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选你想回答的问题。标签说明每篇的定位：零基础版只假定算术；高中数学版假定高三水平的数学，用排列组合、对数与矩阵走得更快；条目导读假定你已读过直觉篇，逐项对照 QEC-Bank 的记录讲：每个数字从哪来、机器核对过什么。 Choose the question you want answered. The tags mark each guide's level: no prerequisites assumes only arithmetic; high-school math assumes final-year school mathematics and moves faster with counting, logarithms and matrices; entry guide assumes you have read the intuition guide and goes through the QEC-Bank records item by item: where each number comes from and what a machine has checked.
- 直觉Intuition→ 从这里开始：把环面剪开铺平；两种边界各自让一种错误链可以「出界」而不触发警报；一轮完整的纠错怎样用帮手比特测检查；平面上的译码游戏；最后在浏览器里现场测出阈值物理错误率的一条临界线：低于它，把码做大，信息被改写的概率就下降；高于它，做大反而更糟。它的数值取决于噪声模型和译码器。。另有高中数学版。Start here: cut the torus open and lay it flat; two kinds of boundary, each letting one kind of error chain leave the patch without setting off an alarm; how one full round of correction measures the checks with helper qubits; the decoding game on a flat patch; and finally a live measurement of the thresholdA critical physical error rate: below it, making the code bigger lowers the chance that the information is rewritten; above it, bigger is worse. Its value depends on the noise model and the decoder. in your browser. Also in a high-school edition.
- 导览Guided tour→ 导览是 QEC-Bank 给每个条目配的一条固定路线：不讲新理论，只把程序真实导出的数据一站站指给你看。这里用 QEC-Bank 发布的最小实例 L = 3（13 个比特，1 个逻辑比特，码距 3；构造器也能造 L = 2 的 [[5,1,2]]，但码距 2 只能发现、不能纠正单个错误，L = 3 是能纠正任意单比特错误的最小一个）：13 个比特的坐标怎样由两条重复码最简单的纠错码：把一个比特抄成 L 份，检查相邻两份是否相同。表面码的版图由两条这样的码「相乘」得到。拼出来；6 条 X 检查和 6 条 Z 检查为什么长那样、为什么互不干扰；把链首尾接上为什么变回环面码；一条错误链从左边界走到右边界时，警报怎样亮起、挪动、熄灭。A guided tour is the fixed route the QEC-Bank attaches to each entry: no new theory, just the data the program actually exported, pointed out stop by stop. Here it uses the smallest instance the QEC-Bank publishes, L = 3 (13 qubits, 1 logical qubit, distance 3; the constructor can also build L = 2, a [[5,1,2]] code, but distance 2 only detects a single error and cannot correct it, so L = 3 is the smallest one that corrects an arbitrary single-qubit error): how the 13 qubit coordinates are assembled from two repetition codesThe simplest error-correcting code: copy one bit L times and check that neighbouring copies agree. The surface-code layout is the “product” of two such codes.; why the 6 X checks and 6 Z checks look as they do and do not interfere; why closing the chains gives back the toric code; how the alarms light up, move and go out as an error chain walks from the left boundary to the right.
- 表示阶梯Representation ladder→ 同一个码的几种写法：星与面的网格图、两张 0/1 检查矩阵、链复形把比特、X 检查、Z 检查三组对象用两张 0/1 矩阵串起来的代数写法。L = 3 时 HX（6 条 X 检查 × 13 个比特）和 HZ（6 条 Z 检查 × 13 个比特）都是 6 × 13，不能直接相乘；「两族检查对易」写成 HXHZT = 0：HZT 是 13 × 6，乘积是 6 × 6，第 i 行第 j 列数的是 X 检查 i 与 Z 检查 j 共有几个比特，模 2（只看奇偶）后必须是 0；在表面码里这个数总是 0 或 2。，以及把它翻译进统计物理（随机键 Ising 模型一种统计物理模型：格点上的小磁针两两耦合，其中一部分耦合（「键」）随机反号。把错误链写成反号的键，译码能不能成功对应这个模型处在有序还是无序的相，阈值就是它的相变点。）、ZX 图一种把量子电路画成图的语言，带一组把图改写成等价图的规则。、quon 图Liu、Wozniakowski 与 Jaffe 提出的一种三维图形语言：量子态和门画成三维空间里的弦与曲面，代数等式变成图形的拓扑变形。与张量网络把一个很大的多比特状态拆成许多小数组（张量）连成的网：每条连线表示相邻两个小数组共用一个下标并对它求和。表面码的码态可以写成这样一张二维网格状的网（PEPS）。的四本「词典」。QEC-Bank 按机器能多直接地执行它，把这些写法排成 R1 到 R5 五级，并注明每条是机器算出来的、照公认定义写的，还是从文献整理来的。Several ways of writing the same code: the grid of stars and faces, two 0/1 check matrices, a chain complexAn algebraic form that links qubits, X checks and Z checks with two 0/1 matrices. At L = 3, HX (6 X checks × 13 qubits) and HZ (6 Z checks × 13 qubits) are both 6 × 13, so they cannot be multiplied directly; “the two check families commute” is written HXHZT = 0: HZT is 13 × 6, the product is 6 × 6, and its entry in row i, column j counts the qubits shared by X check i and Z check j, which must be 0 mod 2 (only the parity counts); in the surface code the count is always 0 or 2., and four “dictionaries” that translate it into statistical physics (the random-bond Ising modelA statistical-physics model: small magnets on a lattice coupled in pairs, with a random share of the couplings (“bonds”) flipped in sign. Write the error chains as the flipped bonds, and whether decoding succeeds corresponds to whether the model sits in its ordered or disordered phase; the threshold is its phase-transition point.), ZX diagramsA language that draws quantum circuits as graphs, with a set of rules for rewriting a graph into an equivalent one., quon diagramsA 3D picture language by Liu, Wozniakowski and Jaffe: quantum states and gates are drawn as strings and surfaces in three-dimensional space, and algebraic identities become topological deformations of the picture. and tensor networksA large many-qubit state split into many small arrays (tensors) joined in a net: each link means two neighbouring arrays share an index that is summed over. The surface code's code states can be written as such a net laid out on a 2D grid (a PEPS).. The QEC-Bank sorts these into five levels, R1 to R5, by how directly a machine can execute them, and marks each as computed by a machine, written from the accepted definition, or collected from the literature.
- 参数与实例Parameters & instances→ 一个旋钮 L 生成一整族码。n、k、d（比特数、逻辑比特数、码距）怎样随 L 变化；QEC-Bank 发布的三个实例 [[13,1,3]]码参数的记法 [[n, k, d]]：n 个物理比特、k 个逻辑比特、码距 d。[[13,1,3]] 就是 13 个比特存 1 个逻辑比特、码距 3。、[[25,1,4]]、[[41,1,5]]；为什么 L = 5 的码距记作 bounded（只登记了上界与下界）而不是 verified（机器独立重算过）；以及算上测量用的帮手比特一共要多少比特。One knob, L, generates a whole family of codes. How n, k, d (number of qubits, number of logical qubits, distance) change with L; the three instances the QEC-Bank publishes, [[13,1,3]]The notation [[n, k, d]] for a code: n physical qubits, k logical qubits, distance d. So [[13,1,3]] stores 1 logical qubit in 13 qubits at distance 3., [[25,1,4]], [[41,1,5]]; why the L = 5 distance is recorded as bounded (only an upper and a lower bound on file) rather than verified (recomputed independently by a machine); and how many qubits you need once the helper qubits used for measurement are counted.
- 形式化 IR 与证书Formal IR & certificates→ 标题里的 IRintermediate representation，中间表示：QEC-Bank 为每个发布的实例导出的一份固定格式的 JSON 文件，把码的矩阵、构造步骤、声明、见证和证书各放进固定字段；核对程序读的是它，而不是散文说明。 是 intermediate representation（中间表示），即 QEC-Bank 给每个实例导出、专供程序读取的那份固定格式的记录。「verified」到底是谁、用什么、对哪个对象核对的：怎样给一个具体的码一个改一个比特就会变的身份编号（语义哈希）；一份「证书」和一篇纸上证明有什么不同；三个实例各拿到了哪几类证书、为什么 L = 5 少一类；以及你怎样在命令行里把这些核对重跑一遍。证书由 Lean一个证明检查程序：只接受每一步都按规则写出的证明，并由它逐步核对。 这类检查程序逐行核对。The IRIntermediate representation: a fixed-format JSON file the QEC-Bank exports for every published instance, with the code's matrices, construction steps, claims, witnesses and certificates each in a fixed field; the checking programs read this file, not the prose. in the title is the intermediate representation, the fixed-format record the QEC-Bank exports for each instance so that programs can read it. Who verified “verified”, with what, and against which object: how a concrete code gets an identity number that changes if a single bit changes (a semantic hash); how a “certificate” differs from a paper proof; which kinds of certificate each of the three instances has and why L = 5 has one kind fewer; and how to rerun these checks yourself from the command line. Certificates are checked line by line by a checking program such as LeanA proof-checking program: it accepts only proofs in which every step is written out by its rules, and checks each step..
- 译码器Decoders→ 从配对游戏到译码器（从警报推断错误的算法）：每个警报要么和另一个警报配成一对，要么配到它所在那类错误链可以出界的边界上（紧贴这种边界的一个错误只亮一个警报，另一端落在边界上不亮），在所有配法里选总长度最短的一种（最小重量完美匹配）；当每个比特独立地、以同一个小概率出错时，为什么最短的配法几乎就是最可能的错误，而各处出错概率不同时，每段长度要换成由该处概率 p 算出的权重 log((1−p)/p)；机器怎样在几亿种配法里快速找到它；贪心配对为什么会输；更快的并查集union-find：一种记录「哪些点已经连成一团」的数据结构，能快速合并两团、查询两点是否同团。并查集译码器让每个警报周围的区域一圈圈长大，两团相碰就合并，直到每团含偶数个警报或碰到边界，再在团内配对。译码器；测量本身也会错时，怎样在「空间 + 时间」里配对；最后把文献里互相打架的阈值数字各自放回它的噪声模型和译码器。From the pairing game to decoders (algorithms that infer the error from the alarms): each alarm is paired either with another alarm or with a boundary where its kind of error chain may leave (a single error right next to such a boundary lights only one alarm, since its other end lies on the boundary and stays dark), and among all pairings the one with the smallest total length is chosen (minimum-weight perfect matching); why, when every qubit fails independently with the same small probability, the shortest pairing is almost the same as the most likely error, while with unequal error probabilities each length must be replaced by a weight log((1−p)/p) computed from the local probability p; how a machine finds that pairing among hundreds of millions; why greedy pairing loses; the faster union-findA data structure that records which points have already joined into one cluster; it merges two clusters and tells whether two points share a cluster quickly. The union-find decoder grows a region around each alarm layer by layer, merges clusters that touch, stops once each cluster holds an even number of alarms or reaches a boundary, and then pairs inside each cluster. decoder; how to pair in “space + time” when the measurements themselves can be wrong; and finally, each of the literature's conflicting threshold numbers put back under its own noise model and decoder.
- 逻辑操作Logical operations→ 剪开、缝合：用格点手术做逻辑门。把两块贴片沿边界缝合，一排只碰邻近比特的新检查连乘起来，就测出了两个逻辑比特的联合奇偶 Z̄Z̄；测完再拆开。一个逻辑 CNOT受控非门：控制比特为 1 时翻转目标比特，为 0 时不动。 要三块贴片：控制 C、目标 T，和一块事先制备成逻辑 |+⟩（0 与 1 等权叠加）的辅助贴片 A。先缝合 C 与 A，测 Z̄CZ̄A；再缝合 A 与 T，测 X̄AX̄T（同样的联合奇偶，换成 ± 基来问）；最后把 A 的比特全部按 Z 读出，A 用完即弃。三个 ±1 读数按一条固定规则记账修正，结果就是 CNOT（逻辑操作篇第 5 章）；最后算清每次缝合为什么要维持 d 轮测量、要花多少时空。Cut and stitch: logical gates by lattice surgery. Stitch two patches together along a boundary, and a row of new checks, each touching only nearby qubits, multiplies out to the joint parity Z̄Z̄ of the two logical qubits; then cut them apart again. A logical CNOTControlled-NOT: flips the target qubit when the control is 1 and leaves it alone when the control is 0. takes three patches: the control C, the target T, and a helper patch A prepared beforehand in the logical |+⟩ (an equal superposition of 0 and 1). First stitch C to A and measure Z̄CZ̄A; then stitch A to T and measure X̄AX̄T (the same joint parity, asked in the ± basis); finally read every qubit of A in the Z basis and discard A. The three ±1 outcomes feed a fixed record-keeping rule, and the result is a CNOT (Logical operations, Chapter 5); finally, why each stitch must be held for d rounds of measurement and what it costs in space and time.
- 用例与实现Use cases & realizations→ 这套码对硬件提出什么要求、适合哪些平台；实验室已经做到了什么：卡片上五条 demonstrated（有实验论文报告过）记录，从 2022 年的祖冲之 2.1 到 2023 年的 Google 与 2024 年的 Willow 超导芯片、可重构中性原子阵列，每个数字带误差与限定。中间一章教你读对一篇实验：每轮逻辑错误率，以及码距每加 2 时错误率缩小的倍数 Λ。What the code demands of hardware and which platforms fit it; what labs have actually shown: the five demonstrated records on the card (each reported by an experimental paper), from Zuchongzhi 2.1 in 2022 to Google's 2023 chip and the 2024 Willow superconducting chip, and reconfigurable neutral-atom arrays, every number with its error bar and qualifier. One chapter teaches you to read an experiment correctly: the logical error rate per round, and Λ, the factor by which that rate shrinks each time the distance grows by 2.
- 关系Relations→ 表面码在 QEC-Bank 的地图上和谁相连。QEC-Bank 只登记了两条关系：它是两条重复码的「超图积」（一种把两条经典码相乘的造法）；它的内部和环面码相同，只是边界不同。本篇在 L = 3 上把两条各算一遍：13 个比特、两组 6 条检查、恰好 1 个逻辑比特；把两条链都首尾接上，1 变成 2。最后是文献里的亲戚（旋转版、XZZX 等），它们各有 QEC-Bank 条目，但还不是这个条目登记的关系。Who the surface code is connected to on the QEC-Bank map. The QEC-Bank records exactly two relations: it is the “hypergraph product” of two repetition codes (a way of multiplying two classical codes), and its interior is the same as the toric code's, only the boundaries differ. This guide works both out at L = 3: 13 qubits, two sets of 6 checks, exactly 1 logical qubit; close both chains and 1 becomes 2. Last come the relatives in the literature (the rotated version, XZZX and others), which have QEC-Bank entries of their own but are not relations recorded on this entry.
- 推导与指南Derivations & guides→ 哪些结论能在纸上证出来、哪些是机器重算的、哪些只是引用文献。五个一步不跳的推导：超图积给出平面贴片；码距恰好等于 L；亮起的警报（症状）恰好是错误链的端点（落在它可以出界的那种边界上的端点不亮）；缝合线测出的是逻辑奇偶；阈值数字各归各的账。外加自测题。Which conclusions can be proved on paper, which a machine recomputed, and which are only cited. Five derivations with no step skipped: the hypergraph product gives the planar patch; the distance is exactly L; the lit alarms (the syndrome) are exactly the endpoints of the error chains (an endpoint on a boundary where that chain may leave stays dark); a seam measures a logical parity; each threshold number goes into its own account. Plus a self-test.
- 参考文献References→ QEC-Bank 里凡不是机器重算的数字，都要挂在一个文献键上（像 dennis2002topological 这样的字符串，指向一篇登记过的论文）。本篇教你读懂一个键：怎么造、登记在哪、被谁引用；把 39 条文献按「它回答什么问题」分成八组；最后从一个具体数字出发，一步步追到原始论文。Every number in the QEC-Bank that a machine did not recompute must hang on a reference key (a string such as dennis2002topological that points to a registered paper). This guide teaches you to read a key: how it is built, where it is registered, who cites it; groups the 39 references into eight groups by the question each answers; and finally traces one concrete number step by step back to the original paper.
建议阅读顺序Suggested order
第一次接触。先读 Toric Code 的直觉篇（零基础版 或 高中数学版），再读本条目的直觉；之后按导览 → 参数与实例 → 译码器 → 逻辑操作走，就把「存住」和「计算」两半都走完了。译码器与逻辑操作两篇用到对数和排列组合，假定读过高中数学版直觉篇。
First contact. Start with the toric-code intuition guide (no-prerequisites or high-school edition), then this entry's Intuition; continue Guided tour → Parameters → Decoders → Logical operations, and both halves, storage and computation, are covered. The decoders and logical-operations guides use logarithms and counting and assume the high-school intuition guide.
核对一个论断。数字在参数与实例与用例与实现；证明在推导与指南；机器核实的范围在形式化证书；出处在参考文献。
Checking a claim. Numbers live in Parameters and Realizations; proofs in Derivations; the machine-verified scope in Formal certificates; sources in References.
往更大的码走。关系篇通向环面码与超图积；系列里的 BB 码 与 qLDPC 码 两篇教程接着讲量子 LDPC 码（低密度奇偶校验码：码越做越大时，每条检查碰的比特数和每个比特参与的检查数都不超过一个固定上限）。表面码自己就是 LDPC 码，但一块只存 1 个逻辑比特；那两篇讲的 BB 码等家族守着同样的上限，一块码却能存很多个逻辑比特。
Toward bigger codes. Relations leads to the toric code and the hypergraph product; the series' BB codes and qLDPC codes tutorials continue with quantum LDPC codes (low-density parity-check codes: as the code grows, the number of qubits each check touches and the number of checks each qubit takes part in both stay below a fixed limit). The surface code is itself an LDPC code, but one block stores only 1 logical qubit; the families those two tutorials cover, such as BB codes, keep the same limits yet store many logical qubits in one block.
QEC-Bank 的构造器（按定义生成检查矩阵的程序）覆盖的是未旋转的平面族 [[2L²−2L+1, 1, L]]：一块 L × L 的贴片上，一组 L² 个比特加一组 (L−1)² 个比特，合计 L² + (L−1)² = 2L²−2L+1 个。条目导读各篇里的参数、证书与推导都针对这个子族。更宽的表面码集合还包括旋转版（图 1）：把 L = 3 贴片四个角上的比特剪掉，13 个数据比特剩 9 个；角旁的三腿边界检查各少一条腿，变成两腿，而每条斜边上相邻的两条两腿检查类型不同、只共用 1 个比特，会互相冲突，所以每条边只留一条；转 45° 看，剩下的 9 个比特排成 3 × 3，最短的逻辑算符仍要 3 个比特，码距不变。一般地，旋转版在同样码距 d 下只用 d² 个数据比特。此外还有 XZZX 版（每条检查混用 X 和 Z）、在贴片上挖洞或放缺陷来存信息的编码等，它们在「关系」篇中单独说明。
The QEC-Bank constructor (the program that builds the check matrices from the definition) covers the unrotated planar family [[2L²−2L+1, 1, L]]: on an L × L patch, one set of L² qubits plus one set of (L−1)² qubits, L² + (L−1)² = 2L²−2L+1 in all. Parameters, certificates and derivations in the entry guides refer to that subfamily. The broader surface-code collection also contains the rotated version (Fig. 1): trim the four corner qubits of the L = 3 patch and 13 data qubits become 9; each three-leg boundary check next to a corner loses a leg and becomes two-leg, and on each slanted edge the two neighbouring two-leg checks are of different types and share just 1 qubit, so they would clash and only one per edge is kept; turned by 45°, the remaining 9 qubits form a 3 × 3 grid, and the shortest logical operator still needs 3 qubits, so the distance is unchanged. In general the rotated version reaches distance d with only d² data qubits. Further members are the XZZX version (each check mixes X and Z), encodings that store information in holes or defects cut into the patch, and others; these are treated separately in “Relations”.
关于这套教程About these guides
每一篇都是单个 HTML 文件，共用同一套页面样式与交互部件：右上角切换中英文，左侧目录随滚动高亮，橙色虚线引用点开即弹出定义卡片，所有格点实验与蒙特卡洛模拟（随机抽很多次噪声、数失败的比例）都在你的浏览器本地运行，不上传任何数据。数字与引用一律沿用 QEC-Bank 的证据词，说明它是怎么来的：verified = 机器独立重算、结果相符；bounded = 只登记了上界与下界；claimed = 文献这样说，QEC-Bank 没有重算；demonstrated = 有实验论文在真实硬件上报告过；proposed = 设计已具体到可以模拟或估算成本，但还没有硬件执行过。
Each guide is a single HTML file sharing the same page style and interactive components: switch language top right, the left rail follows your scroll, amber dashed citations pop up definition cards, and every lattice experiment and Monte-Carlo simulation (sample the noise many times at random, count the fraction of failures) runs locally in your browser with nothing uploaded. Numbers and citations keep the QEC-Bank evidence words, which say how each was obtained: verified = recomputed independently by a machine, and it matched; bounded = only an upper and a lower bound are recorded; claimed = a paper says so, the QEC-Bank has not recomputed it; demonstrated = an experimental paper has reported it on real hardware; proposed = the design is concrete enough to simulate or cost, but no hardware has run it.