This book focuses on the design methodologies of various quantum circuits, DNA circuits, DNA-quantum circuits and quantum-DNA circuits. It considers the merits, and challenges of multivalued logic circuits in quantum, DNA, quantum-DNA and DNA-quantum computing Multiple-Valued Computing in Quantum Molecular Biology: Arithmetic and Combinational Circuits is volume one of a two volume set. This book discusses from fundamentals to advanced levels of different Multiple-Valued Logic DNA-quantum and quantum-DNA circuits. The book consists of four parts. Part one introduces Multiple-Valued Quantum Computing and DNA Computing. It contains the basic understandings of Multiple-Valued Quantum Computing, Multiple-Valued DNA Computing, Multiple-Valued Quantum-DNA Computing and Multiple-Valued DNA-Quantum Computing. Part two examines heat calculation, speed calculation, heat transfer, data conversion and data management in multivalued quantum, DNA, quantum-DNA and DNA-quantum computing. Part three discusses multiple-valued logic operations in quantum and DNA computing such as ternary AND, NAND, OR, NOR, XOR, XNOR, multiple-valued arithmetic operations such as adder, multiplier, divider etc. Quantum-DNA and DNA-quantum multiple-valued arithmetic operations are also explained in this section. Part four explains multiple-valued quantum and DNA combinational circuits such as multiple-valued DNA-quantum and quantum-DNA multiplexer, demultiplexer, encoder, and decoder. This book will be a great help for researchers and students in Quantum Computing, DNA Computing, Quantum-DNA Computing and DNA-Quantum Computing researchers.