定义在环面周期边界上的二维拓扑稳定子码，以星算符和面算符产生 syndrome，并由非平凡同调环承载逻辑算符。
作为 surface-code 家族的周期边界叶子单独建模；planar 与 rotated patch 继续作为 surface 聚合项的处置记录。
code/topological/toricCode 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。
使用 n=2L²、k=2、d=L 的标准周期边界 toric family，取 L=3。
L×L square torus with qubits on edges
L×L square torus with qubits on edges
L×L square torus with qubits on edges
star and plaquette checks
homological logical operators
anyon-pair decoding
作为 surface-code 家族的周期边界叶子单独建模；planar 与 rotated patch 继续作为 surface 聚合项的处置记录。
Toric L=3 [[18,2,3]] Execution Route：周期方格、星/面稳定子、同调错误链与双逻辑读出。当前路线绑定 MWPM on torus、Union-Find decoder，并覆盖 Repeated star/plaquette checks、Spacetime detection events。
选 L=3 作为最小非平凡可比点。
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。
2002Topological quantum memory2003Fault-tolerant quantum computation by anyons2021Quantum Low-Density Parity-Check CodesDeep 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。
Toric 码是把表面码放在二维环面上的周期边界 CSS 码。正方晶格的边承载数据量子比特，顶点星算子与面算子分别检测两类 Pauli 误差。由于环面有两条独立的非收缩基本循环，标准 Toric 码编码两个逻辑量子比特。
The toric code is the periodic-boundary CSS surface code on a two-dimensional torus. Data qubits live on lattice edges and vertex-star and plaquette checks detect the two Pauli sectors. The two independent noncontractible cycles yield two logical qubits.
在 周期正方晶格上，每条边放置一个量子比特，并定义
所有校验两两对易，但星算子全集与面算子全集各有一个乘积约束，因此独立稳定子数为 。非收缩的原晶格与对偶晶格闭合弦分别给出逻辑 与逻辑 算子。
On an periodic square lattice, place a qubit on every edge and define
开放的 弦在端点产生 型星缺陷，开放的 弦在端点产生 型面缺陷。局域误差只能成对产生并移动这些激发；只有误差弦绕过环面的非平凡循环后闭合，才会在无综合征的情况下作用为逻辑算子。保护尺度由最短非收缩循环决定。
Open strings create pairs of electric star defects and open strings create magnetic plaquette defects. Local errors create and move excitations in pairs. A syndrome-free logical action occurs only when a string closes around a noncontractible cycle.
标准正方环面族满足
本站的 实例因此是 。校验权重和量子比特参与度均保持常数，但编码率 随系统尺寸下降。这个族适合展示拓扑距离与局域性，却不是高编码率 qLDPC 的代表。
两对逻辑 Pauli 算子可选为沿环面两条基本循环的原格与对偶格闭合弦；相交一次的弦反对易。逻辑信息可通过弦形算子、任意子输运和拓扑形变理解。把封闭流形模型用于实际门操作时，需要明确如何引入边界、缺陷、测量或全局晶格对称。
Two logical Pauli pairs can be represented by primal and dual loops around the fundamental cycles; loops intersecting once anticommute. Converting this closed-manifold picture into hardware gates requires explicit boundaries, defects, measurements, or lattice symmetries.
独立 Pauli 噪声下，解码分别处理星缺陷和面缺陷，并寻找最可能的误差链同调类。最小权完美匹配是经典基线，其他方法包括并查集、重整化、张量网络与机器学习。周期边界意味着恢复还必须判断两个方向的绕行同调类；测量有噪声时则在时空晶格上解码。
Under Pauli noise, star and plaquette defects are decoded separately by estimating the most likely error-chain homology class. Minimum-weight perfect matching is a standard baseline, with union-find, renormalization, tensor-network, and learned alternatives. Periodic boundaries require tracking winding classes; noisy measurements require spacetime decoding.
Toric 码常用于研究代码容量、现象学噪声和电路级噪声阈值，但这些数字属于不同统计模型。综合征电路的门序、辅助量子比特、相关故障和解码权重都会改变有效阈值。环面还隐含周期耦合，因此模拟结果不能不加说明地移植到平面芯片。
The toric code is a standard model for code-capacity, phenomenological, and circuit-level thresholds, but those numbers belong to different models. Gate ordering, ancillas, correlated faults, and decoder weights change the threshold. Periodic connectivity also prevents direct transfer to planar hardware without qualification.
Toric 码主要作为拓扑序、任意子与解码算法的理论和数值基准，也可在量子模拟器或具备合适连接的实验设置中演示。大多数门模型硬件更常实现带边界的表面码补丁。本站因此把 Toric 实例定位为结构与算法教学对象，而不是默认硬件布局。
The toric code is primarily a theoretical and numerical benchmark for topological order, anyons, and decoding, with demonstrations possible on suitable simulators and platforms. Gate-model hardware more commonly implements planar surface-code patches, so Atlas treats this as a structural teaching object.
Toric 码是表面码的周期边界特例，也是二维拓扑稳定子码与量子双模型的核心例子。切开环面并选择粗糙、光滑边界可得到平面表面码；改变晶格、局域基或规范自由度又连接到彩色码、XZZX 码和子系统拓扑码。
The toric code is the periodic-boundary specialization of the surface code and a central topological stabilizer and quantum-double model. Cutting the torus with rough and smooth boundaries gives planar surface codes; lattice, basis, and gauge changes connect to color, XZZX, and subsystem codes.
读取 Toric 码数据时要区分线性尺度 与物理量子比特数 ，并确认是否采用正方晶格、是否计入综合征辅助量子比特。任何阈值都必须附带噪声层级、错误偏置、解码器和有限尺寸外推方法。本站 QECDB 永久链接已核对为 family=toric 的 [[18,2,3]] 精确实例。
Distinguish linear size from the data-qubit count and check lattice and ancilla conventions. Every threshold requires a noise level, bias, decoder, and finite-size method. The Atlas QECDB permanent link is verified against a family=toric record.
All checks commute. Each complete star and plaquette set has one global product constraint, leaving independent stabilizers. Noncontractible primal and dual loops realize logical and .
The standard square-torus family has
The Atlas instance is . Check weight and qubit degree remain constant, while rate decreases with size. The family illustrates locality and topological distance rather than high-rate QLDPC behavior.