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

Quantum decoherence is one of the most substantial challenges on the way to fullyfledged quantum technology. Noise mitigation based on dynamical control techniques, aside from error correction, is known to be another effective approach to protect qubits from decoherence. In this thesis, we studied the dynamics of a spin qubit interacting with a disordered spin bath in different dimensions. By modeling the environmental spins from fundamental dipolar couplings and employing Monte-Carlo simulations, this research provides an insight into the precise driving and control of a noisy spin qubit, including the noise distribution, decoherence mechanism, driving error, gate fidelity, and performance of dynamical decoupling sequence. This knowledge will be helpful to the future design of noise-robust quantum gates and potential decoupling protocols of spin qubits.

In this thesis we introduce a variation on the quantum random walk to discuss shifts in an arbitrary range. The concept of Hadamard coin was therefore generalised to a higher order. By a Fourier transform method and a tensor product decomposition of the evolution matrix the long-range quantum random walk was found to converge in distribution to a random variable, different for every range. The limiting random variable consists of three parts: one part fast decaying with the range size, a non-convergent part and a convergent part. Lastly, an introduction was made into the topic of trapped quantum random walks. As a starting point, the survival probability of such a walk on a 3-cycle was calculated and found to scale as 2^(-n), as does the classical trapped random walk on this topology.