In this work we tackle the challenge of designing quantum unitary operators which represent solutions to optimization problems. We start with a novel method which combines an evolutionary algorithm known as an Evolution Strategy (ES) with a method to randomly generate unitary operators. With this new method, a quantum operator is represented for the first time using real-valued vectors and can be "evolved" or designed to meet certain target criteria. This criteria could be the solution to an optimization problem. With the ability to evolve quantum operators, we attempt to evolve various known single and multi-qubit quantum gates as well as quantum oracles. We evolve quantum operators which solve instance problems of a known NP-Hard problem and even attempt to evolve a generalized solution operator. We evolve multiple operators with varying size and investigate their properties through eigenanalysis methods as well as by synthesizing them into quantum logic gates using the quantum compiler Qubiter. We also present a new quantum logic algebra which offers a new way to represent quantum circuits and demonstrate its immediate uses in quantum computing.