剪开环面，得到
Surface Code Cut the torus open:
the Surface Code

环面码速览：L×L 周期网格，边上比特（n = 2L²）；星 A_v = XXXX 与面 B_p = ZZZZ 因「角连两边」全体对易；∏A = ∏B = I 两条恒等式给 k = 2；Z-错误连成链，症状 = 链的端点（∂链）；可收缩闭圈 = 面检查乘积 = 恒等，缠绕圈 = 逻辑算符 Z̄、X̄，交叉一次故反对易；码距 d = L。两副眼镜与沙盘，一分钟热身：

The toric code at a glance: an L×L periodic grid with qubits on edges (n = 2L²); stars A_v = XXXX and plaquettes B_p = ZZZZ all commuting by “a corner touches two sides”; the identities ∏A = ∏B = I give k = 2; Z errors form chains with syndrome = endpoints (∂chain); contractible closed loops = products of plaquettes = identity, winding loops = the logicals Z̄, X̄, crossing once hence anticommuting; distance d = L. One minute of warm-up with the two lenses and the sandbox:

代价清单先摆出来。环面的 k = 2 依赖两条恒等式 ∏A_v = ∏B_p = I，而恒等式依赖「每条边恰被两颗星、两块面分享」的完美均匀——剪开边缘后立刻失效。此外边缘的检查会缺腿：是把缺腿检查扔掉、还是收编？三步手术看全程；注意第 ③ 步的收边分工——它不是美化，而是设计：

Price list first. The torus's k = 2 rested on the identities ∏A_v = ∏B_p = I, which rested on the perfect uniformity “every edge shared by exactly two stars and two faces” — dead on arrival at a cut edge. Rim checks also lose legs: discard them or enlist them? Watch the three-step surgery, and mind step ③'s division of labour — it is design, not cosmetics:

对易定理安然无恙：「角连两边」是纯局部论证，三腿检查照样适用（可在下方开启检查显示逐对核验）。记账用尺寸参数 d：

The commutation theorem survives intact: “a corner touches two sides” is purely local and applies to three-leg checks unchanged (toggle the checks below and verify pair by pair). The ledger, with size parameter d:

下面是 d = 3 的完整蓝图：13 个比特全部编号，四条边界标了名字。点上下两排的顶点看三腿星（光边界），点最左、最右一列的方块看三腿面（毛边界）；「显示 Z̄」「显示 X̄」给出第 3 章的逻辑串。

Below is the complete d = 3 blueprint: all 13 qubits numbered and the four boundaries named. Click a vertex in the top or bottom row for a three-legged star (smooth boundary), a square in the leftmost or rightmost column for a three-legged face (rough boundary); “Show Z̄” and “Show X̄” display the logical strings of Chapter 3.

机理照旧是碰撞定理：Z-链的端点警报来自「被奇数次碰到的星」。让链一路画到左侧悬边并停在那里——悬边外侧本该有的下一颗星根本不存在，于是那一端零警报。分工严格：Z-链只能死在毛边界（左右）；把端点搭到光边界的边上，那里的三腿星照样报警。X-链镜像：只能死在光边界（上下）。在沙盘里验证两种命运：

The mechanism is still the collision theorem: a Z-chain's endpoint alarm is “a star struck an odd number of times”. Draw the chain onto a left-side dangler and stop — the next star beyond the dangler's tip simply does not exist, so that end alarms nothing. The division is strict: Z-chains may die only on the rough boundary (left/right); land an endpoint on a smooth-boundary edge and its three-leg star alarms as usual. X-chains mirror: they die only on the smooth boundary (top/bottom). Verify both fates in the sandbox:

推到极端：两端都死在毛边界的 Z-串——从最左悬边横穿到最右悬边。零警报、通过全部检查；它是检查的乘积吗？用对易性审计：这条横跨串与竖跨的 X̄-串（上光边界到下光边界）恰好交叉一次，反对易；而任何面检查的乘积都与 X̄ 对易（X̄ 与每块面共享 0 或 2 边）。反对易 ≠ 对易——矛盾，故它不是乘积。按定义，它就是逻辑算符 Z̄；同一论证给出 X̄，且二者反对易——恰好一对，与 k = 1 吻合。最短横跨用 d 条边 ⇒ 码距 d（没有更短逻辑算符的理由见补课 3.A）。

Push to the extreme: a Z-string dying on the rough boundary at both ends — leftmost dangler clean across to a rightmost one. Zero alarms, passes every check; is it a product of checks? Audit with commutation: this spanning string crosses the vertical X̄-string (smooth top to smooth bottom) exactly once — anticommutes; yet any product of face checks commutes with X̄ (X̄ shares 0 or 2 edges with every face). Anticommuting ≠ commuting — contradiction, so it is no product. By definition it is the logical Z̄; the same argument yields X̄, and the two anticommute — exactly one pair, matching k = 1. The shortest crossing uses d edges ⇒ distance d (why no logical operator is shorter: Catch-up 3.A).

「有横跨串」还不等于「是逻辑算符」，要数奇偶。看两条平行的横跨串：第 1 行的 Z₁Z₂Z₃ 与第 2 行的 Z₄Z₅Z₆（编号见第 2 章的蓝图）。每条单独看都从左毛边界走到右毛边界、零警报，都是 Z̄。两条一起做呢？把夹在它们中间的第 1 行 3 块面检查乘起来：Z₁Z₄Z₁₀ · Z₂Z₅Z₁₀Z₁₁ · Z₃Z₆Z₁₁，比特 10、11 各出现两次而抵消（Z·Z = I），剩下的恰好是 Z₁Z₂Z₃Z₄Z₅Z₆。所以两条横跨串合起来是检查的乘积，对编码态毫无作用：做两次 Z̄ 等于什么都没做。判据因此是奇偶：一条零警报的 Z 串与 X̄（比特 2、5、8）共享奇数个比特，它就是 Z̄（乘上若干面检查）；共享偶数个，它就是面检查的乘积，无害。上面两条各与 X̄ 共享 1 个比特，合起来共享 2 个（比特 2 和 5），偶数。这个数为什么靠得住：每块面检查与 X̄ 共享 0 或 2 个比特，乘进来不改变奇偶，而 Z̄ 共享 1 个；零警报的 Z 串只有这两类（补课 3.A 末尾数了一遍）。按下「两条横跨串」看这个例子：

“Contains a spanning string” is not yet “is a logical operator”; you have to count parity. Take two parallel spanning strings: Z₁Z₂Z₃ along row 1 and Z₄Z₅Z₆ along row 2 (numbering as in Chapter 2's blueprint). Each on its own runs from the left rough boundary to the right one with zero alarms, and each is Z̄. What about both together? Multiply the 3 face checks of row 1 that sit between them: Z₁Z₄Z₁₀ · Z₂Z₅Z₁₀Z₁₁ · Z₃Z₆Z₁₁; qubits 10 and 11 appear twice and cancel (Z·Z = I), leaving exactly Z₁Z₂Z₃Z₄Z₅Z₆. So the two spanning strings together are a product of checks and do nothing to encoded states: applying Z̄ twice is the same as doing nothing. The criterion is therefore a parity: an alarm-free Z string that shares an odd number of qubits with X̄ (qubits 2, 5, 8) is Z̄ (times some face checks); one that shares an even number is a product of face checks and harmless. The two strings above share 1 qubit each with X̄, 2 together (qubits 2 and 5), which is even. Why this count can be trusted: every face check shares 0 or 2 qubits with X̄, so multiplying one in does not change the parity, while Z̄ shares 1; and alarm-free Z strings come in exactly these two kinds (counted at the end of Catch-up 3.A). Press “Two spanning strings” to see the example:

先看整机。译码器现在多一种配法：把警报「嫁给」最近的己方边界（第 3 章的单警报正需要它）。修复后残留 = 错误⊕修复，闭合或贴边；它无害当且仅当它与 X̄ 交叉偶数次，也就是含偶数条横跨串（0 条也算偶数；两条横跨串合起来是面检查乘积，见第 3 章）。看四拍循环自动运转：

The whole machine first. The decoder gains one new move: marrying an alarm to the nearest boundary of its own kind (exactly what Chapter 3's lone alarms require). The residual = error ⊕ repair is closed or boundary-anchored; it is harmless iff it crosses X̄ an even number of times, that is, contains an even number of spanning strings (zero counts as even; two spanning strings together are a product of faces, see Chapter 3). Watch the four beats run:

谁去测 B_p = ZZZZ？没有仪器能一口气读四比特联合奇偶。工程答案：给每条检查配一个辅助比特ancilla：不存数据、专职收集一条检查的奇偶并被读出的量子比特。芯片上数据比特与辅助比特约一比一。。以 Z-检查为例：辅助比特制备为 |0⟩，与四个数据比特依次做一个「受控翻转」两比特门（CNOT，数据为控制端），最后单独测量辅助比特——它的 0/1 恰好携带四者的联合奇偶，而数据比特全程未被直接测量。

Who measures B_p = ZZZZ? No instrument reads a four-qubit joint parity in one gulp. Engineering's answer: give every check its own ancillaA qubit storing no data, dedicated to collecting one check's parity and being read out. Chips carry roughly one ancilla per data qubit.. For a Z-check: prepare the ancilla in |0⟩, apply a “controlled flip” two-qubit gate (CNOT, data as control) with each of the four data qubits in turn, then measure the ancilla alone — its 0/1 carries exactly the four-fold joint parity, while no data qubit is ever measured directly.

调高 p 玩几轮，你会体感到两件事：警报密度随 p 线性涨，而「配错所酿成的横跨」增长得快得多——像在预演一条相变曲线。它就是下一章。

Raise p for a few rounds and two things become tactile: alarm density grows linearly in p, while spanning accidents from mispairing grow far faster — a rehearsal of a phase-transition curve. It is the next chapter.

两件记号先备好：路径的长度用 ℓ 表示（花体小写 L，读「ell」，免得和 1、I 混淆）；给某条具体路径起名用 γ（希腊字母，读「伽马」）。模型：每轮里每个比特独立以概率 p 遭 Z 错误；测症状，然后做最小重量修复：在所有能产生这组警报的 Z 串里挑边数最少的一条当修复（有几条并列时任取一条）；残留 = 错误 ⊕ 修复若与 X̄ 交叉奇数次，本轮失败，记失败概率 P_L(d, p)。定理：用最小重量修复时，p < 1/36 则 P_L(d, p) 随 d 指数衰减。四步证明：

Two notations first: path length is written ℓ (a script lowercase L, read “ell”, so it can't be mistaken for 1 or I); a specific path gets the name γ (Greek, read “GAM-ma”). Model: each round every qubit independently suffers a Z error with probability p; syndromes are measured, followed by a minimum-weight repair: among all Z strings that produce the observed alarms, take one with the fewest edges (any one, if several tie). The round fails if the residual = error ⊕ repair crosses X̄ an odd number of times; call the probability P_L(d, p). Theorem: with minimum-weight repair, for p < 1/36, P_L(d, p) decays exponentially in d. Four steps:

第 ① 步为什么成立。记错误为 E、修复为 C，都看成边的集合；残留 R = E ⊕ C 是一条零警报的 Z 串。失败时 R 与 X̄ 交叉奇数次，而闭圈和同侧两端的路都是面检查乘积、与 X̄ 交叉偶数次（补课 3.A），所以 R 里至少有一段从左毛边界到右毛边界的路；把其中绕回同一顶点的小圈剪掉，得到一条不重复经过顶点的横跨路径 γ，它的每条边都在 R 里，长 ℓ ≥ d。γ 两端都停在毛边界的悬边外端，本身零警报，所以 C ⊕ γ 与 C 产生同一组警报，也是一个合格的修复。C 是边数最少的修复，于是 |C ⊕ γ| ≥ |C|（|·| = 边数）。把 γ 的 ℓ 条边分成两类：在 C 里的 a 条，不在 C 里的 ℓ − a 条。C ⊕ γ 删掉前一类、添上后一类，边数是 |C| − a + (ℓ − a)，所以 ℓ − a ≥ a。又因为 γ 的边都在 R = E ⊕ C 里，每条恰好属于 E、C 之一：不在 C 里的那 ℓ − a 条就是真出了错的边。所以 γ 上至少一半的边真出了错。小例子：d = 3，错误 E = Z₄Z₅ 只在比特 6 左端的顶点报一次警，最短修复是 C = Z₆（1 条边，配到右毛边界），残留 Z₄Z₅Z₆ 是 Z̄，本轮失败；γ = 第 2 行，3 条边里 2 条出了错，确实过半。在 d = 3 上，计算机把全部 2¹³ 种错误、每种错误的每个最短修复都试了一遍，每次失败都找得到这样的 γ。这一步只用了「C 最短」这一件事，所以下面的 1/36 只对总是给出最短修复的译码器成立。

Why step ① holds. Write the error as E and the repair as C, both as sets of edges; the residual R = E ⊕ C is an alarm-free Z string. On failure R crosses X̄ an odd number of times, while closed loops and paths with both ends on the same side are products of faces and cross X̄ an even number of times (Catch-up 3.A); so R contains at least one path from the left rough boundary to the right one. Cut out any small loop that returns to a vertex already visited, and you get a spanning path γ that visits no vertex twice, with every edge in R and length ℓ ≥ d. Both ends of γ stop at the outer ends of rough-boundary danglers, so γ itself sets off no alarm, and C ⊕ γ produces the same alarms as C: it is also a valid repair. C has the fewest edges of all repairs, so |C ⊕ γ| ≥ |C| (|·| = number of edges). Split the ℓ edges of γ in two: a of them lie in C, ℓ − a do not. C ⊕ γ removes the first kind and adds the second, so it has |C| − a + (ℓ − a) edges, hence ℓ − a ≥ a. And since every edge of γ lies in R = E ⊕ C, each belongs to exactly one of E and C: the ℓ − a edges not in C are exactly the edges that really erred. So at least half of γ's edges really erred. A small example: d = 3, the error E = Z₄Z₅ alarms only once, at the vertex at the left end of qubit 6; the shortest repair is C = Z₆ (1 edge, matched to the right rough boundary); the residual Z₄Z₅Z₆ is Z̄ and the round fails; γ = row 2, and 2 of its 3 edges erred, more than half. On d = 3 a computer tried all 2¹³ errors with every shortest repair of each, and every failure has such a γ. This step uses only “C is shortest”, so the 1/36 below holds only for decoders that always return a shortest repair.

读结论：只要 6√p < 1，即 p < 1/36 ≈ 2.8%，括号 (6√p)^d 就随 d 指数式坍塌——前面的因子 d 拦不住它。对最小重量修复、码容量噪声（只有数据比特出 Z 错误、测量完美），阈值存在，且至少 2.8%。证明里每件工具——不回头路径的计数、C(n,k) 的二项式和、公比小于 1 的等比数列——都来自你的高中书包。

Reading it: whenever 6√p < 1, i.e. p < 1/36 ≈ 2.8%, the factor (6√p)^d collapses exponentially in d — the prefactor d cannot save it. For minimum-weight repair under code-capacity noise (only data qubits suffer Z errors, measurements are perfect), a threshold exists, and it is at least 2.8%. Every tool in the proof — counting non-backtracking paths, the binomial sum for C(n,k), a geometric series with ratio below 1 — comes from your high-school satchel.

左端曲线按 d=7 < d=5 < d=3 分层且间距近似等比——这正是压制律suppression law：阈值之下逻辑失败率随 d 指数下降的经验公式。 P_L ≈ A一个与 d 无关的常数因子。·(p/p_th阈值 threshold：诸尺寸失败曲线的交叉点；p 低于它，加大 d 才有益。)^⌈d/2⌉ 的指纹：d 每增加 2，失败率乘一次同一因子 p/p_th。交叉带即阈值。实验设定：只有数据比特出 Z 错误、测量完美（码容量噪声）；译码器在警报不超过 14 个时做精确的最小重量完美匹配（可配到边界），更多时退为贪心，所以交叉点略低于 10.3% 的文献值。6.1 的 1/36 下界只覆盖精确配对的那部分；贪心不一定给出最短修复，第 ① 步对它不成立，本页没有给它证明。这条压制律就是造大规模量子计算机的全部经济学：p 压到 p_th 之下若干倍后，逻辑错误率随 d 指数下降，而成本只随 d² 多项式上升。

The left-end curves stack d=7 < d=5 < d=3 with near-constant ratios — the fingerprint of the suppression lawThe empirical formula for how logical failure falls exponentially in d below threshold. P_L ≈ AA constant prefactor independent of d.·(p/p_thThe threshold: where the failure curves of different sizes cross; below it, growing d helps.)^⌈d/2⌉: each +2 in d multiplies failure by the same factor p/p_th. The crossing band is the threshold. The setting: only data qubits suffer Z errors and measurements are perfect (code-capacity noise); the decoder does exact minimum-weight perfect matching (boundaries allowed) up to 14 alarms and greedy matching beyond, so the crossing sits somewhat below the literature's 10.3%. The 1/36 bound of §6.1 covers only the exact-matching part; greedy matching need not return a shortest repair, step ① does not apply to it, and this page gives it no proof. This law is the entire economics of scalable quantum computing: push p a few-fold below p_th, and logical errors fall exponentially in d while cost rises only polynomially, as d².

现实进展（截至本教程编写时）：2023 年 Google 首次在硬件上验证 d=5 优于 d=3（每轮逻辑错误率 2.914% 对 3.028%，arXiv 2207.06431）；2024 年底 Willow 芯片达成 d=7，d 每加 2 逻辑错误率降为 1/2.14（Λ = 2.14 ± 0.02），d = 7 时每轮 0.143%（arXiv 2408.13687）——首次明确运行在阈值之下。这些是电路级真实噪声与作者自己的译码器下的数字；随后中性原子、离子阱等平台相继展示低于阈值的行为，另有多个团队研发比表面码更省比特的 qLDPC 码。你刚在浏览器里画出的那族曲线，正在真实实验室里逐条落地。

Reality check (as of this writing): in 2023 Google first verified d=5 beating d=3 on hardware (logical error per round 2.914% versus 3.028%, arXiv 2207.06431); in late 2024 the Willow chip reached d=7, each +2 in d dividing the logical error rate by 2.14 (Λ = 2.14 ± 0.02), with 0.143% per round at d = 7 (arXiv 2408.13687) — the first clear below-threshold operation. These numbers come from real circuit-level noise and the authors' own decoders; neutral-atom and trapped-ion platforms have since shown below-threshold behaviour too, while several teams develop qLDPC codes thriftier than the surface code. The family of curves you just drew in a browser is landing, one by one, in real laboratories.

从矩阵乘法那个不肯交换的 −1，到亲手证出并实测的阈值：你已握有稳定子纠错的完整语言——公设、投影、碰撞定理、同调类、配对译码、压制律。当代论文的三条延伸路：其一，更省的版图：把网格旋转 45° 裁边即「旋转表面码」，同码距只用 d² 个数据比特，是实验室的实际部署版。其二，逻辑门：存住只是仓库，还要运算——拼接与切分两块贴片的「格点手术」（lattice surgery）承担 CNOT 类操作，配合为顽固门准备的「魔法态蒸馏」构成通用计算，这是当前工程化的主战场。其三，更省的码：qLDPC 家族用更巧的检查图谱把编码率提升一个量级，是近年最活跃的理论前沿。入口文献均列于下。

From the −1 that refused to commute, to a threshold you proved and then measured: you now hold stabilizer error correction's full language — postulates, projectors, the collision theorem, homology classes, matching decoders, the suppression law. Three roads current papers extend: one, thriftier layouts — rotate the grid 45° and trim to get the “rotated surface code”, same distance from only d² data qubits, the version labs actually deploy. Two, logical gates — storage is a warehouse; computation merges and splits patches via “lattice surgery” for CNOT-class operations, joined by magic-state distillation for the stubborn gates: today's main engineering front. Three, thriftier codes — the qLDPC family lifts encoding rates by an order of magnitude with cleverer check graphs, the liveliest theory frontier of recent years. Entry points below.