Information locality of a quantum locally recoverable code

2026-08-04

AI Summary

This work addresses a critical limitation in existing quantum locally recoverable codes, whose symbol-based locality definitions overestimate the number of code symbols required for recovery in multi-erasure scenarios and fail to accurately capture information-theoretic dimensions. The paper introduces, for the first time, the notion of information locality tailored to quantum stabilizer codes, demonstrating that naively extending classical locality definitions to the quantum setting can either over- or under-estimate repair overhead. By leveraging Hermitian and Euclidean orthogonal constructions, optimizing repair groups via linear algebraic techniques, analyzing truncated dimensions, and minimizing decoding observables, the authors construct tighter repair sets. Illustrative constructions expose the shortcomings of conventional locality definitions and validate that the proposed framework enables more precise assessment and reduction of quantum error-correction resource requirements.

Abstract

A classical linear code $C$ of length $n$ is said to have symbol locality $(r, δ)$ if for any index $j$ there exists a repair group $J_j \subseteq \{1, \ldots, n\}$ with $j\in J_j$ and $|J_j| \leq r+δ-1$ such that any $δ-1$ or fewer erasures in $J_j$ can be corrected by using codeword symbols only in $J_j$. Later it turned out that this way of defining $r$ overestimates the number of necessary codeword symbols for multiple-erasure correction, and information locality was proposed to define $r$ as the dimension of the punctured code of $C$ onto $J_j$. Recently locality $(r,δ)$ was proposed for quantum error-correcting codes by following the original definition of symbol locality $(r, δ)$. We propose a quantum counterpart of the information locality for quantum stabilizer codes constructed by Hermitian orthogonality, and a linear algebraic procedure computing a smaller repair group predicted by the proposed information locality and simultaneously reducing the number of measured observables in decoding to its minimum possible value. Then we demonstrate that the previously proposed definition of quantum locality $(r,δ)$ has the same drawback of overestimating the number of necessary codeword symbols for erasure correction by providing an explicit example of a quantum stabilizer code. Finally, we will give another example of a quantum stabilizer code constructed by Euclidean orthogonality and two different linear codes, with which a natural translation of the classical information locality into the quantum setting underestimates the number of necessary codeword symbols for erasure correction.

Problem

• quantum locally recoverable code
• information locality
• erasure correction
• stabilizer code
• locality

Innovation

• quantum locally recoverable codes
• information locality
• stabilizer codes
• erasure correction
• repair group