The primary focus of this dissertation is the development of a framework for the design and analysis of quantum circuits and quantum algorithms based on a parameterization of primitive quantum operators. Parameterized operators are characterized by the linear combination of two logically opposing operators. Single qubit operators can be viewed as a partial identification and negation. Similarly, conditional operators act partially in terms of a given condition and partially in terms of the logical negation of that condition. In support of this logic, several sets of basis operators are defined for n qubit computation and a hierarchy of these operators is established. A set of elementary n qubit operators is then placed within the general hierarchy of n qubit operators.
Within the framework, operators are further characterized by the encoding of binary information about their input to the phase of their output. This encoding is expressed as a boolean function on the n bit strings that correspond to the basis states of an n qubit state. Furthermore, interference between operators is characterized in terms of these encoding functions with specific relationships between the encoding functions producing a special case of interference dubbed decoding. When decoding occurs between two operators then the decoding operator effectively interprets the information encoded in the phase of a quantum state and produces a new state relative to the decoded information. The encoding and decoding properties of single qubit operators, elementary n qubit operators, and basic n qubit composite operators is developed. Finally, Deutsch's algorithm is analyzed and generalized by way of the parameterized operator and phase encoding/decoding framework.