Quantum computers can solve specific problems with practical applications efficiently faster than classical computers. Spin qubits in semiconductor quantum dots are one of the most promising physical realizations of the quantum computers. This thesis aims to investigate the dynamics of semiconductor spin qubits in their actual environment. Specifically, we aim to understand how the actual environment of the spin qubits give rise to nonlinear response of the qubits to external driving, crosstalk, dephasing (T2 processes), and the temperature-dependence of the qubit frequency.
Chapter 3 reports on experimental observation of the nonlinear response of the spin qubits to external driving as well as the crosstalk effect, where the Rabi frequency of an adjacent qubit changes as the target qubit is driven. We propose a phenomenological model that relates the external drivings to the observed dynamics of the spin qubits. The physical mechanism that give rise to these phenomena could not be reproduced in our analysis.
Given the progress in reducing noise sources in the spin qubits environment, it is pertinent to investigate the dephasing of spin qubits in a sparse bath of defects. In Chapter 4, we theoretically investigate the qubit dephasing, as measured in the Ramsey and Hahn echo experiments, in a sparse bath of two-level fluctuators (TLFs) with 1/f spectral density. We find that although the spectral density remains approximately unchanged, the coherence times become more variable as the bath becomes more sparse. We also find that in a sparse bath the qubit decoherence is dominated by only a fraction of TLF defects. Removing these defects results in a significant improvement of the coherence times.
Chapter 5 explores the potential of a bath of TLFs in elucidating the frequency shifts of spin qubits with temperature and the temperature insensitivity of Ramsey and echo decay times. These effects have been observed in experiments. By tuning the bath parameters, we are able to replicate the observed qubit frequency shift. However, our simulations reveal a decrease in qubit decoherence with temperature, which is inconsistent with the experimental findings.
On the whole, Chapters 3 and 5 aim to refine the models that we use to describe the dynamics of spin qubit in their environment. On the other hand, the theoretical work in Chapter 4 is inspired by the experimental observation of the variability of qubit decoherence and offers suggestions to improve coherence times in certain parameter regimes.@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.

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.

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.