Long operation of quantum computers, on the scale of days, weeks, or months, is essential for the realisation of large-scale quantum algorithms. However, qubits interact with their environment, leading to decoherence and the loss of quantum information. These environmental interactions are collectively referred to as noise, and are commonly modelled as stochastic processes. A key assumption in many such models is stationarity, meaning that the statistical properties of the noise do not depend on the absolute time at which the noise is measured.

Recent experiments on semiconductor quantum dots have shown that this assumption may not hold over long timescales. Such non-stationarity may arise when the system has not had sufficient time to equilibrate before measurement, or when fluctuations initialise the system far from equilibrium. In this thesis, we investigate how non-stationarity affects the power spectral density, which describes how the noise power is distributed across frequencies.

We introduce non-stationarity by conditioning the initial state of the noise process away from equilibrium. We first study the conditioned Ornstein-Uhlenbeck process, showing that non-stationarity modifies the amplitude of the ideal power spectral density while preserving its Lorentzian shape. We then study a conditioned ensemble of two-level fluctuators, a common model for charge noise in semiconductor qubits. Surprisingly, we find that non-stationarity can change the low-frequency spectrum from the usual $1/f$ behaviour to a $1/f^2$ dependence. Similar effects have been reported in recent experiments, and our model suggests one possible mechanism by which they may arise.

Finally, we investigate several spectral estimation methods to examine how finite observation times and non-stationarity affect PSD estimation. Overall, this thesis shows that non-stationarity can significantly affect measured noise spectra, and should be considered when interpreting long-timescale noise measurements in qubit devices. ... Read more

We study how global system parameters and local realizations of quenched disorder shape the dynamics of a classical many-body spin system and how classical indicators of chaos, Lyapunov's exponents, relate to quantum signatures of chaos. In particular, we focus on the classical analogue of the quantum spin glass shards model in a random transverse magnetic field studied by Georgeot and Shepelyansky. The classical spin phase space is constructed as a symplectic manifold and we evolve trajectories with a second-order Suzuki–Trotter integrator. This symplectic structure-preserving scheme enables reliable computation of Lyapunov's exponents via standard repeated QR-based orthogonalization of tangent vectors, yielding accurate finite-time Lyapunov spectra for trajectories in different regions of phase space.

Using these tools, we examine the dynamics of trajectories sampled from different regions of phase space while varying two global system parameters: the relative strength of the spin-spin coupling and the transverse magnetic field. We find that both the strong spin-spin coupling and strong magnetic field limits are nearly integrable, with maximal chaos emerging at intermediate coupling–field ratios. When initial spin configurations are sampled uniformly over the Bloch sphere, different disorder realizations do not qualitatively change whether dynamics are chaotic or integrable, while configurations concentrated near the $x$- or $z$-axis are highly sensitive to the specific disorder realization and can exhibit either almost fully integrable or strongly chaotic behavior under identical global parameters.

For \(17\) spins, all observed trajectories in the classical system are chaotic when the spin-spin coupling is approximately three times stronger than the transverse field. A comparison with the quantum level-spacing statistics of the corresponding quantum model shows qualitative agreement regarding which choices of global parameters lead to integrable or chaotic dynamics. However, there is a quantitative mismatch in which global parameter values produce the {strongest} chaotic dynamics. This demonstrates that the relationship between classical Lyapunov exponents and quantum energy-level spacing statistics is complex and non-trivial.
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Spin-photon interfaces play a crucial role in the realization of large-scale quantum computation and quantum communication. In this thesis, we investigate two types of spin-photon interfaces, Group-IV color centers in diamond and trapped Rubidium atoms, and evaluate their potential to enable long-distance quantum communication and modular quantum computation. ... Read more

This paper covers the perfect indirect quantum measurement, specifically in the context of repeated measurement. The indirect measurement is useful as it allows information to be obtained from quantum systems without inflicting much disturbance on them. We restrict ourselves to cases with no evolution of the measured system between measurements and to perfect measurements, that is, measurements from which no outgoing information is missed and no extra information is added. In this case we can make use of the work by M. A. Nielsen (2005). It says that the expected amount of information following a perfect indirect measurement is larger than the information before the measurement. We make use of this result to show that the repeated indirect perfect measurement of a quantum state has two mutually exclusive outcomes. The first outcome is that the measured state becomes a pure state almost surely. The second is that the measurement eventually stops resulting in information being revealed. In the latter case, further measurements on the system result in the state switching through spaces with the same dimension, and thus it does not become a pure state. This paper builds on the work by Maassen and K¨ummerer from 2005, which already proved this, by expanding their proofs and adding additional theorems and proofs to create a more self-contained result. Further studies might look at the rate at which states become pure, and what might influence this rate. ... Read more

Shuttling is expected to play a vital role in scaling up quantum computing based on semiconductor spin qubits. Decoherence, caused by interactions between the qubit and its environment, remains a major obstacle to maintaining qubit fidelity. For stationary qubits, decoherence is typically described using standard random processes. However, trajectories such as forth-back shuttling give rise to random processes with nontrivial correlations. We develop and benchmark a numerical method to analyse the effect of 3D random magnetic fields on shuttled spins, particularly relevant to Ge-based devices where the direction of the effective magnetic field varies significantly. Building on this, we design new dynamical decoupling (DD) sequences for shuttling, incorporating a central pulse to suppress the effect of mirror symmetry between the forward and backwards paths. Using realistic parameters for Si and Ge devices with the developed numerical method, we evaluate the effectiveness of DD sequences at suppressing decoherence during shuttling. In particular, we find that the XY8-Z sequence significantly improves fidelity during forth-back shuttling, and outperforms the standard XY8 sequence commonly used for stationary qubits. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more