  For years, the quantum/reversible circuit community has been convinced that:
a) the addition of auxiliary qubits is instrumental in constructing a smaller
quantum circuit; and, b) the introduction of quantum gates inside reversible
circuits may result in more efficient designs. This paper presents a systematic
approach to optimizing reversible (and quantum) circuits via the introduction
of auxiliary qubits and quantum gates inside circuit designs. This advances our
understanding of what may be achieved with a) and b).
