AI 中文总结
本文针对量子稳定子码提出量子信息局部性概念及对应线性代数方法，证明现有量子局部性定义高估删除纠正所需符号数，还给出经典信息局部性的量子转换低估该数值的实例。
AI 中文摘要
长度为 $n$ 的经典线性码 $C$ 若对任意索引 $j$，存在修复组 $J_j \subseteq \{1, \ldots, n\}$ 满足 $j \in J_j$ 且 $|J_j| \leq r+\delta-1$，使得 $J_j$ 内任意 $\delta-1$ 个或更少的删除可仅用 $J_j$ 内的码字符号纠正，则称其具有符号局部性 $(r, \delta)$。后来发现，这种定义 $r$ 的方式会高估多删除纠正所需的码字符号数量，因此提出信息局部性，将 $r$ 定义为 $C$ 在 $J_j$ 上的删余码的维度。最近，人们遵循符号局部性 $(r, \delta)$ 的原始定义，为量子纠错码提出了局部性 $(r, \delta)$。我们针对通过厄米正交性构造的量子稳定子码，提出了量子信息局部性的量子对应概念，以及一种线性代数方法，该方法可计算由所提信息局部性预测的更小修复组，同时将译码中测量的可观测量数量降至其最小可能值。随后，我们通过一个量子稳定子码的具体例子证明，先前提出的量子局部性 $(r, \delta)$ 的定义同样存在高估删除纠正所需码字符号数量的缺陷。最后，我们将给出另一个由欧几里得正交性和两个不同线性码构造的量子稳定子码的例子，对于该码，经典信息局部性到量子场景的自然转换会低估删除纠正所需的码字符号数量。
英文摘要
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.
Commentseprint source includes a C program verifying mathematical claims in Section 4 by brute-force computation
