Construction of Quantum Rank-Metric Codes Using Hermitian Orthogonality

2026-05-04

AI Summary

This work addresses the limitations of conventional CSS-based quantum Gabidulin codes, which are constrained by algebraic conditions to square-stack quantum memories with odd side lengths, thereby restricting architectural flexibility. The authors propose a unified construction framework that integrates symplectic and Hermitian self-orthogonality, leveraging the Matsumoto–Uyematsu method to successfully construct novel quantum Gabidulin codes in even-dimensional stacked architectures. This approach overcomes the prior restriction to odd dimensions and, while preserving code rate, nearly doubles the ratio of minimum rank distance to the number of physical qubits. Consequently, it significantly enhances both the relative minimum rank distance and error-correcting capability, substantially broadening the applicability of quantum rank-metric codes in practical stacked quantum storage systems.

Abstract

Stacked quantum memory is an architecture in which multiple layers of qubits are stacked. Quantum rank-metric codes are effective for error correction in stacked quantum memories. However, the previously proposed quantum Gabidulin codes based on the CSS construction had a problem: due to algebraic constraints, the applicable memory layouts were strictly limited to square shapes of odd length. In this paper, we first propose a framework for constructing quantum rank-metric codes from classical linear codes with symplectic self-orthogonality. Building upon this, we propose a new construction method for quantum Gabidulin codes by combining the Hermitian self-orthogonality of classical Gabidulin codes--utilizing the self-dual basis that exists when the extension degree of the finite field is even--with the quantum code construction method using Hermitian orthogonality by Matsumoto and Uyematsu. The proposed method succeeds in approximately doubling the ratio of the minimum rank distance to the number of physical qubits while maintaining the code rate. Furthermore, it eliminates the restriction of the conventional method that requires the number of cells and layers of the stacked memory to be odd, realizing the construction of quantum rank-metric codes applicable to memories with an even number of cells and layers. This construction improves the relative error correction capability of the stacked quantum memory architecture and increases the degree of freedom in design while preserving the code rate.

Problem

• quantum rank-metric codes
• stacked quantum memory
• Hermitian orthogonality
• Gabidulin codes
• memory layout constraints

Innovation

• quantum rank-metric codes
• Hermitian orthogonality
• Gabidulin codes
• stacked quantum memory
• symplectic self-orthogonality