Aspects of Reversible and Quantum Computing in a $p$ -Valued Domain

This paper presents basic aspects of non-binary reversible and quantum computing in a p-valued environment, since there are no physical reasons for quantum computing to be necessarily binary. A “quantum technology” is not yet available, but different alternatives at the level of laboratory experiments are promising. A theoretical background is needed to face the challenge of designing circuits for quantum computing. Pauli matrices are introduced in the p-valued domain and their properties are explained. The Vilenkin-Chrestenson matrix is shown to produce superposition of states. Entanglement of quantum states in the p-valued environment is introduced and its measurement effect is illustrated. It is shown that in the p-valued domain, the binary Toffoli gate may be given different generalizations with different functionalities. An ancillary-free realization of a new generalized p-valued Toffoli gate based on Muthukrishnan-Stroud elementary gates is presented, its functionality is extended to two kinds of control, and a proof of its performance is given.