Interacting electrons on material lattices can build up strong quantum correlations, which in turn can lead to the emergence of a wide range of novel and potentially useful magnetic and electronic material properties. Our understanding of this physics, however, is severely limited by the exponential growth in complexity with system size, which leads all classical methods to fall fundamentally short. In this thesis, I show how artificial lattices of conduction band electrons in semiconductors, so-called quantum dot arrays, can be used to directly emulate and therefore elucidate such Fermi-Hubbard physics. To this end, I focus on two approaches. A top-down approach allows to scale easily, but lacks to ability to control or measure individual sites. A bottom-up approach on the other hand utilizes the small devices employed by the community for qubit experiments, in which the control of individual sites is both a blessing and a curse. We address the issue of control to the point where mapping to relevant models is possible and efficiently calibrating larger devices becomes feasible. These results open up the inherently well-suited and scalable platform of quantum dots to emulate novel quantum states of matter.@en

For scaling up the qubits in silicon quantum computers, it is vital to determine crosstalk effects that can lower the fidelity of the computer. In this computational project, we examine single-qubit gate-fidelities in the presence of crosstalk for uncoupled spin qubits that are driven with X-gates via electron dipole spin resonance (EDSR). We introduce two models: the first model introduces the AC Stark shift and the novel second model expands on this by adding a resonance frequency shift on top. We assume the latter resonance frequency shift to be due to heating effects. We optimize the gate-fidelity for a qubit coupled to up to six drives as a function of the overall driving time and -frequency of a single drive for both models using the Nelder-Mead algorithm. Using the AC Stark shift model, we still obtain 0.99999 fidelity if we do not account for the crosstalk. However, when using the second model, the fidelity drops to 0.69 in the presence of two drives when we do not correct for the heating-induced resonance frequency shift and the AC Stark shift. Furthermore, the fidelity decreases linearly with the number of drives coupled to the qubit, implicating that the resonance frequency shift will become a significant problem for the scalability of silicon quantum computers. We find that we can correct for the resonance frequency shift entirely by using optimized driving time and -frequency, where most gain comes from optimizing the driving frequency. Moreover, we discover that there is a linearly increasing dependence of the resonance frequency shift at the theoretical driving time as a function of the total drives. Up to a translation factor of 0.5 MHz, we discover the same linear relationship for the correction needed on the theoretical driving frequency to hit maximum fidelity as a function of the total drives.

With the rapid development of Quantum Computers (QC) and QC Simulators, there will be an increased demand for functioning Quantum Algorithms in the near future. Some of the most ubiquitously useful algorithms are solvers for linear systems of equations. Since the conception of the Quantum Linear Solver Algorithm (QLSA) by Harrow, Hassidim and Lloyd (HHL) in 2009, many improvements have been made, although a generic implementation for arbitrary matrices and vectors is still not available. In this thesis a variant of the HHL QLSA is studied, and the open challenges are investigated. Solutions for two of the challenges, namely the Eigenvalue Inversion subroutine and the Higher-Order Ancilla Rotation subroutine, are discussed. As part of the thesis project, these subroutines have been implemented in the QX Quantum Computer Simulator, and the subroutines are combined to form a complete Quantum Linear Solver (QLS), with the restraint that the implementation for the vector and Hamiltonian of the matrix must be provided by the user. A proof-of-concept QLS by Cao et al. is also implemented in the QX simulator, and using the implementation of the vector and Hamiltonian of Cao et al. the complete solver is tested. In the process of this thesis, a framework for basic Quantum Arithmetic is built providing three variants of Integer Adders, two variants of Integer Subtracters, one Integer Multiplier and one Integer Divider. In addition, gates not natively available in the QX simulator are implemented, and a number of improvements and extensions of algorithms presented in the literature are given, making the described algorithms function on the QX simulator and extending features.

Electron spins trapped in quantum dots have recently proven to be a promising technology for the implementation of qubits, already demonstrating high fidelity single- and two-qubits gates. The next step towards fault-tolerant quantum computing is to increase the number of so-called spin qubits on the processor. However, this poses several challenges, one of which being how to implement single-qubit gates in multi-qubit systems, with high fidelity. This work focused on two aspects. The first one was to identify the physical phenomena and limitations that hinder the realisation of high fidelity single-qubit gates in multi-qubit environments. The second objective was to investigate, implement and optimise methods in order to minimise the impact of these perturbing phenomena. The identified perturbing effects include unwanted driving between a pulse and the neighbouring qubits, as well as frequency shifts induced by drive-spin coupling and non-ideal pulse in experimental setups. Crosstalk was addressed using pulse shaping (i.e. amplitude or phase modulation of the driving pulse to tailor its spectral characteristics and reduce the energy transmitted at unwanted frequencies). The performance of shapes taken from both NMR and signal processing literature was investigated, for various pulse parameters, through an extensive grid search. In addition, a frequency correction algorithm was devised: off-resonant drive frequencies are selected so that the shifted qubits are resonantly driven. It currently accounts for two phenomena: the AC Stark and Bloch-Siegert shifts. The algorithm furthermore tracks the changes caused by the time-dependent characteristics of the shaped pulses. The proposed frequency correction algorithm was shown to almost entirely negate the effects of the considered shifts, leading to unitary fidelity improvements by up to 55%. In addition, pulse shaping was demonstrated to noticeably improve the fidelity of simultaneous single-qubit rotations compared to unshaped driving. Rotations with fidelities as high as 99.98% were obtained for pi/2 rotations on two-qubits systems. Moreover, shapes whose Fourier transform is narrow and sharp, associated with low Rabi frequencies, were demonstrated to generally provide the highest fidelities of the tested configurations. Lastly, the trends and guidelines highlighted by these results were shown to scale to systems with larger numbers of qubits. The correction techniques investigated in this work have proven promising for the implementation of high fidelity single-qubit gates in multi-qubit systems. In particular, the guidelines for selecting a well-performing pulse shape should also be useful for the design of optimised driving schemes, regardless of the number of qubits involved. Additionally, the proposed frequency shift correction algorithm is expected to be able to handle arbitrary shifts, and so to be easily adaptable to use in experiments.

Quantum computers promise an exponential speed-up over their classical counterparts for certain tasks relevant to various fields including science, technology, and finance. To unlock this potential, the technology must be scaled up and the errors at play must be reduced. As developments in scalable quantum computation devices advance, the demand for scalable benchmarking techniques that are able to reliably assess the fidelity – the complement of the error rate – of a device has increased significantly. Randomized benchmarking offers a single, concise number that reflects the average fidelity of multi-qubit operations performed on a quantum device. While this method is robust against state preparation and measurement errors, it still suffers from scalability issues. In this thesis, we present a protocol that efficiently predicts the multi-qubit fidelity obtained from randomized benchmarking by only benchmarking single- and two-qubit subspaces, greatly increasing the scalability. The protocol uses simultaneous randomized benchmarking with the aim of catching cross-talk effects while at the same time reducing the number of required benchmarking sequences. We have run numerical simulations of the protocol under two noise models, one depolarizing and one dephasing, to verify its performance. The results of these noisy simulations are promising and suggest that our protocol could offer a valuable tool on the road to developing large-scale quantum computers.

This thesis investigates intrinsic frustrated magnetic systems on an insulating square lattice. The frustrated systems were successfully engineered from magnetic iron atoms on top of an insulating square c(2x2) reconstruction of nitrogen on Cu3Au(100) (copper-gold) surface. Vertical atom manipulation was used to position the magnetic atoms atomically precise on the surface and methods were found to increase the success rate of this. For the investigation of frustration, a numerical simulator was made in Python for modelling frustrated spin structures based on already existing spin models and the following experimental measurements were taken: topography, spectroscopy and current-time traces. The simulations of a frustrated spin system show low energy excited states and the measurements show switching of the system between states.

To harness the potential of quantum mechanics for quantum computation applications, one of the main challenges is to scale up the number of qubits. The work presented in this dissertation is concerned with several aspects that are relevant in the quest of scaling up quantum computing systems based on spin qubits in silicon. Few-qubit experiments are maturing quickly, but simultaneously the lacuna between them and large-scale quantum computers is filled with a combination of science and engineering challenges. The challenges that are addressed in this dissertation are reliable and reproducible sample fabrication, qubit resilience to temperature, spatial correlations in the noise affecting the qubits, and co-integration of qubits with classical control electronics. I start with describing the development of an integration scheme for silicon spin qubits in an academic cleanroom environment, as several research groups have demonstrated over the last years. This has allowed them to successfully fabricate and operate silicon spin qubit devices. The development of such a scheme is crucial for the fabrication of proof-of-principle devices, and the testing of several design variations for more and more complex qubit devices, before transferring the optimal designs to industrial foundries that are generally less flexible. Moreover, it is essential for performing paramount few-qubit experiments in the near term. The developed scheme has been successfully implemented in the next chapter of this thesis. In the first experiment, we investigate the effect of temperature on the spin lifetime, as a first step towards higher temperature operation of silicon spin qubits. Spin qubit operation at elevated temperatures will be required to allow for co-integration of qubits with classical control electronics on a single chip, since the heat load associated with this electronics will be too much to deal with at the current qubit operation temperature of ∼10 mK. At a temperature of ∼1-4 K, significantly more cooling power is available (see for example CERN's Large Hadron Collider). Such co-integration would alleviate the interconnect bottleneck and facilitate the implementation of local control in large-scale devices. We find only a modest temperature dependence and measure a spin relaxation time of 2.8 ms at 1.1 K (still much longer than the record spin dephasing time measured in such a system). In addition, we present a theoretical model and use it in combination with our experimentally obtained parameters to demonstrate that the spin relaxation time can be enhanced by low magnetic field operation and by employing high-valley-splitting devices. Together with more recent work, this experiment demonstrates no fundamental limitations to prevent high-temperature operation of silicon spin qubits. Simultaneously, bringing classical control electronics to lower temperatures also is an active research area. The second experiment uses maximally entangled Bell states of two qubits to study spatial correlations in the noise acting on those two qubits. Spatial correlations in qubit errors hinder quantum error corrections schemes that will be required for fault-tolerant large-scale quantum computers, as these schemes are commonly derived under the assumption of negligible correlations in qubit errors. Therefore, it is important to know to what extent the noise causing these errors is correlated. We find only modest spatial correlations in the noise and gain insight in their origin. The data is in accordance with decoherence being dominated by a combination of nuclear spins and multiple distant charge fluctuators coupling asymmetrically to the two qubits. We recommend to perform similar experiments in isotopically purified silicon to eliminate the effect of nuclear spins and in isolation study spatial correlations in charge noise. Furthermore, our insights show how correlations can be either maximized or minimized through qubit device design. For these reasons, the prospects for the development and implementation of quantum error correction schemes in fault-tolerant large-scale quantum computers are promising. Finally, after having studied several aspects that are relevant to determine the suitability of silicon spin qubits for large-scale quantum computation in the preceding experiments, we propose a concrete physical implementation of co-integrated spin qubits with classical control electronics in a sparse spin qubit array. While the community usually claims compatibility of silicon spin qubits with conventional CMOS fabrication, existing proposals make assumptions that remain to be validated. Implementing quantum error correction protocols in a sparse array has been studied, but the description of a physical implementation was largely missing. The sparseness of the array allows for integration of local control electronics, as shown to be promising earlier in this thesis. Specifically, we propose to implement sample-and-hold circuits alongside the qubit circuitry that would allow to offset inhomogeneity in the qubit array. This enables individual local control and shared global control, resulting in an efficient line scaling. The scalable unit cell design fits 220 (≈106) qubits in ∼150 mm2. We assess the feasibility of the proposed scheme, as well as its physical implementation and the associated footprint, line scaling and interconnect density.@en

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