Quantum computing and DNA computing are gaining attention as alternatives to classical computing systems. There exist some quantum algorithms which are significantly faster than their conventional counterparts and these processes establish quantum computing as a superior future technology that involves quantum circuits and quantum gates. Traditional silicon computers consume much more power as compared to the computing systems, based on Deoxyribonucleic Acid (DNA), whereas DNA-based logic gates are stable and reusable. DNA computation might save a billion times the energy of an electrical computer while storing data in a trillion times less space. DNA computing is a mechanism for accurately storing and retrieving information inside a DNA sequence. A novel logic gate design based on chemical reactions is presented in which observance of double-stranded sequences indicates a truth evaluation. Both of these properties might be captured by integrating quantum-DNA computing. In this chapter, combined quantum and DNA computing are discussed, novel theoretical methods for designing different memory devices considering a binary logic system. This is an entirely new way of merging these two technologies and representing memory devices in a computer architecture. In quantum biocomputing or quantum-DNA computing, this chapter will cover all four memory devices.