This paper presents a novel theory for quantum computing program compilation and optimization. The core of this theory lies in translating high-level quantum programs into efficient quantum circuit implementations. This is achieved through the utilization of an abstract syntax tree (AST) derived from the quantum program. A compilation algorithm is designed to traverse the AST and generate corresponding quantum circuits, incorporating optimization strategies throughout the process. The work addresses a significant challenge within quantum computing – the complexity of program compilation – and proposes a new theoretical framework to facilitate the development of more practical and performant quantum algorithms. The theoretical framework outlined here provides a foundation for future research and development in quantum compiler design.