Qutrit Error Correction Codes for Quantum Communications and Computations

This paper proposes simple quantum [[9],[1],[3]] error correction codes for quantum systems beyond qubits, focusing on quantum ternary digits (qutrits) that represent the simplest higher-dimensional systems for future quantum technologies. Although qutrit-based systems offer larger capacity information over 2-dimensional systems, the challenge of identifying commuting stabilizers for constructing error correction codes becomes more difficult as the dimensionality increases or when working with a large number of qutrits. We design qutrit codes based on the generalized Pauli group and investigate the underlying structure, including shift, phase, and rotation operators. We successfully create suitable stabilizer codes that detect and correct qutrit errors. Furthermore, we (i) identify operators in qutrit, which are selected from various matrix interpretations for higher-dimensional systems, (ii) propose stabilizers for 9-qutrit error correction codes, and (iii) design a new 9-qutrit quantum circuit for communications. Our study is expected to contribute a foundational basis for possible future theoretical and experimental advancements in quantum computing, communications, and code-based cryptography in higher dimensional systems.