Graphical Stabilizer Decompositions for Multi-Control Toffoli Gate Dense Quantum Circuits

In this thesis, we study concepts in quantum computing using graphical languages, specifically using the ZX-calculus. The core of the research revolves around (graphical) stabilizer decompositions. The first major focus is on the decomposition of non-stabilizer states created from star edges. We discuss previous results and then present novel decompositions that yield a theoretical improvement. The second major focus is on weighting algorithms, applied to the special class of multi-control Toffoli gate dense quantum circuits. The representation of the corresponding gates is based on star edges. The applicability of known methods, such as CNOT-grouping, traditionally used for other classes, is examined in the context of this specific class. We then present a novel weighting algorithm that attempts to determine the best vertex to decompose. A refined version is implemented to simulate a known class of quantum querying algorithms, which is used to search for causal configurations of multiloop Feynman diagrams. For this case, as well as for a generalized benchmark consisting of randomly generated quantum circuits, we demonstrate occasional improvements in the final number of terms against traditional methods. These results are discussed by considering different simplification strategies. This thesis also provides a brief but broad outline of the important preliminaries.