发表机构
University of Maryland; Joint Institute for Quantum Information and Computer Science, University of Maryland/NIST（马里兰大学; 马里兰大学/美国国家标准与技术研究院量子信息与计算科学联合研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究基于错误集模型开发近似量子纠错理论，虽完全Knill-Laflamme线性不成立，但受限形式仍能扩展相关结构特征，导出保真度保证，通过分区构造展示模型范围，产生多种系统的渐近良好代码族。
AI 中文摘要
我们基于错误集模型开发了一种近似量子纠错（QEC）理论，并辅以代码构造的通用方法。精确QEC具有强大的错误集结构，通过Knill-Laflamme条件，纠正给定错误集的代码能自动抵御克劳斯算子位于其线性跨度内的每个信道。长期以来的观点认为这些特性不适用于AQEC，我们表明虽然完全的Knill-Laflamme线性不成立，但一种受限形式仍然存在并足以将所有三个结构特征扩展到近似设置。具体而言，一个共同的错误集标准控制着信道族，其克劳斯算子是给定错误集的线性组合且系数矩阵满足谱约束。我们在两个新的代码参数方面为这些族导出了统一的保真度保证。为展示该模型的范围，我们在不同量子系统和几何结构中开发了基于分区的构造，将精确和近似校正置于平等地位。这些构造导致了希尔伯特空间、度量和错误族的度量 - 错误对齐层次结构，进而表征了所得的恢复保证。它们产生了第一个已知的费米子系统、一维里德堡阻塞系统和删除错误的渐近良好代码族，并扩展到其他物理平台。
英文摘要
We develop a theory of approximate quantum error correction (QEC) based on the error-set model, complemented by general methods for code construction. Exact QEC has a powerful error-set structure: by the Knill-Laflamme conditions, a code correcting a given error set automatically protects against every channel whose Kraus operators lie in their linear span. This linearity gives rise to code distance, the equivalence between erasures and general errors, and a theory of asymptotically good codes. A longstanding view has been that these features do not extend to AQEC, leaving the theory essentially channel-by-channel. We show instead that, although full Knill--Laflamme linearity fails, a restricted form survives and suffices to extend all three structural features to the approximate setting. Specifically, a common error-set criterion governs families of channels whose Kraus operators are linear combinations of a given error set and whose coefficient matrices satisfy a spectral constraint. Using the Bény-Oreshkov worst-case and Petz average-case frameworks, we derive uniform fidelity guarantees for these families in terms of two new code parameters--the \emph{environment-leakage distance}, controlling worst-case performance, and the \emph{Knill-Laflamme Hellinger distance}, characterizing the average-case performance of Petz recovery. To demonstrate the scope of this model, we develop partition-based constructions across diverse quantum systems and geometries, placing exact and approximate correction on equal footing. These constructions lead to a metric--error alignment hierarchy for Hilbert spaces, metrics, and error families, which in turn characterizes the resulting recovery guarantees. They yield the first known asymptotically good code families for fermionic systems, one-dimensional Rydberg-blockaded systems, and deletion errors, and extend to other physical platforms.
Comments97 pages, 3 figures. v2: Corrected an indexing error inherited from the literature; results unchanged. v3: Added experimental data to support the theory
