From Quantum Circuits to Quantum Computational Logics

Abstract

The theory of quantum logical circuits has naturally inspired new forms of quantum logic that have been termed quantum computational logics. From a semantic point of view, any formula of the language of a quantum computational logic is supposed to denote a piece of quantum information that lives in a Hilbert space whose dimension depends on the linguistic complexity of the formula in question. At the same time, the logical connectives are interpreted as special examples of quantum logical gates. Accordingly, any formula of a quantum computational language can be regarded as a synthetic logical description of a quantum circuit. In this way, linguistic formulas acquire a characteristic dynamic meaning, representing possible computational actions. The most natural semantics for quantum computational logics is a form of holistic semantics, where the puzzling entanglement-phenomena can be used as a logical resource. The concept of logical consequence, defined in this semantics, characterizes a weak form of quantum logic, where many important logical arguments (which are valid either in classical logic or in Birkhoff and von Neumann’s quantum logic) are possibly violated.