The differences between T1 and T2 in real-world quantum computing platforms underscore the importance of studying the thresholds of quantum error-correcting codes under biased noise, also spurring active searches for error-correcting codes with thresholds exceeding the hashing bound under biased noise. Recently, new error-correcting codes such as the XZZX code and the holographic seven-qubit tailored code have exhibited a 50% threshold under pure Pauli noise. Notably, the XZZX code achieves a threshold exceeding the hashing bound in cases of high bias.

This work reports on a holographic quantum error-correcting code, the HaPPY code, which also exhibits a 50% threshold under pure Pauli noise and surpasses the hashing bound threshold under high biased noise. Additionally, this work also explores the threshold of the holographic Steane code under biased noise for comparison.

In addition to studying thresholds under biased noise, this work also investigates the thresholds of various codes, including the Hyper-Invariant Tensor-Network code (HTN code), holographic Reed-Muller code, and some heterogeneous holographic codes, under quantum erasure channels and depolarizing channels.

This work has developed an automated quantum tensor network operator push program, which supports the automated generation of stabilizers and complete logical operators for tensor network quantum error-correcting codes. This greatly enhances the research efficiency of holographic codes, and the program is now ready to be made available to the open-source community.

This thesis investigates the robustness of Gate Set Tomography (GST) under the influence of time-correlated (non-Markovian) noise. GST is a widely used protocol for characterizing quantum gates, yet its efficacy traditionally relies on the assumption of Markovian noise, where errors are memoryless. This research challenges this assumption by introducing non-Markovian noise into simulated GST experiments using a custom-developed Python library. The study com- pares baseline GST performance under Markovian noise with GST results un- der various non-Markovian noise conditions. The findings highlight significant discrepancies in GST’s error reporting and reveal a distinct trade-off between GST’s accuracy and consistency in practical, non-Markovian settings. This work contributes to the broader understanding of error characterization in quantum computing and provides a robust framework for future studies in quantum gate fidelity under realistic noise conditions.

Quantum Markov Semigroups (QMS) describe the evolution of a quantum system by evolving a projection or density operator in time. QMS are generated by a generator obeying the well-known Lindblad equation. However, this is a difficult equation. Therefore, the result that the Lindblad form greatly simplifies in the case of the generator commuting with the modular automorphisms group, is useful. Unfortunately, the proof only works for finite dimensional Hilbert spaces, which is why the aim of this thesis is to generalise this result to countably infinite dimensional Hilbert spaces. To this end, the Lindblad equation is derived from both a mathematical and physical perspective. Where the former relies on rigorous proof and the latter relies on approximations.   In the rigorous case the theory of unital completely positive maps is used. Furthermore, multiple topologies are considered which put less stringent conditions on the operators of interest than the norm topology. Additionally, the Haar measure is used on the unitaries of the bounded linear operators to construct the explicit Lindblad form. To derive the result by employing physical assumptions the interaction picture is used. The physical derivation starts from the Von Neumann equation and uses multiple assumptions to obtain the final Lindblad form. The most important physical assumptions are: the Born approximation, the Markov approximation and the rotating wave approximation.   Furthermore, the main result is the generalisation of the simplified Lindblad form. This simplified form holds for generators commuting with the modular automorphisms group in case the Hilbert spaces are countably infinite dimensional. However, this requires the domain of the generator to be restricted to trace class operators with the identity operator artificially added. Additionally, the generator needs to map strongly convergent sequences to weakly convergent sequences. It also needs to be self-adjoint with respect to the Hilbert-Schmidt inner product. Lastly, the generator is assumed to be self-adjoint with respect to the Gelfand-Naimark-Segal (GNS) inner product <X, Y>=Tr(σ X*Y) for σ a density operator. This last assumption implies that the generator commutes with the modular automorphisms group, which is the symmetry we are considering. Hence, the two previous assumptions are the additional requirements needed to generalise the result, besides the restriction of the domain. Therefore, it is recommended for further research to generalise the result for the domain extended to the bounded operators B(H). It should be noted that the proof heavily relies on the Hilbert space structure induced by the Hilbert-Schmidt inner product. Consequently, the generalisation for the bounded operators would probably require a different approach. Another recommendation is to try and lift the sequence and self-adjoint requirements on the generator. In addition, it is interesting to investigate which physical systems actually have the symmetry of generators commuting with the modular automorphisms group.

This thesis analyzes the occurence of the quantum Zeno effect in a qubit in different situations. A system with a particle with spin 1/2, which represents a qubit, and detector is considered. The detector is modeled by a coordinate q, which has a Gaussian distribution with dispersion σ. There is the free evolution of the qubit and the interaction with the detector. Moreover, there are calculated algebraic expressions for the probabilities of the qubit to be in one of its states at certain moments in time by considering the wave function and density matrix at that time and consequently tracing out the detector coordinate q. Furthermore, there is done an analysis using plots of the evolution of these probabilities in time for different situations. In the situation with one continuous measurement and no free evolution, the qubit remains in its initial state, as expected.In the situation where periods of only free evolution and only measurements alternate, the probabilities keep the same value during the measurement, so then the evolution of the system freezes, and the probabilities evolve in a sine form during the free evolution, in line with the expectation.To neglect free evolution during the measurement, tm << tev is assumed, so there are no series of fast subsequent measurements or a continuous measurement. In this case, the quantum Zeno effect does not occur.Furthermore, we consider the situation where periods of only free evolution and periods with a measurement during free evolution alternate. Now the oscillations continue in time, due to the ongoing free evolution. The larger the influence of the interaction between the qubit and detector during the measurement and the smaller the influence of the magnetic field of the free evolution, the higher the equilibirium position of the oscillations of the probability for the qubit to remain in its initial state is in time. Moreover, the amplitude of these oscillations is smaller. However, due to the free evolution the qubit always has a probability to undergo a transition to its other state. The continuous measurement does not freeze the evolution of the system totally. The quantum Zeno effect does not occur.In the situation where the dispersion σ of the detector coordinate q goes to 0, there is a perfect measurement. The larger σ, the larger the measurement error in the detector resulting in dissipation of the system. The oscillations of the probabilities damp in time.Recommendations for further research include plotting the evolution of the probabilities for a longer period in time with another integration tool. Also, it would be interesting to work out the assumptions done in this analysis to more realistic conditions. Furthermore, another distribution of the detector coordinate q and other qubit states might be interesting to work out in follow-up research.

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.