An Error Correctable Implication Algebra for a System of Qubits

TL;DR

This paper introduces a Lukasiewicz logic-based implication algebra for qubit systems, enabling quantum error correction and algorithm implementation using three-valued logic and indeterminate states.

Contribution

It embeds three-valued Lukasiewicz logic into quantum stabilizer codes and characterizes errors, facilitating logic-based quantum computation.

Findings

Lukasiewicz logic can be embedded in quantum stabilizer codes

Characterization of non-trivial quantum errors up to group isomorphism

Algorithms consistent with Lukasiewicz logic can run on quantum systems

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.