The exponential nature of Moore's law has inadvertently created huge data storage complexes that are scattered around the world. Data elements are continuously being searched, processed, erased, combined and transferred to other storage units without much regard to power consumption. The need for faster searches and power efficient data processing is becoming a fundamental requirement. Quantum computing may offer an elegant solution to speed and power through the utilization of the natural laws of quantum mechanics. Therefore, minimal cost quantum circuit implementation methodologies are greatly desired.

This thesis explores the decomposition of group functions and the Walsh spectrum for implementing quantum canonical cascades with minimal cost. Three different methodologies, using group decomposition, are presented and generalized to take advantage of different quantum computing hardware, such as ion traps and quantum dots. Quantum square root of swap gates and fixed angle rotation gates comprise the first two methodologies. The third and final methodology provides further quantum cost reduction by more efficiently utilizing Hilbert spaces through variable angle rotation gates. The thesis then extends the methodology to realize a robust quantum circuit synthesis tool for single and multi-output quantum logic functions.