Syntax and Semantic Structure, and Philosophical Consideration of Quantum Logic

Vladimir L. Vasyukov

Mingsheng Ying

Raoni Wohnrath Arroyo, Gilson Olegario da Silva

Bas C. van Fraassen

Viktoras Rimsha, Laimontas Rimsha

In this paper we provide a structural derivation of Birkhoff–von Neumann quantum logic starting from an extended statistical interpretation of quantum mechanics. Quantum logic is not introduced as an independent logical hypothesis, but is shown to arise naturally from the non-classical structure of quantum probability, in particular from interference, non-additivity, and the priority of the whole over its alternatives. We demonstrate that the failure of Boolean logic in quantum mechanics is not a semantic or philosophical anomaly, but a direct consequence of the probabilistic structure encoded in Hilbert space. Within this framework, quantum logic appears as the logical form of quantum probability itself. This interpretation clarifies the physical meaning of quantum logic and highlights its contemporary relevance in quantum information and quantum computation.

David Chamizo, Jose Garcia-Alonso, Juan M. Murillo

Current quantum programs are mainly designed at the level of quantum gates acting on individual qubits; on a large scale and for complex problems this may involve a high cognitive load on the programmer, making the program specification nontrivial and error-prone. In this context, providing quantum programming with higher abstraction mechanisms will assist in making this task more manageable and robust against design errors. In this work, a conceptual framework is addressed following the notion of the whole quantum computation as a structure composed of quantum registers representing each an undivided entity. Thus, computation progresses through semantically well-defined transformations that act on, or entangle, quantum registers, thereby modifying the global state. Ultimately, the program reaches the desired state by following a specific composition strategy. With this in mind, high-level syntax is presented through an algebraic formalism that bridges them with their low-level semantics. Proposed syntax is based on certain well-know operations used on quantum algorithms that apply phase shifts upon logical condition satisfaction or leverage on parallel evaluation. Based solely on the formalized operations, a quantum satisfiability modulo theories (SMT) solver can be designed. At its core, this work contributes to establishing some methodological principles towards realizing a high-level quantum structured programming.