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.

In the past few years, the search for good quantum low density parity check (qLDPC) codes suddenly took flight, and many different constructions of these codes have since been presented, including many product constructions. As these code constructions have a natural interpretation in the language of homology, this thesis studies the interplay between homological algebra and various recent product constructions of qLDPC codes. First, we provide an overview of the theory of singular homology, cellular homology, and homological algebra over vector spaces, and use this theory to analyse two product constructions: the hypergraph product construction and the distance balancing procedure. These constructions can be interpreted as tensor products of chain complexes over vector spaces. We present new proofs for results from the literature regarding the distance and the number of encoded qubits of these product codes. Secondly, we survey the theory of homology over ring modules, and use this theory to interpret and analyse another product code construction (the lifted product code construction), which is a generalisation of the hypergraph product construction. We prove a Künneth theorem for these codes, and use this theorem to prove a formula for the number of qubits such codes encode. Thirdly, we investigate the homology of fibre bundles. To this end, we provide an overview of the theory of covering spaces and of fibre bundles, as well as an overview of the theory of homology with local coefficients and of spectral sequences. We then calculate the homology of a specific class of fibre bundles using three different methods. After this discussion, we consider two product code constructions: the fibre bundle product construction and the balanced product construction. We explicate their mathematical foundations, and use these insights to prove two new results for the number of qubits that these product codes encode. Finally, we explain under which conditions these two code constructions and the lifted product construction coincide. Lastly, we consider a specific example of fibre bundle product codes: the twisted toric code. We determine an analytic expression for the distance of such a code and verify this expression using numerical simulations. Furthermore, we perform an extensive numerical study on these codes to determine how performing a twist alters the scaling of the logical error rate of such codes. We present a new analytic method to explain the scaling of these codes on a small domain, and verify the validity of these calculations by comparing them with the results of our numerical simulations.

Quantumcomputation is the modern version of Schrödinger’s cat experiment. It is backed up in principle by the theory and thinking about it can make people equally uncomfortable and excited. Besides, its practical realization seems so extremely challenging that some people even doubt it is possible. On the other hand, we are nowadays much closer to realizing quantum computation and in addition, it has much more implications than Schrödinger’s original cat experiment. One of the major difficulties in realizing quantum computation is the inevitable presence of noise in realistic quantum devices which makes the direct realization of quantum computers impossible. In order to protect quantum information and quantum processes against noise, quantum error correction and fault-tolerance have been devised. Although the gap between experiments and the requirements of fault-tolerance is still daunting, the field of quantum error correction and fault-tolerance extends and influences architectural decisions from the hardware to the ideal quantum programs that we want to run. That is why it has the potential to make or break the practicality of quantum computation and a lot of research effort goes into this field. In this thesis we investigate and improve several aspects of fault-tolerant schemes and quantum error correction. We implement an experiment which validates on a small device the usefulness of fault-tolerance for quantum computation. We investigate the advantages of harnessing quantum continuous degrees of freedom present in the lab to protect discrete quantum information in a scalable way. We establish a framework to analyze the fault-tolerant properties of code deformation techniques which are versatile techniques to process quantum information protected by an error correcting code. We also present some novel code deformation techniques with the potential to increase reliability. Finally we define a new class of quantum error correcting codes, quantum pin codes, with built in capabilities for fault-tolerant quantum gates. We give some practical constructions and show some protocols with interesting parameters. The roads towards universal and fault-tolerant quantum computation are still steep but research efforts are pushing in the right directions.@en

In this work, we first analyse a theoretical technique based on entropic inequalities, which in principle can be used to compare classical and quantum algorithms running on noisy quantum devices. The technique has been used in the past for depolarizing noise, and has shown that for NISQ variational quantum algorithms (VQAs) to give an advantage over purely classical methods, it is essential that the structure of the problem is close to the quantum hardware the VQA is run on. Here, we use this technique for relaxation and pure dephasing noise and examine whether we can deduce the same results for the Quantum Approximate Optimization Algorithm (QAOA) running on a noisy quantum device. In the later part of the thesis, we study the properties of the steady state of the Lindblad Master Equation for a 1D array of transmon qubits coupled by cross-resonance type interactions for relaxation and pure dephasing noise. For n=2,3 number of qubits we solve exactly the master equation, while for a general number of n qubits we approximate the solution by using two different approaches, namely perturbation theory and mean-field theory. Finally, we assess how well those approximation methods compare to the exact solution for n=2,3 qubits.

Quantum systems are in general not e_ciently simulatable by classical means. If one wishes to determine (some of) the eigenvalues of a Hamiltonian H that is associated with a quantum system, there are two favoured strategies: Quantum simulation and quantum Monte Carlo schemes. The former strategy uses an experimentally well-controllable quantum system that emulates the system of interest, in a digital (i.e. universal) or analog manner. The latter, albeit with a limited range of applicability, uses classical stochastic processes to e_ciently obtain (often low-lying) eigenvalues of H. Quantum Monte Carlo methods may su_er from a sign problem when simulating fermionic or frustrated bosonic systems. This yields, for a given accuracy, a simulation time that scales exponentially in the system size and the inverse temperature.

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.

The concept of entanglement is one of the distinguishing features in quantum mechanics. Information about one particle can determine the state of another particle. This information and entanglement in subsystems is quantified by the entanglement entropy. The entanglement entropy has become an important research topic in recent years. Entanglement entropy is expected to grow with the boundary size for systems described by Hamiltonians with local interactions, described by the area law. For one dimension there exists a bound quantifying this area law S = O (21/ΔE) in terms of the energy gap ΔE between the ground-state energy and the first excited-state energy. It is an open problem whether the area law holds in higher spatial dimensions. Therefore it is of great interest to study the entanglement entropy in systems that violate the area law. In this thesis spin chain models with local interactions are studied for arbitrary spin. Hamiltonians with a unique ground state are constructed by mapping spin chains onto (coloured) Dyck and (coloured) Motzkin paths. We show that these models express logarithmic or power law violations of the area law for the bipartite entanglement entropy. These results are presented in the table below. The known bound of the area law in one dimensional systems is dependent on the energy gap of these models. The energy gap of the Motzkin-path model is investigated by constructing the an orthogonal excited state with small energy. In doing so we associate the Hamiltonian with the Laplacian of a graph. We present an alternative proof for the bound on the energy gap of the colourless Motzkin-path model than S. Bravyi et al and R. Movassagh et al. We show that the energy gap ΔE scales in terms of the chain length n as ΔE = O(n-2) in the limit n →∞. Therefore ΔE → 0 in this limit, resulting in area-law violations of the entanglement entropy while maintaining the known bound on the entanglement entropy.

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.

Bell inequalities are certain probabilistic inequalities that should hold in the context of quantum measurement under assumption of a local hidden variable model. These inequalities can be violated according to the theory of quantum mechanics and have also been violated experimentally. Bell inequalities were therefore historically used to disprove local hidden variable models as an interpretation of quantum mechanics. An interesting question is to what degree Bell inequalities can be violated according to the theory of quantum mechanics. The degree to which a particular Bell inequality can be violated is quantified by the largest violation of the Bell inequality. A central question we address in this thesis is how large the largest violation can become when considering all possible Bell inequalities. In particular the CHSH-inequality, a well-known Bell inequality, has a largest violation equal to the square root of 2 and we want to find a Bell inequality with largest violation exceeding the square root of 2. This thesis consists of two main parts. In the first part we formally define Bell inequalities and the largest violation and prove several theorems about Bell inequalities. In the second part we search for Bell inequalities with largest violation exceeding the square root of 2. We do this by first approximating the largest violation of a given Bell inequality using numerical optimization. Next, using this approximation, we use numerical optimization to maximize the largest violation. Due to run-time restrictions we were only able to consider Bell inequalities with a relatively small number of terms and were unable to find any among them with largest violation exceeding the square root of 2.

In this research, the implementations of quantum random walks in superconducting circuit-QED are studied. In particular, a walk that moves across the Fock states of a quantum harmonic oscillator by a Jaynes-Cummings model is investigated, which is difficult to implement because of different Rabi frequencies for different Fock states. Theoretically, the lower boundary vacuum state of the harmonic oscillator causes a reflection of the probability amplitudes in the distribution. A walk that moves across a grid of coherent states |nα+imα〉 in phase space is then investigated. A setup for a 1D and 2D quantum random walk is suggested, using controlled displacements of a resonator dispersively coupled to one or two superconducting transmon qubits in circuit-QED, followed by Hadamard gates. From numerical simulations it was observed that the 2D walk commutes for α·β = 0 mod π/2 for which the variance is proportional to the number of steps t squared. For other values of α·β the horizontal and vertical displacements do not commute, resulting in extra phase factors. The numerical simulations showed that for most values of α· β with a larger distance to 0 mod π/2 than 0.01, the probability distribution of the walk collapses to a distribution centered around origin within t = 100 steps, similar to a classical random walk. Exceptions are the 2D walks for α·β = ±π/6 mod π/2 or ± π/4 mod π/2, for which the variance is still proportional to the number of steps squared.

In this report the goals is to theoretically construct a shift resistant encoding in a rotor system. The report starts with an overview of the rotor space. In the context of which the difference between PVM and POVM is highlighted. After this overview, the shift resistant encodings are discussed. These two concepts are then combined and a shift resistant encoding in the rotor space is presented. Lastly a system which hints at an implementation is presented.. Of this system the Hamiltonian is discussed.

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.

In this thesis, the repetition code for bit flip errors is examined. Based the stabilizer measurements outcome of a run of the repetition code, one does not know exactly which errors have occurred. Statistics can be used to estimate the probability of all possible error events. This probability estimation is investigated for simulated data assuming phenomenological and circuit level noise. Experiments on the repetition code are performed by the DiCarlo group at the Delft University of Technology on a superconducting quantum computer. For this data, some error probabilities are estimated to be nonphysical. The goal of this thesis is to investigate these nonphysical probabilities and to determine whether they result from sampling noise or a problem with the assumed error model that determines the estimation of the probabilities. By estimating the standard deviation using the bootstrap method it is shown that the nonphysical probabilities are not due to sampling noise. Therefore, it is possible that non-conventional errors, such as leakage or crosstalk, are affecting these estimates. In a further analysis on the error probabilities and their standard deviation, it is shown that the standard error for space and time edges in the experiment is consistent with the standard error from phenomenological noise initialised with the average error data and ancilla error probabilities, even when these may be affected by non-conventional errors.