用一维冗余和同基奇偶校验保护单一 Pauli 错误通道，是 syndrome 循环、距离标度与硬件噪声表征的基础实验码。
明确其只保护 bit-flip 或 phase-flip 一类错误，不能被描述为任意单量子比特错误的完整量子存储码。
code/block/repetitionCode dossier
这里汇总参数景观中的结构化记录、码族谱系中的构造说明、执行路线与外部知识库，作为完整主题页的代码专属延伸。
总体量子码距为 1；受保护的单一 Pauli 通道方向距离为 15。散点 d 使用完整量子码的最小距离，避免把偏置方向距离误作通用距离。
15-qubit linear repetition code with nearest-neighbor parity checks
15-qubit linear repetition code with nearest-neighbor parity checks
15-qubit linear repetition code with nearest-neighbor parity checks
single-Pauli protection
repeated parity checks
distance-scaling benchmark
明确其只保护 bit-flip 或 phase-flip 一类错误，不能被描述为任意单量子比特错误的完整量子存储码。
Quantum Repetition [[15,1,1]] Execution Route：最近邻奇偶测量、检测事件链与单错误扇区多数表决。当前路线绑定 Majority vote、1D matching decoder，并覆盖 Repeated nearest-neighbor parity、Detection-event differences。
明确区分总体 d=1 与受保护方向距离 15。
以下来源分别约束该码的定义、构造或性能结论；参数与工程判断应回溯到具体实例、噪声模型和评测口径。
2018A repetition code of 15 qubits2025Quantum error correction below the surface code threshold2021Exponential suppression of bit or phase errors with cyclic error correctionDeep reference
定义、代数构造、保护能力与执行证据均按来源段落独立维护；中文稿经过术语整理，英文稿保留用于逐段对照。
量子重复码把选定正交基中的一个逻辑比特编码为
在计算基中它保护比特翻转，在 Hadamard 基中保护相位翻转。该映射对叠加态产生 GHZ 型纠缠，并不是复制未知量子态，因此不违反不可克隆定理。
A quantum repetition code encodes one logical qubit in a chosen orthogonal basis as
比特翻转版本可取稳定子
长度 的比特翻转重复码可纠正至多 个 错误，却不能检测单个 错误；相位翻转版本交换 与 。因此它沿受保护 Pauli 方向的距离为 ，但作为同时面对任意 Pauli 错误的量子码，其完整距离只有 。
完整量子码参数写作 [[n,1,1]]，编码率为 。本站的 15 量子比特实例因此绘制为 [[15,1,1]]，并另行记录受保护方向距离 15。把后者直接填入普通 字段会错误暗示该码能纠正任意组合的 、、 错误。
The full quantum parameters are with rate . Atlas plots the 15-qubit instance as and records protected-sector distance 15 separately. Putting the directional distance in ordinary would falsely imply protection against arbitrary , , and errors.
逐比特 Hadamard 在比特翻转与相位翻转版本之间变换，但它不是固定重复码空间内的通用横向逻辑门。逻辑 需要作用于全部数据量子比特，而任意单个 在码空间内代表逻辑 。这种极端非对称性使它适合偏置噪声基准，却不构成通用量子存储。
理想测量时，邻接奇偶校验定位翻转域壁，解码等价于多数表决或一维最可能链推断。测量有噪声时，连续轮次的校验差分形成时空探测事件，可用匹配、动态规划、有限状态机或面向具体线路的解码器恢复。最终读出也必须与历史综合征联合处理。
With ideal measurements, adjacent parity checks locate domain walls and decoding reduces to majority vote or one-dimensional maximum-likelihood inference. With measurement noise, check differences across rounds form spacetime detector events decoded by matching, dynamic programming, finite-state, or circuit-specific methods.
每个权重二奇偶校验通常由测量辅助量子比特反复提取。数据错误、测量错误和复位错误在时空图上产生不同事件模式；只有执行足够轮次并使用电路级噪声模型，才能评价逻辑抑制。编码 GHZ/猫态需要受控门网络或测量式制备，制备深度与连接约束应单独报告。
Weight-two parity checks are repeatedly extracted with measurement ancillas. Data, measurement, and reset faults create different spacetime patterns; logical suppression must be evaluated over enough rounds with a circuit-level model. GHZ or cat-state preparation also needs an explicit circuit with reported depth and connectivity.
量子重复码是硬件校准、综合征循环、泄漏与串扰诊断、实时解码和随码距缩放研究的常用基准。实验中报告的“指数抑制”通常只针对选定的比特翻转或相位翻转通道，不能直接解释为完整量子纠错。15 量子比特实例适合复现实验工作流和解码接口。
代表文献
Quantum repetition codes are common benchmarks for calibration, syndrome cycling, leakage and crosstalk diagnosis, real-time decoding, and distance scaling. Experimental exponential suppression normally refers to one selected Pauli sector, not full quantum error correction. A 15-qubit instance is useful for reproducing workflows.
它是经典重复码的量子嵌入，也是 GHZ 态与猫态的稳定子描述。猫态可作为测量高权 Pauli 算子的辅助资源，但“猫态辅助测量”不等于把数据编码成重复码。量子重复码还可视为最简单的非对称稳定子码和一维综合征实验平台。
The code is a quantum embedding of the classical repetition code and a stabilizer description of GHZ or cat states. Cat states can serve as ancillas for high-weight Pauli measurements, but that use is distinct from encoding data in a repetition code. The family is also a basic asymmetric stabilizer code and one-dimensional testbed.
看到重复码的“距离”或“阈值”时，先问它指完整量子距离、受保护 Pauli 方向距离，还是时空重复测量的有效距离；再核对边界、轮数和是否包含测量错误。QECDB 当前没有本站可验证的 [[15,1,1]] 精确条目，QECirc 也没有对应实例页，因此两者保持不可用而不回退到通用首页。
When a repetition-code result quotes distance or threshold, determine whether it means full quantum distance, protected-sector distance, or effective spacetime distance, then check boundaries, round count, and measurement faults. Atlas has no verified QECDB or QECirc page for this instance and exposes no generic fallback.
The computational-basis version protects against bit flips and the Hadamard-basis version protects against phase flips. Superpositions become GHZ-type entangled states; the map does not clone an unknown quantum state.
逻辑算子可选为 与 。对所有量子比特施加 Hadamard 可得到相位翻转版本，此时校验变为 。最近邻链表示的校验权重为二，适合演示重复综合征测量和时空解码。
For the bit-flip version, choose stabilizers
Logical operators may be and . Applying Hadamard to every qubit yields the phase-flip version with checks. The nearest-neighbor chain has weight-two checks and is well suited to repeated-syndrome demonstrations.
A length- bit-flip repetition code corrects up to errors but cannot detect a single error; the phase-flip version exchanges and . Its directional distance is , while its full quantum distance against arbitrary Pauli errors is only .
Bitwise Hadamard exchanges bit-flip and phase-flip encodings rather than providing a universal gate set in one fixed code. Logical acts on all data qubits, while any single represents logical . This asymmetry makes the code useful for biased-noise benchmarks but not a universal memory.