An Error Correctable Implication Algebra for a System of Qubits
Morrison Turnansky
TL;DR
The paper connects three-valued Lukasiewicz logic to quantum computation by embedding the $MV_3$ algebra into the stabilized space of Pauli-based stabilizer codes. It shows that a cubic lattice realization faithfully represents the Lukasiewicz logic and that the associated projections in stabilizer codes realize a 3-valued semantics, with a formal correspondence between stabilizer structures and Lukasiewicz algebras. It further characterizes non-trivial errors via automorphisms of the cubic lattice, using centralizers and normalizers to capture error structure, and demonstrates how any Lukasiewicz-consistent algorithm can be deployed on a quantum system, exemplified by a fault-tolerant Renyi–Ulam game. The results provide a semantic and algebraic framework for reasoning about observables and errors in quantum codes, with potential implications for fault-tolerant quantum computation and logic-guided quantum algorithms.
Abstract
We present the Lukasiewicz logic as a viable system for an implication algebra on a system of qubits. Our results show that the three valued Lukasiewicz logic can be embedded in the stabilized space of an arbitrary quantum error correcting stabilizer code. We then fully characterize the non trivial errors that may occur up to group isomorphism. Lastly, we demonstrate by explicit algorithmic example, how any algorithm consistent with the Lukasiewicz logic can immediately run on a quantum system and utilize the indeterminate state.