With the rapid development of Quantum Computers (QC) and QC Simulators, there will be an increased demand for functioning Quantum Algorithms in the near future. Some of the most ubiquitously useful algorithms are solvers for linear systems of equations. Since the conception of the Quantum Linear Solver Algorithm (QLSA) by Harrow, Hassidim and Lloyd (HHL) in 2009, many improvements have been made, although a generic implementation for arbitrary matrices and vectors is still not available. In this thesis a variant of the HHL QLSA is studied, and the open challenges are investigated. Solutions for two of the challenges, namely the Eigenvalue Inversion subroutine and the Higher-Order Ancilla Rotation subroutine, are discussed. As part of the thesis project, these subroutines have been implemented in the QX Quantum Computer Simulator, and the subroutines are combined to form a complete Quantum Linear Solver (QLS), with the restraint that the implementation for the vector and Hamiltonian of the matrix must be provided by the user. A proof-of-concept QLS by Cao et al. is also implemented in the QX simulator, and using the implementation of the vector and Hamiltonian of Cao et al. the complete solver is tested. In the process of this thesis, a framework for basic Quantum Arithmetic is built providing three variants of Integer Adders, two variants of Integer Subtracters, one Integer Multiplier and one Integer Divider. In addition, gates not natively available in the QX simulator are implemented, and a number of improvements and extensions of algorithms presented in the literature are given, making the described algorithms function on the QX simulator and extending features.

This thesis investigates intrinsic frustrated magnetic systems on an insulating square lattice. The frustrated systems were successfully engineered from magnetic iron atoms on top of an insulating square c(2x2) reconstruction of nitrogen on Cu3Au(100) (copper-gold) surface. Vertical atom manipulation was used to position the magnetic atoms atomically precise on the surface and methods were found to increase the success rate of this. For the investigation of frustration, a numerical simulator was made in Python for modelling frustrated spin structures based on already existing spin models and the following experimental measurements were taken: topography, spectroscopy and current-time traces. The simulations of a frustrated spin system show low energy excited states and the measurements show switching of the system between states.