Qubit arrays in germanium

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.QCD/Veldhorst La

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.QCD/Veldhorst LabBusiness DevelopmentQCD/Scappucci La

Single-hole pump in germanium

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.QCD/Veldhorst LabQuTechBUS/TNO STAFFQN/Veldhorst LabQCD/Scappucci La

A two-dimensional array of single-hole quantum dots

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. QCD/Veldhorst LabQCD/Vandersypen LabBUS/TNO STAFFQCD/Scappucci La

Probing resonating valence bonds on a programmable germanium quantum simulator

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.QCD/Veldhorst LabBUS/TNO STAFFQCD/Scappucci LabQN/Veldhorst La

Low percolation density and charge noise with holes in germanium

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 quietoperation 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 arraysQCD/Scappucci LabQuTechQCD/Veldhorst LabQCD/Vandersypen LabQN/Vandersypen LabBUS/TNO STAFFQN/Veldhorst La

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.QCD/Veldhorst LabQuTechQCD/Vandersypen LabBusiness DevelopmentQCD/Scappucci La

Ballistic supercurrent discretization and micrometer-long Josephson coupling in germanium

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.QCD/Veldhorst LabQuTechBusiness DevelopmentQCD/Scappucci La

Electrical operation of planar Ge hole spin qubits in an in-plane magnetic field

Hole spin qubits in group-IV semiconductors, especially Ge and Si, are actively investigated as platforms for ultrafast electrical spin manipulation thanks to their strong spin-orbit coupling. Nevertheless, the theoretical understanding of spin dynamics in these systems is in the early stages of development, particularly for in-plane magnetic fields as used in the vast majority of experiments. In this work, we present a comprehensive theory of spin physics in planar Ge hole quantum dots in an in-plane magnetic field, where the orbital terms play a dominant role in qubit physics, and provide a brief comparison with experimental measurements of the angular dependence of electrically driven spin resonance. We focus the theoretical analysis on electrical spin operation, phonon-induced relaxation, and the existence of coherence sweet spots. We find that the choice of magnetic field orientation makes a substantial difference for the properties of hole spin qubits. Specifically, we find that (i) EDSR for in-plane magnetic fields varies nonlinearly with the field strength and weaker than for perpendicular magnetic fields. (ii) The EDSR Rabi frequency is maximized when the a.c. electric field is aligned parallel to the magnetic field, and vanishes when the two are perpendicular. (iii) The orbital magnetic field terms make the in-plane g-factor strongly anisotropic in a squeezed dot, in excellent agreement with experimental measurements. (iv) Focusing on random telegraph noise, we show that the effect of noise in an in-plane magnetic field cannot be fully mitigated, as the orbital magnetic field terms expose the qubit to all components of the defect electric field. These findings will provide a guideline for experiments to design ultrafast, highly coherent hole spin qubits in Ge.QCD/Veldhorst LabQN/Veldhorst LabQCD/Scappucci La

Phase flip code with semiconductor spin qubits

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.QCD/Veldhorst LabQCD/Vandersypen LabBUS/TNO STAFFQCD/Terhal GroupQuantum ComputingQCD/Scappucci LabQN/Veldhorst La