabrication process is an innovative lithographic procedure based on direct laser patterning of the semiconductor surface, allowing the gap between atomic-scale STM-patterned structures and the outside world to be bridged. Using this fabrication process, we show electron confinement in a 5 nm wide phosphorus-doped nanowire in single-crystal Ge. At cryogenic temperatures, Ohmic behavior is observed and a low planar resistivity of 8.3 kΩ/□ is measured.

Preparation of the Ge(001) surface towards fabrication of atomic-scale germanium devices.

We demonstrate the preparation of a clean Ge(001) surface with minimal roughness (RMS ~0.6 Å), low defect densities (~0.2% ML) and wide mono-atomic terraces (~80-100 nm). We use an ex situ wet chemical process combined with an in situ anneal treatment followed by a homoepitaxial buffer layer grown by molecular beam epitaxy and a subsequent final thermal anneal. Using scanning tunneling microscopy, we investigate the effect on the surface morphology of using different chemical reagents, concentrations as well as substrate temperature during growth. Such a high quality Ge(001) surface enables the formation of defect-free H-terminated Ge surfaces for subsequent patterning of atomic-scale devices by scanning tunneling lithography. We have achieved atomic-scale dangling bond wire structures 1.6 nm wide and 40 nm long as well as large, micron-size patterns with clear contrast of lithography in STM images.

Atomic-scale patterning of hydrogen terminated Ge(001) by scanning tunneling microscopy.

In this paper we demonstrate atomic-scale lithography on hydrogen terminated Ge(001). The lithographic patterns were obtained by selectively desorbing hydrogen atoms from a H resist layer adsorbed on a clean, atomically flat Ge(001) surface with a scanning tunneling microscope tip operating in ultra-high vacuum. The influence of the tip-to-sample bias on the lithographic process have been investigated. Lithographic patterns with feature-sizes from 200 to 1.8 nm have been achieved by varying the tip-to-sample bias. These results open up the possibility of a scanning-probe lithography approach to the fabrication of future atomic-scale devices in germanium.