The minimum vertex cover problem (MinVertexCover) is an important optimization problem in graph theory, with applications in numerous fields outside of mathematics. As MinVertexCover is an NP-hard problem, there currently exists no efficient algorithm to find an optimal solution on arbitrary graphs. We consider quantum optimization algorithms, such as the Quantum Alternating Operator Ansatz (QAOA+), to find good minimum vertex covers.

This thesis presents new 'second degree' mixing Hamiltonians for MinVertexCover in the QAOA+ framework, which allow mixing between solutions Hamming distance 2 apart. The performance of these new Hamiltonians is evaluated on a small graph. Methods to extend this idea by constructing Hamiltonians which allow mixing between solutions at Hamming distance n are also presented, along with a generalization of second degree mixing Hamiltonians to other optimization problems.

Currently there are no efficient quantum algorithms for the addition of real-valued numbers. In classical computers addition is performed by using barrel shifters, a concept proven to be very inefficient as a quantum circuit due to its many garbage outputs when the barrel shifter is made reversible. This thesis aims to design a quantum algorithm able to perform floating-point arithmetic. It uses the new Posit format as its number format so the algorithm can be built on a very small scale, which makes it possible to easily implement the entire algorithm. The designed Quantum Posit Adding Algorithm uses a tablebase approach, examining each number checking if it changes during addition. An optimized version of the algorithm is also designed, removing any unnecessary controls. Finally a method to extend the algorithm is proposed along with an approach to building a similar subtractor, also presenting some non-working ideas.