Novel convolution-based multiple-stream error-control coding and decoding methods and their corresponding circuits are introduced. The new coding method applies the reversibility property in (1) the convolution-based encoder for multiple-stream error-control encoding and (2) in the new reversible Viterbi decoding algorithm for multiple-stream error-correction decoding. The complete synthesis of quantum circuits for the quantum realization of the new quantum Viterbi cell in the quantum domain (Q-domain) is also introduced, and the associated quantum computing representations and operations are presented. In quantum mechanics, a closed system is an isolated system that cannot exchange energy or matter with its surroundings and does not interact with other quantum systems. Closed quantum systems obey the unitary evolution and thus they are reversible. Reversibility property in error-control coding can be important for the following main reasons: (1) reversibility is a basic requirement for low-power circuit design in future technologies such as in closed-system quantum computing (QC), (2) reversibility leads to super-speedy encoding/decoding operations because of the superposition and entanglement properties that exist in the reversible closed-system quantum computing circuits and systems, and (3) the reversibility relationship between multiple-streams of data can be used for the correction of errors that are usually uncorrectable using the implemented decoding algorithm such as in the case of triple-errors that are uncorrectable using the irreversible Viterbi algorithm.