This is a new idea to be introduced in this chapter called quantum biocomputing or quantum-DNA computing. The combination of quantum computing and DNA computing is called quantum-DNA computing system. This idea is introduced here for the first time in modern history. An arithmetic circuit in quantum-DNA computing will contain two parts; one is quantum part and another is DNA part. The inputs are qubit and the outputs are DNA molecular sequences. Some necessary things like data conversion, data management, heat transfer, or management are also needed in these quantum-DNA circuits, which will be discussed in this chapter.