In recent years, various systems have been developed which integrate a linear accelerator with an MR system, enabling high quality imaging during radiotherapy. However, these systems produce strong magnetic fields that cause the secondary electrons to deflect. This leads to significant changes in the dose distribution since the path of the electrons is altered in this magnetic field. Several methods have been developed to accurately determine the influence of a magnetic field on the dose distribution, however, these methods are impractical due to their long computation times. In this work we develop a deterministic DGFEM method for solving the Linear Boltzmann transport equation (LBTE) in magnetic fields for photon therapy. For this purpose, we first developed a deterministic Boltzmann solver based on the discrete ordinate methods. This algorithm was extended to use the DGFEM method and finally a magnetic field term was implemented to determine the influence of a magnetic field on the dose distribution.

The results acquired with the algorithm based on the DGFEM method were compared to exact solutions, these results were consistent with the exact solutions and reported high levels of accuracy. The accuracy of these methods was comparable to those achieved by using discrete ordinates. Furthermore, the cost of the DGFEM algorithm were compared to those of the discrete ordinate method, here it has been shown that the DGFEM algorithm is only slightly more computationally expensive.

The DGFEM based solution algorithm was extended by implementing the magnetic field operator into the algorithm. The deterministic results in the presence of a magnetic field were compared against the MCNP and TOPAS Monte Carlo codes. These results showed similar dose distributions compared to MCNP, however, the deterministic results were not in accordance with the TOPAS simulation. It is suspected that the discrepancy in dose distribution originates from the difference in source spectrum between the two methods.

In order to investigate the influence of a magnetic field, dose distributions were determined with a magnetic field perpendicular to the photon beam. The results showed that the buildup region decreases for stronger magnetic fields and that higher values for the dose are formed at the boundaries between materials with different densities. This increased dose is caused by the electron return effect and becomes more condensed for stronger magnetic fields. Furthermore, a lateral shift in the dose distribution has been observed in the direction of the Lorentz force. These results show that the developed deterministic Boltzmann solver is able to generate accurate dose distributions in the presence of a magnetic field.

Random walks have been used in a number of different fields for a long time. With the rise of of quantum random walks a lot of these applications have been made more efficient. Recently there have been a lot of improvements in the realization of these quantum random walks in real world systems.
In this research, the effect of uncertainty in the measurement time and measurement frequency are studied on three problems that make use of a quantum random walk. These problems are the network centrality problem, the graph isomorphism problem and the spatial search problem.
These effects are studied by first looking at the theory behind these problems after which the theoretical results of these three problems are calculated. These theoretical results will then be compared to simulated results that have a certain level of uncertainty in the measurement time or frequency.
For both the network centrality problem and the spatial search problem we concluded that only a small error is made when uncertainties are introduced in the system. This implies that these problems are solvable in realizations of the quantum random walk in real world systems. For the Graph isomorphism problem we saw that the error that is made when uncertainties are introduced was large which implies that this problem would only be solvable for small uncertainties.