Quantum conservative many-valued computing

We extend the basic principles and results of conservative logic to include the main features of many-valued logics with a finite number of truth values. Different approaches to many-valued logics are examined in order to determine some possible functionally complete sets of logic connectives. As a result, we describe some possible finite-valued gates which realize a functionally complete set of fundamental connectives. One of the purposes of this work is to show that the framework of reversible and conservative computation can be extended toward some non-classical “reasoning environments”, originally proposed to deal with propositions which embed imprecise and/or uncertain information, that are usually based upon many-valued and modal logics. We also describe a possible quantum realization of the proposed gates, using creation and annihilation operators. In such realization the gates are expressed as formulas that are obtained using three techniques: a “brute force” technique, an extension of the Conditional Quantum Control method introduced by Barenco, Deutsch, Ekert and Jozsa [Conditional quantum control and logic gates, Phys. Rev. Lett. 74 (1995) 4083], and a new technique that we call the Constants method. We show the Constants method allows one to reduce the number of local operators in the formulas which correspond to the proposed gates.