Simulations using highly tunable quantum systems may enable investigations of condensed matter systems beyond the capabilities of classical computers. Quantum dots and donors in semiconductor technology define a natural approach to implement quantum simulation. Several material platforms have been used to study interacting charge states, while gallium arsenide has also been used to investigate spin evolution. However, decoherence remains a key challenge in simulating coherent quantum dynamics. Here, we introduce quantum simulation using hole spins in germanium quantum dots. We demonstrate extensive and coherent control enabling the tuning of multi-spin states in isolated, paired, and fully coupled quantum dots. We then focus on the simulation of resonating valence bonds and measure the evolution between singlet product states which remains coherent over many periods. Finally, we realize four-spin states with s-wave and d-wave symmetry. These results provide means to perform non-trivial and coherent simulations of correlated electron systems.

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Highly uniform quantum systems are essential for the practical implementation of scalable quantum processors. While quantum dot spin qubits based on semiconductor technology are a promising platform for large-scale quantum computing, their small size makes them particularly sensitive to their local environment. Here, we present a method to electrically obtain a high degree of uniformity in the intrinsic potential landscape using hysteretic shifts of the gate voltage characteristics. We demonstrate the tuning of pinch-off voltages in quantum dot devices over hundreds of millivolts that then remain stable at least for hours. Applying our method, we homogenize the pinch-off voltages of the plunger gates in a linear array for four quantum dots, reducing the spread in pinch-off voltages by one order of magnitude. This work provides a new tool for the tuning of quantum dot devices and offers new perspectives for the implementation of scalable spin qubit arrays.

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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.

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A strained Ge quantum well, grown on a SiGe/Si virtual substrate and hosting two electrostatically defined hole spin qubits, is nondestructively investigated by synchrotron-based scanning X-ray diffraction microscopy to determine all its Bravais lattice parameters. This allows rendering the three-dimensional spatial dependence of the six strain tensor components with a lateral resolution of approximately 50 nm. Two different spatial scales governing the strain field fluctuations in proximity of the qubits are observed at <100 nm and >1 μm, respectively. The short-ranged fluctuations have a typical bandwidth of 2 × 10-4 and can be quantitatively linked to the compressive stressing action of the metal electrodes defining the qubits. By finite element mechanical simulations, it is estimated that this strain fluctuation is increased up to 6 × 10-4 at cryogenic temperature. The longer-ranged fluctuations are of the 10-3 order and are associated with misfit dislocations in the plastically relaxed virtual substrate. From this, energy variations of the light and heavy-hole energy maxima of the order of several 100 μeV and 1 meV are calculated for electrodes and dislocations, respectively. These insights over material-related inhomogeneities may feed into further modeling for optimization and design of large-scale quantum processors manufactured using the mainstream Si-based microelectronics technology.

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The efficient control of a large number of qubits is one of the most challenging aspects for practical quantum computing. Current approaches in solid-state quantum technology are based on brute-force methods, where each and every qubit requires at least one unique control line—an approach that will become unsustainable when scaling to the required millions of qubits. Here, inspired by random-access architectures in classical electronics, we introduce the shared control of semiconductor quantum dots to efficiently operate a two-dimensional crossbar array in planar germanium. We tune the entire array, comprising 16 quantum dots, to the few-hole regime. We then confine an odd number of holes in each site to isolate an unpaired spin per dot. Moving forward, we demonstrate on a vertical and a horizontal double quantum dot a method for the selective control of the interdot coupling and achieve a tunnel coupling tunability over more than 10 GHz. The operation of a quantum electronic device with fewer control terminals than tunable experimental parameters represents a compelling step forward in the construction of scalable quantum technology.

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The fault-tolerant operation of logical qubits is an important requirement for realizing a universal quantum computer. Spin qubits based on quantum dots have great potential to be scaled to large numbers because of their compatibility with standard semiconductor manufacturing. Here, we show that a quantum error correction code can be implemented using a four-qubit array in germanium. We demonstrate a resonant SWAP gate and by combining controlled-Z and controlled-S⁻¹ gates we construct a Toffoli-like three-qubit gate. We execute a two-qubit phase flip code and find that we can preserve the state of the data qubit by applying a refocusing pulse to the ancilla qubit. In addition, we implement a phase flip code on three qubits, making use of a Toffoli-like gate for the final correction step. Both the quality and quantity of the qubits will require significant improvement to achieve fault-tolerance. However, the capability to implement quantum error correction codes enables co-design development of quantum hardware and software, where codes tailored to the properties of spin qubits and advances in fabrication and operation can now come together to advance semiconductor quantum technology.

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Quantum dots fabricated using methods compatible with semiconductor manufacturing are promising for quantum information processing. In order to fully utilize the potential of this platform, scaling quantum dot arrays along two dimensions is a key step. Here, we demonstrate a two-dimensional quantum dot array where each quantum dot is tuned to single-charge occupancy, verified by simultaneous measurements using two integrated radio frequency charge sensors. We achieve this by using planar germanium quantum dots with low disorder and a small effective mass, allowing the incorporation of dedicated barrier gates to control the coupling of the quantum dots. We measure the hole charge filling spectrum and show that we can tune single-hole quantum dots from isolated quantum dots to strongly exchange coupled quantum dots. These results motivate the use of planar germanium quantum dots as building blocks for quantum simulation and computation.

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Spin quantum bits (qubits) defined in semiconductor quantum dots have emerged as a promising platform for quantum information processing. Various semiconductor materials have been studied as a host for the spin qubit. Over the last decade, research focussed on the group‐IV semiconductor silicon, owing to its compatibility with semiconductor manufacturing technology and the ability to eliminate magnetic noise through isotope purification. However, to this end, hole states in germanium can be considered as well. Furthermore, their low effective mass and high carrier mobility allow for well‐controlled devices, the lack of valley states ensures a well‐defined qubit manifold and the intrinsic spin‐orbit coupling enables all‐electric control. In this thesis, we study strained planer germanium quantum wells, with a focus on applications for quantum information processing.In Chapter 5, we discuss the material platform growth and properties. We show that starting from a silicon wafer, using a reverse grading process, defect‐free, undoped, strained, and shallow germanium quantum wells can be grown, as confirmed by transmission electron microscopy, secondary ion mass spectrometry, and x‐ray measurements. Using heterostructure field‐effect transistors, we characterise the transport properties of the material and find a carrier mobility of μ > 500,000 cm2/Vs. Furthermore, we study the effect of the quantum well depth on the quantum mobility and charge noise sensitivity (Chapter 6) and observe an improvement in both parameters when the quantum well depth is increased from 20 nm to 60 nm.The spin qubit is defined by a hole spin confined in a gate‐defined quantum dot. In Chapter 7 we study the properties of a quantum dot in planar germanium. We describe the nanofabrication process we use to define gate‐controllable quantum dots, contacted by metallic ohmic leads. A nearby quantum dot is used as a charge sensor, which can be read out using high‐bandwidth reflectometry measurements. This allows us to deplete a two‐by‐two quantum dot array to the single‐hole charge occupation, as a host for the spin qubits.Having established a fabrication integration scheme to define quantum dots and ohmic regions, we move to qubit operation in Chapter 8. We measure a double quantum dot in transport and observe a blockade of the transport current for certain hole occupation numbers. This is found to be caused by Pauli spin blockade and can be used to perform the spin‐to‐charge conversion. When a microwave tone resonant with the magnetic field induced Zeeman splitting is applied, the blockaded transport current recovers. This is the result of an induced spin flip, mediated by electric dipole spin resonance (EDSR). Using a tailored measurement technique to increase the signal‐to‐noise ratio of the transport measurements, we demonstrate coherent rotations of the spins in both quantum dots at a Rabi frequency of up to 100 MHz. By operating at the point of the lowest charge noise sensitivity, we find qubit dephasing times beyond 800 ns and a single qubit control fidelity above 99 %. To form a universal quantum gate set, an entangling operation is needed as well. We implement a two‐qubit conditional rotation gate, mediated by the exchange interaction between the qubits. Using the dedicated tunnel barrier gate, we can set the exchange interaction as high as 60 MHz, enabling fast and coherent two‐qubit rotations.Transport measurements only allow for sampling of the average measurement outcome over an ensemble of individual shots. In Chapter 9 we establish single‐shot measurements of a single‐hole spin qubit by making use of a separate radio‐frequency charge sensor. This allows us to isolate the qubits from their hole reservoirs, and we find increased spin relaxation times of over 1 ms. Furthermore, we observe a strong electric modulation of the hole g‐factor that can be attributed to the spin‐orbit coupling and ensures individual qubit addressability.Practical quantum computing applications require large numbers of qubits and many proposals rely on two‐dimensional (2D) layouts to achieve this. As a first step towards 2D grids of spin qubits, we operate a two‐by‐two qubit array in Chapter 10. A latched readout process is implemented to increase the readout visibility and overcome spin relaxation during spin‐to‐charge conversion. Fast single‐qubit gates are achieved using EDSR, with control fidelities of over 99 % for all four qubits. By implementing dynamical decoupling sequences, low‐frequency noise can be mitigated and the phase coherence of the qubit can be increased by several orders of magnitude, up to 100 μs.Harnessing the electric control over the quantum dot coupling, we show the gate‐controlled isolation and coupling of all four qubits, enabling one‐, two‐, and threefold conditional qubit rotations. The large range of control over the exchange interaction also allows performing a controlled phase (CZ) two‐qubit gate in only 10 ns. Implementing a quantum circuit based on CZ gates between all qubits, we coherently entangle and disentangle the four qubits in a Greenberger‐Horne‐Zeilinger (GHZ) state.Finally, in Chapter 11 we study the integration of superconductors into the platform and define gate‐controlled Josephson junctions. We observe a supercurrent through the quantum well over a length up to 6 μm. The critical current of the junction can be modulated using the top gate, up to a maximum IcRN of 17 μV. We demonstrate the Josephson nature of the supercurrent by showing the presence of both the dc and ac Josephson effect. From multiple Andreev reflection and excess current measurements, we extract a characteristic superconducting gap size of 0.2 meV and a junction transparency of 0.6. Finally, we define a superconducting quantum point contact and observe discretisation of the supercurrent, showing superconducting transport restricted to individual channels.@en

The prospect of building quantum circuits^1,2 using advanced semiconductor manufacturing makes quantum dots an attractive platform for quantum information processing^3,4. Extensive studies of various materials have led to demonstrations of two-qubit logic in gallium arsenide⁵, silicon^6–12 and germanium¹³. However, interconnecting larger numbers of qubits in semiconductor devices has remained a challenge. Here we demonstrate a four-qubit quantum processor based on hole spins in germanium quantum dots. Furthermore, we define the quantum dots in a two-by-two array and obtain controllable coupling along both directions. Qubit logic is implemented all-electrically and the exchange interaction can be pulsed to freely program one-qubit, two-qubit, three-qubit and four-qubit operations, resulting in a compact and highly connected circuit. We execute a quantum logic circuit that generates a four-qubit Greenberger−Horne−Zeilinger state and we obtain coherent evolution by incorporating dynamical decoupling. These results are a step towards quantum error correction and quantum simulation using quantum dots.

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Single-charge pumps are the main candidates for quantum-based standards of the unit ampere because they can generate accurate and quantized electric currents. In order to approach the metrological requirements in terms of both accuracy and speed of operation, in the past decade there has been a focus on semiconductor-based devices. The use of a variety of semiconductor materials enables the universality of charge pump devices to be tested, a highly desirable demonstration for metrology, with GaAs and Si pumps at the forefront of these tests. Here, we show that pumping can be achieved in a yet unexplored semiconductor, i.e. germanium. We realise a single-hole pump with a tunable-barrier quantum dot electrostatically defined at a Ge/SiGe heterostructure interface. We observe quantized current plateaux by driving the system with a single sinusoidal drive up to a frequency of 100 MHz. The operation of the prototype was affected by accidental formation of multiple dots, probably due to disorder potential, and random charge fluctuations. We suggest straightforward refinements of the fabrication process to improve pump characteristics in future experiments 2021 The Author(s). Published by IOP Publishing Ltd.

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We engineer planar Ge/SiGe heterostructures for low disorder and quiet hole quantum dot operation by positioning the strained Ge channel 55 nm below the semiconductor/dielectric interface. In heterostructure field effect transistors, we measure a percolation density for two-dimensional hole transport of 2.1 × 10 10 cm−2 , indicative of a very low disorder potential landscape experienced by holes in the buried Ge channel. These Ge heterostructures support quiet
operation of hole quantum dots and we measure an average charge noise level of √SE = 0.6 μeV/√Hz at 1 Hz, with the lowest level below our detection limit√SE = 0.2 μeV/√Hz. These results establish planar Ge as a promising platform for scaledtwo-dimensional spin qubit arrays@en

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.

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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.

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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 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.

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Electrons and holes confined in quantum dots define excellent building blocks for quantum emergence, simulation, and computation. Silicon and germanium are compatible with standard semiconductor manufacturing and contain stable isotopes with zero nuclear spin, thereby serving as excellent hosts for spins with long quantum coherence. Here, we demonstrate quantum dot arrays in a silicon metal-oxide-semiconductor (SiMOS), strained silicon (Si/SiGe), and strained germanium (Ge/SiGe). We fabricate using a multi-layer technique to achieve tightly confined quantum dots and compare integration processes. While SiMOS can benefit from a larger temperature budget and Ge/SiGe can make an Ohmic contact to metals, the overlapping gate structure to define the quantum dots can be based on a nearly identical integration. We realize charge sensing in each platform, for the first time in Ge/SiGe, and demonstrate fully functional linear and two-dimensional arrays where all quantum dots can be depleted to the last charge state. In Si/SiGe, we tune a quintuple quantum dot using the N + 1 method to simultaneously reach the few electron regime for each quantum dot. We compare capacitive crosstalk and find it to be the smallest in SiMOS, relevant for the tuning of quantum dot arrays. We put these results into perspective for quantum technology and identify industrial qubits, hybrid technology, automated tuning, and two-dimensional qubit arrays as four key trajectories that, when combined, enable fault-tolerant quantum computation.

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We fabricate Josephson field-effect transistors in germanium quantum wells contacted by superconducting aluminum and demonstrate supercurrents carried by holes that extend over junction lengths of several micrometers. In superconducting quantum point contacts we observe discretization of supercurrent, as well as Fabry-Pérot resonances, demonstrating ballistic transport. The magnetic field dependence of the supercurrent follows a clear Fraunhofer-like pattern, and Shapiro steps appear upon microwave irradiation. Multiple Andreev reflections give rise to conductance enhancement and evidence a transparent interface, confirmed by analyzing the excess current. These demonstrations of ballistic superconducting transport are promising for hybrid quantum technology in germanium.

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Buried-channel semiconductor heterostructures are an archetype material platform for the fabrication of gated semiconductor quantum devices. Sharp confinement potential is obtained by positioning the channel near the surface; however, nearby surface states degrade the electrical properties of the starting material. Here, a 2D hole gas of high mobility (5 × 10⁵ cm² V⁻¹ s⁻¹) is demonstrated in a very shallow strained germanium (Ge) channel, which is located only 22 nm below the surface. The top-gate of a dopant-less field effect transistor controls the channel carrier density confined in an undoped Ge/SiGe heterostructure with reduced background contamination, sharp interfaces, and high uniformity. The high mobility leads to mean free paths ≈ 6 µm, setting new benchmarks for holes in shallow field effect transistors. The high mobility, along with a percolation density of 1.2 × 10¹¹cm⁻², light effective mass (0.09m_e), and high effective g-factor (up to 9.2) highlight the potential of undoped Ge/SiGe as a low-disorder material platform for hybrid quantum technologies.

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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.

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We define single electron spin qubits in a silicon metal-oxide-semiconductor double quantum dot system. By mapping the qubit resonance frequency as a function of a gate-induced electric field, the spectrum reveals an anticrossing that is consistent with an intervalley spin-orbit coupling. We fit the data from which we extract an intervalley coupling strength of 43 MHz. In addition, we observe a narrow resonance near the primary qubit resonance when we operate the device in the (1,1) charge configuration. The experimental data are consistent with a simulation involving two weakly exchanged-coupled spins with a Zeeman energy difference of 1 MHz, of the same order as the Rabi frequency. We conclude that the narrow resonance is the result of driven transitions between the T- and T+ triplet states, using an electron spin resonance signal of frequency located halfway between the resonance frequencies of the two individual spins. The findings presented here offer an alternative method of implementing two-qubit gates, of relevance to the operation of larger-scale spin qubit systems.

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