从理想症状
到真实电路 From ideal syndromes
to real circuits

记号：n = 物理量子比特数，k = 逻辑量子比特数，d = 码距（最轻逻辑算符触碰的比特数），L = 环面的边长。[[n, k, d]] 是量子码的名片写法。本篇和 QEC-Bank 一样，k 直接记逻辑比特数。

Notation: n = number of physical qubits, k = number of logical qubits, d = distance (qubits touched by the lightest logical operator), L = side length of the torus. [[n, k, d]] is the calling-card notation for a quantum code. Here, as in the QEC-Bank, k counts logical qubits directly.

取三个数据比特 b₁、b₂、b₃，只看一条检查：「b₁ ⊕ b₂ = 0 吗」（⊕ 读「异或」，就是2.3 节的模 2 加法：0⊕0 = 0，0⊕1 = 1，1⊕1 = 0）。Toric Code 教程直接把答案写在纸上；真实机器不能，它要用一个帮手比特ancilla / helper qubit：一个额外的比特，先置 0，依次与要检查的数据比特做「异或进来」的操作（量子里叫 CNOT），最后单独读出它。读出的值就是这条检查的答案。 h 去收集：h 先置 0，再把 b₁ 异或进 h，再把 b₂ 异或进 h，最后读 h。读到 0 就是「偶，没警报」，读到 1 就是「奇，警报」。这就是一条检查的提取电路syndrome-extraction circuit：制备帮手、逐个 CNOT、读出、复位，这一小段步骤合起来叫提取电路。。现在假设机器每一轮都跑一遍这段电路，我们把读数按轮记下来。

Take three data bits b₁, b₂, b₃ and look at one check only: “is b₁ ⊕ b₂ = 0?” (⊕ reads “xor”, the mod-2 addition of section 2.3: 0⊕0 = 0, 0⊕1 = 1, 1⊕1 = 0). The toric-code guide simply writes the answer on paper; a real machine cannot, it collects it with a helper qubitancilla: an extra qubit, first set to 0, then xor-ed with each data qubit to be checked in turn (a CNOT in quantum language), and finally read out on its own. The value read is the check's answer. h: set h to 0, xor b₁ into h, xor b₂ into h, read h. Reading 0 means “even, no alarm”; reading 1 means “odd, alarm”. This is the extraction circuitsyndrome-extraction circuit: prepare the helper, one CNOT per data qubit, read out, reset; these few steps together are the extraction circuit. of one check. Now suppose the machine runs this circuit every round and we write the readings down round by round.

看第二行和第三行的第 1 轮：都是 1。只看这一轮，译码器无法区分「b₁ 真的翻了」和「b₁ 好好的，是帮手报错了」。前者需要把 b₁ 翻回去，后者什么都不该做——做了反而制造一个错误。这就是测量错误measurement error：数据比特没错，但检查的读数错了。单独一轮里与真警报无法区分。的全部麻烦。第四行提醒一件事：这条检查根本不看 b₃，所以 b₃ 翻了它不报——在真正的码里，b₃ 会被别的检查盯着。

Look at rows two and three in round 1: both read 1. From that round alone a decoder cannot tell “b₁ really flipped” from “b₁ is fine, the helper reported wrongly”. The former needs b₁ flipped back; the latter needs nothing done, and doing something would create an error. That is the whole trouble with a measurement errorThe data qubits are fine but the check's reading is wrong. Indistinguishable from a real alarm within a single round.. Row four is a reminder: this check never looks at b₃, so a flip of b₃ goes unreported here; in a real code other checks watch b₃.

表里第二行和第三行在第 2 轮分开了：真的数据错误让读数持续为 1，帮手的一次误读只影响一轮。于是不看读数本身，看读数有没有变。定义探测器detector：同一条检查相邻两轮读数的异或，D(t) = s(t) ⊕ s(t−1)。读数不变时为 0，只有「变了」才亮。第 0 轮之前约定读数为 0。 D(t) = s(t) ⊕ s(t−1)：s(t) 是第 t 轮的读数。把上表改写成探测器：

Rows two and three separate in round 2: a real data error keeps the reading at 1 persistently, one misread of the helper affects one round. So instead of the reading itself, look at whether it changed. Define the detectorThe xor of the same check's readings in two consecutive rounds, D(t) = s(t) ⊕ s(t−1). It is 0 while the reading is unchanged and lights only on a “change”. The reading before round 0 is taken to be 0 by convention. D(t) = s(t) ⊕ s(t−1), where s(t) is the reading in round t. Rewrite the table in detectors:

记号：t = 轮数（1, 2, 3, …）；s(t) = 第 t 轮这条检查的读数（0 或 1）；D(t) = 第 t 轮的探测器；⊕ = 异或。

Notation: t = round number (1, 2, 3, …); s(t) = this check's reading in round t (0 or 1); D(t) = the detector of round t; ⊕ = xor.

在真正的码里，一个数据错误同时点亮它两端的两条检查（第 0 章事实 4），所以数据错误的探测器花样是「同一轮、相邻两条检查」的一对；测量错误的花样是「同一条检查、相邻两轮」的一对。两种一对，方向不同，同一种配对游戏可以一起处理——这是《译码器 · 高中数学版》5.1 节和5.2 节的时空译码。人话：测量错误不需要新思想，只需要新维度：把时间画成第二根轴。但请记住第三行：探测器只告诉你「哪一轮变了」，不告诉你「谁变的」；决定谁变的仍是译码器的猜测。

In a real code one data error lights the two checks at its ends at once (Chapter 0, fact 4), so a data error's detector pattern is a pair “same round, two neighbouring checks”, while a measurement error's is a pair “same check, two consecutive rounds”. Two kinds of pair in two directions, and one pairing game handles both: that is the spacetime decoding of sections 5.1 and 5.2 of “Decoders · High-School Edition”. In plain words: measurement errors need no new idea, only a new dimension: draw time as a second axis. But keep row three in mind: a detector only tells you “which round changed”, not “who changed”; deciding who is still the decoder's guess.

图注：每格是一个探测器 D(t) 的值，列 = 检查（c1 问 b1 ⊕ b2，c2 问 b2 ⊕ b3），行 = 轮次；填色、写 1 的格子是亮起的探测器，连线标出一对。左：b2 同时被 c1、c2 盯着，它翻了，两条检查在同一轮各亮一次。右：c1 第 2 轮多读出一个 1，第 3 轮读数弹回，D(2)、D(3) 各亮一次。两个例子都是本篇手算的。

Caption: each cell is the value of one detector D(t); columns = checks (c1 asks b1 ⊕ b2, c2 asks b2 ⊕ b3), rows = rounds; filled cells marked 1 are lit detectors, and the line marks the pair. Left: b2 is watched by both c1 and c2; when it flips, both checks light once in the same round. Right: c1 reads an extra 1 in round 2 and the reading snaps back in round 3, so D(2) and D(3) light once each. Both examples were worked out by hand for this guide.

「一个码在错误率 p 下存得好不好」这句话里，「错误率 p」指什么，取决于你允许哪些东西出错。文献里分三层，越往下越像真机器，也越难：

In the sentence “how well a code stores at error rate p”, what “error rate p” means depends on what you allow to fail. The literature distinguishes three levels, each more like a real machine, and harder, than the last:

三层不是同一道题的三种难度，是三道不同的题，答案（阈值）不能互相比。《译码器 · 高中数学版》的账本给了表面码在三层下的数字；每个数字里的 p 指什么，要先说清。前两层用同一种数据错误：每个比特每轮各自独立地以概率 p 发生 X 翻转，并独立地以概率 p 发生 Z 翻转；X、Z 两类分开译码，所以 p 是「正在译的这一类翻转」的概率，不是「发生任意一种 Pauli 错误」的总概率。（对照：总概率为 p 的去极化噪声让 X、Y、Z 各以 p/3 出现，X 类翻转来自 X 或 Y，概率是 2p/3。）在这个约定下，完美测量、最小重量匹配约 10.3%；再让每条检查的读数每轮以概率 q = p 读反，约 2.9%。这两个数都来自 Wang、Harrington 与 Preskill 2003 年的数值模拟（5.3 节）。电路级约 0.5%–1%，依赖具体电路与「一步出错」的定义（5.4 节）。一个具体的例子是 Fowler 等 2012 年的模型：一轮提取电路分 8 步；空等或 Hadamard 之后以 p/3 的概率各插入 X、Y、Z；制备与读出各以概率 p 出错；每个 CNOT 之后以 p/15 的概率各插入 15 种非平凡的两比特 Pauli；用最小重量匹配，得 0.57%，p 按每一步计。区间的上端来自别的电路与噪声约定，本篇没有逐一核对。三个数字都是文献里数值模拟得到的估计，译码器都是最小重量匹配，不是本篇或 QEC-Bank 算的。它们差了十几倍，码没变，变的是「什么算一次错误」。6.1 节说任何阈值数字至少要带五个限定项：错误信道、症状模型、电路约定、译码器、归一化。本篇后面每次出现数字，都会说它属于哪一行。

The three levels are not three difficulties of one problem but three different problems, and their answers (thresholds) cannot be compared across rows. The ledger of “Decoders · High-School Edition” gives the surface code's numbers at the three levels; first be clear what p means in each. The first two levels use the same data errors: in every round each qubit independently suffers an X flip with probability p and, independently, a Z flip with probability p; X and Z are decoded separately, so p is the probability of a flip of the type being decoded, not the total probability of some Pauli error. (For comparison: depolarizing noise of total probability p applies X, Y, Z with p/3 each; an X-type flip comes from X or Y, with probability 2p/3.) With this convention, perfect measurement and minimum-weight matching give about 10.3%; letting every check's reading also flip with probability q = p per round gives about 2.9%. Both come from the numerical simulations of Wang, Harrington and Preskill (2003) (section 5.3). Circuit level gives about 0.5%–1%, depending on the circuit and on how “one step fails” is defined (section 5.4). One concrete case is the model of Fowler et al. (2012): an extraction round has 8 steps; after an idle step or a Hadamard, X, Y or Z is inserted with probability p/3 each; preparation and readout each fail with probability p; after every CNOT one of the 15 nontrivial two-qubit Paulis is inserted with probability p/15 each; with minimum-weight matching this gives 0.57%, with p counted per step. The upper end of the range comes from other circuits and noise conventions that this guide did not check one by one. All three are estimates from numerical simulations in the literature, all with minimum-weight matching as the decoder, not computed by this guide or the QEC-Bank. They differ by more than tenfold; the code did not change, what counts as “one error” did. Section 6.1 says any threshold number needs at least five qualifiers: error channel, syndrome model, circuit convention, decoder, normalisation. Every time a number appears below, this guide says which row it belongs to.

记号：p = 码容量与现象学两层里，每个数据比特每轮发生 X 翻转（另算 Z 翻转）的概率；电路级里，按所用模型定义的每个位置每一步出错的概率；q = 每条检查读数每轮被读反的概率；阈值 = 低于它时把码做大能让逻辑错误率变小的那个 p。

Notation: p = in the code-capacity and phenomenological levels, the probability that a data qubit suffers an X flip (Z flips counted separately) per round; at circuit level, the per-step failure probability of one location as the chosen model defines it; q = the probability that a check's reading is flipped per round; threshold = the p below which making the code larger makes the logical error rate smaller.

Wanted 板的原文点了四种真实故障的名字：measurement、routing、leakage、correlated circuit faults。每个给一个一行的玩具：

The Wanted board names four kinds of real fault: measurement, routing, leakage, correlated circuit faults. One one-line toy for each:

第四行的最后一句值得停一下。BB 码条目的推导页把三个「距离」分开写（BB 译码器教程 4.3 节原文照录）：static code distance（矩阵的 d：最轻的逻辑泡利串）、circuit distance（电路里最少多少个故障能不触发任何探测器却造成逻辑错误）、decoder-found failure weight（某个译码器在模拟里找到的最轻失败）。原文接着说：「A static distance of 12 is compatible with a smaller circuit distance under a bad schedule.」人话：矩阵说 12，坏的电路顺序可以让少于 12 个故障就够。具体少多少，要对那张电路逐一验证；推导页没有给出数字，本篇也不编一个。所以「真实电路下存得好不好」这道题，连码距都要重新算。

The last sentence of row four deserves a pause. The BB entry's derivation page keeps three “distances” apart (quoted in section 4.3 of the BB decoders guide): static code distance (the matrix's d: the lightest logical Pauli string), circuit distance (the fewest faults in the circuit that trip no detector yet cause a logical error), decoder-found failure weight (the lightest failure some decoder found in simulation). The source continues: “A static distance of 12 is compatible with a smaller circuit distance under a bad schedule.” In plain words: the matrix says 12, a bad circuit order can make fewer than 12 faults enough. How many fewer must be checked for that particular circuit; the derivation page gives no number, and this guide does not invent one. So for the question “how well does it store in a real circuit”, even the distance must be recomputed.

逐词翻译。replayable：可重放——别人拿同一份程序、同一个随机种子，跑出一模一样的数字（和码距篇里 seed 5、iters 2000 的搜索是同一种要求）。decoder：译码器，从探测器历史猜出该修什么的程序。benchmark adapter：基准适配器——一个「插头」，把这个码族接到 QEC-Bank 的基准测试台上，让它的表现能和别的码放进同一张表比（BB 译码器教程 3.1 节的三条记录就是那张表的几行）。repeated syndrome extraction：反复提取症状，即 1.2 节的多轮测量。sparse：稀疏——每条检查只碰少数比特（qLDPC 教程 1.2 节）。nongeometric：非几何——检查连接的比特不在平面上挨着；一条检查的 6 个比特可能分散得很远，于是 1.4 节的 routing 问题不可避免。qLDPC：量子低密度奇偶检查码。measurement, routing, leakage, and correlated circuit faults：1.4 节的四个词。rather than only perfect syndrome bits：而不是只有理想症状。人话：请为「检查连得很远」的那类码，造一台把测量错、路上错、跑出字母表、一错两处全算进去的可重放译码器，并把它插到 QEC-Bank 的比较表上。

Word by word. replayable: someone else, with the same program and the same random seed, gets identical numbers (the same demand as the seed-5, iters-2000 search of the distance guide). decoder: the program that guesses what to repair from the detector history. benchmark adapter: a “plug” connecting this code family to the QEC-Bank benchmark bench so its performance can sit in the same table as other codes (the three records of section 3.1 of the BB decoders guide are rows of that table). repeated syndrome extraction: the many-round measurement of 1.2. sparse: each check touches only a few qubits (qLDPC guide section 1.2). nongeometric: the qubits a check connects are not neighbours in a plane; a check's 6 qubits may be far apart, so the routing problem of 1.4 is unavoidable. qLDPC: quantum low-density parity-check codes. measurement, routing, leakage, and correlated circuit faults: the four words of 1.4. rather than only perfect syndrome bits: not just ideal syndromes. In plain words: for the codes whose checks reach far, build a replayable decoder that counts misreads, delays, alphabet escapes and one-fault-two-places, and plug it into the QEC-Bank comparison table.

三个名字，一句否定。small-set flip：量子扩展码自带的局部译码器（qLDPC 教程 6.4 节）：反复找一小组比特，翻转它们能让亮起的检查数严格减少就翻。BP-OSD：置信传播（BP）让每个比特和碰它的检查之间来回传「这个比特错了的可能有多大」的估计，传若干轮后按估计猜错误；猜出的错误对不上症状（卡住）时，用 OSD（ordered statistics decoding：把比特按「多可疑」排序，再按这个顺序解线性方程）收尾（BB 教程 6.3 节）。algebraic CSS baselines：代数基线——直接解「哪些比特翻了能给出这个症状」的线性方程组，不管概率。Perfect-syndrome 修饰全部三个：它们都工作在 1.3 节的第一行。none is a family-specific circuit-level threshold workflow：没有一个是「为这一族码专门做的、电路级的、能给出阈值的完整流程」。人话：QEC-Bank 手里有三把在理想症状下能用的工具，没有一把是给真实电路的。

Three names, one negation. small-set flip: the local decoder that comes with quantum expander codes (qLDPC guide section 6.4): repeatedly find a small set of qubits whose flip strictly reduces the number of lit checks, and flip it. BP-OSD: belief propagation (BP) has each qubit and the checks touching it exchange estimates of “how likely this qubit is to be wrong”, and after some rounds guesses the error from those estimates; when the guess does not reproduce the syndrome (it stalls), OSD (ordered statistics decoding: sort the qubits by how suspicious they are, then solve the linear equations in that order) finishes the job (BB guide section 6.3). algebraic CSS baselines: solve the linear system “which flipped qubits give this syndrome” directly, ignoring probabilities. Perfect-syndrome qualifies all three: they all work in row one of 1.3. none is a family-specific circuit-level threshold workflow: none is a complete pipeline made for this family, at circuit level, delivering a threshold. In plain words: the QEC-Bank holds three tools that work under ideal syndromes and none for real circuits.

Wanted 板的 related 字段列了两个条目：quantum_expander 与 bivariate_bicycle。先看第一个。QEC-Bank 的量子扩展码条目（在 QEC-Bank 中查看 →）登记了两个确定性的有限实例，instances.yaml 原文：

The board's related field lists two entries: quantum_expander and bivariate_bicycle. The first one first. The QEC-Bank quantum-expander entry (open in the QEC-Bank →) registers two deterministic finite instances; instances.yaml verbatim:

两个实例的来历在qLDPC 教程 3.4 节：第一个是完全图 K₅ 的顶点–边关联矩阵（5 行 10 列）做超图积，第二个是四阶仿射平面的直线–点关联矩阵。条目页上 d 那一列分别写 verified 和 claimed：3 是穷举验出来的，5 是引文（Tillich 与 Zémor 2009，tillich2009quantum，已登记）。请注意这两个词说的都是静态码距——1.4 节三个距离里的第一个。它们对「真实电路下的距离」一个字也没说。

Where the two instances come from is in qLDPC guide section 3.4: the first is the hypergraph product of the vertex–edge incidence matrix of the complete graph K₅ (5 rows, 10 columns), the second uses the line–point incidence matrix of the affine plane of order 4. The entry page writes verified and claimed in the d column respectively: 3 was verified by exhaustion, 5 is a citation (Tillich and Zémor 2009, tillich2009quantum, registered). Note that both words speak of the static distance, the first of the three distances of 1.4. They say nothing about “distance in a real circuit”.

下面逐字转录 codes/quantum_expander/card.yaml 的 decoders 字段（渲染出的译码器页面 →）。每条记录有四个字段：id（译码器类型）、noise（它工作在什么噪声设定下）、status（成熟度）、notes（对这个码的说明）。

Below is the decoders field of codes/quantum_expander/card.yaml, transcribed word for word (the rendered decoders page →). Each record has four fields: id (decoder type), noise (the noise setting it works in), status (maturity), notes (what it means for this code).

三条记录对应 Wanted 板原文的三个名字，只是顺序不同。第一条 css-pivot 就是「algebraic CSS baselines」：渲染页解释它「Solves the X- and Z-syndrome linear systems by left-to-right GF(2) elimination with free variables fixed to zero」——用消元法解方程，能算但不聪明，「ignores likelihoods and degeneracy」（不管哪种错误更可能，也不管多个错误等价）。它的 notes 里藏着一句关键话：small-set flip 这个家族专属算法「has no distinct benchmark adapter」——QEC-Bank 的基准测试台上没有它的插头，它只以「view」（一段讲解）的形式存在。第二条 bp-osd 的噪声设定写明「code-capacity … perfect syndrome」，1.3 节第一行。第三条的 id 和 status 都是 none-known，noise 一栏写的正是本篇标题里的「真实电路」：数据、测量、路由、泄漏、相关硬件故障，一个不少。notes 里两件事都说了：截至 2026-08-09 的文献检索没找到这一族的电路级阈值或实验译码器；这个空缺不否定完美症状下的 small-set-flip 定理。第三条记录还带一个 gap 字段，指向 wanted:qldpc-circuit-decoding——就是本篇。

The three records match the three names on the board, in a different order. Record one, css-pivot, is the “algebraic CSS baselines”: the rendered page explains that it “Solves the X- and Z-syndrome linear systems by left-to-right GF(2) elimination with free variables fixed to zero”, solving equations by elimination, workable but not clever, and “ignores likelihoods and degeneracy” (neither which error is more probable nor that several errors are equivalent). Its notes hide a key sentence: the family-specific small-set flip “has no distinct benchmark adapter”; the QEC-Bank benchmark bench has no plug for it, and it exists only as a “view” (an explanation). Record two, bp-osd, states “code-capacity … perfect syndrome” in its noise setting: row one of 1.3. Record three has none-known as both id and status, and its noise column is exactly the “real circuit” of this guide's title: data, measurement, routing, leakage, correlated hardware faults, none missing. The notes say two things: the primary-source search completed on 2026-08-09 found no circuit-level threshold or experimental decoder for this exact family; and this gap does not erase the perfect-syndrome small-set-flip theorem. Record three also carries a gap field pointing at wanted:qldpc-circuit-decoding, this very guide.

量子扩展码的检查矩阵是「非几何」的，卡片的一条 view（类型 nongeometric-syndrome-circuits，来源 curated，引文 delfosse2021measurement）原文：「Bounded check weight does not imply bounded-depth nearest-neighbor extraction in a low-dimensional layout. Expansion distributes Tanner neighbors non-geometrically, so a device needs long-range couplers, routing, or growing local depth.」人话：每条检查只碰 6 个比特（稀疏）不等于这 6 个比特在芯片上挨着；扩展性恰恰要求它们散得开。（「扩展」具体指什么、指哪张图，见下一段。）在平面上只连近邻的电路做不到这种散开：一块区域向外的连线只与它的周长成正比，跟不上区域的面积（qLDPC 教程 1.4 节的 Delfosse–Beverland–Tremblay 界）。所以机器要么有长距离的连线，要么让帮手跑路（routing），要么每轮多花很多步。同一张卡片的 realizations 字段补了一句日期：「No repeated-syndrome laboratory realization of the exact family was found as of 2026-08-09.」这就是为什么 Wanted 板的 statement 把 routing 单独点名：对表面码它不是问题，对这一族它是电路设计的核心。

The check matrix of a quantum expander code is “nongeometric”; one view on the card (type nongeometric-syndrome-circuits, origin curated, citing delfosse2021measurement) reads: “Bounded check weight does not imply bounded-depth nearest-neighbor extraction in a low-dimensional layout. Expansion distributes Tanner neighbors non-geometrically, so a device needs long-range couplers, routing, or growing local depth.” In plain words: a check touching only 6 qubits (sparse) does not make those 6 qubits neighbours on the chip; expansion demands precisely that they be spread out. (What “expansion” means, and on which graph, is the next paragraph.) A circuit with neighbour-only connections in a plane cannot spread things out like this: the wires leaving a region grow only with its perimeter and cannot keep up with its area (the Delfosse–Beverland–Tremblay bound of the qLDPC guide, section 1.4). So the machine needs long-range wiring, or a travelling helper (routing), or many more steps per round. The card's realizations field adds a dated sentence: “No repeated-syndrome laboratory realization of the exact family was found as of 2026-08-09.” That is why the board's statement names routing explicitly: for the surface code it is a non-issue, for this family it is the core of the circuit design.

「扩展」说的是哪张图。量子扩展码是一条经典码 H 与自己做超图积得到的（这里 H 是 K5 的 5 × 10 关联矩阵，2.1 节）。small-set-flip 定理（Leverrier、Tillich、Zémor 2015 的定理 2）的假设落在经典因子码的 Tanner 图上：比特各画一个圆，检查各画一个方块，检查读某个比特就连一条线。设每个比特连 Δ 条检查，集合 S 的邻居 Γ(S) 是至少连着 S 中一个比特的那些检查。S 最多能有 Δ·|S| 个邻居（各比特的线全落在不同的检查上）；「左扩展」要求：凡是不超过全体固定比例的比特集合 S，都有 |Γ(S)| ≥ (1 − δ)·Δ·|S|，即几乎没有两条线撞在同一条检查上；检查一侧也要有同样的性质（「右扩展」），定理要求两侧的 δ 都小于 1/6。乘出来的量子码自己的 Tanner 图严格说并不是扩展图，论文的说法是它「保留了足够的扩展结构」，译码算法才能工作。

Which graph “expansion” is about. A quantum expander code is the hypergraph product of a classical code H with itself (here H is the 5 × 10 incidence matrix of K5, section 2.1). The hypothesis of the small-set-flip theorem (Theorem 2 of Leverrier, Tillich and Zémor 2015) is about the Tanner graph of the classical factor code: one circle per bit, one square per check, a line whenever a check reads a bit. Let every bit touch Δ checks; the neighbours Γ(S) of a set S of bits are the checks joined to at least one bit of S. S can have at most Δ·|S| neighbours (all its lines landing on different checks); “left expansion” demands that every set S of bits up to a fixed fraction of all bits has |Γ(S)| ≥ (1 − δ)·Δ·|S|, so almost no two lines land on the same check; the check side must have the same property (“right expansion”), and the theorem needs δ below 1/6 on both sides. The quantum code's own Tanner graph is, strictly speaking, not an expander; in the paper's words it retains enough expanding structure for the decoding algorithm to work.

数一数。K5 的码里比特 = 边（e01 连顶点 0 与 1，依此类推），检查 = 顶点，Δ = 2。下图左：e01 与 e23 没有公共顶点，邻居 v0、v1、v2、v3 共 4 个，正好是最多的 2 × 2。下图右：e01 与 e02 都连着 v0，邻居只有 v0、v1、v2 三个。δ < 1/6 要求两个比特至少有 (5/6) × 4 ≈ 3.33 个邻居，也就是全部 4 个，所以只要这样的两比特集合在「固定比例」之内，K5 的图就不满足定理的假设；三条边围成的三角形 {e01, e02, e12} 更糟，6 条线只落在 3 个检查上。所以 [[125,37,3]] 这样的小实例不在定理的适用范围里：定理说的是比特数趋于无穷的一族码，其中的图要有足够好的扩展（随机抽取的图以高概率满足要求，但没有已知的高效确定性构造，也没有高效的方法验证一张给定的图满足要求，这是 Fawzi、Grospellier、Leverrier 2018 自己写明的局限）。

Count it. In the K5 code the bits are edges (e01 joins vertices 0 and 1, and so on), the checks are vertices, and Δ = 2. Left in the figure: e01 and e23 share no vertex, so their neighbours are v0, v1, v2, v3, four, exactly the maximum 2 × 2. Right: e01 and e02 both touch v0, so their neighbours are only v0, v1, v2, three. δ < 1/6 requires two bits to have at least (5/6) × 4 ≈ 3.33 neighbours, i.e. all four, so as soon as such two-bit sets fall within the “fixed fraction”, the K5 graph fails the theorem's hypothesis; the triangle {e01, e02, e12} is worse, its 6 lines landing on only 3 checks. So a small instance such as [[125,37,3]] lies outside the theorem's reach: the theorem is about a family whose number of bits goes to infinity, with graphs that expand well enough (randomly sampled graphs qualify with high probability, but no efficient deterministic construction is known, nor an efficient way to certify that a given graph qualifies, a limitation that Fawzi, Grospellier and Leverrier (2018) state themselves).

related 的第二个条目 bivariate_bicycle 是「有答案长什么样」的最近参照。它的卡片 decoders 字段第一条记录 bp-osd 的噪声设定是「memoryless Pauli and phenomenological qLDPC benchmarks」，状态 standard——1.3 节的前两行；第二条记录 bp-osd 的噪声设定是「circuit detector model for the seven-CNOT-layer extraction schedule」，状态 research-grade——第三行，并且带着文献里最常被引用的那个数字：0.8%。BB 教程 6.4 节把它的五个限定项逐条列过，BB 译码器教程 3.3 节再讲了一遍。QEC-Bank 的推导页 syndrome-circuits-and-decoding.md 第 5 节则列出 0.8% 不是什么（原文照录）：

The second related entry, bivariate_bicycle, is the nearest reference for “what having an answer looks like”. The first record of its card's decoders field, bp-osd, has noise setting “memoryless Pauli and phenomenological qLDPC benchmarks”, status standard: the first two rows of 1.3. The second record, bp-osd, has noise setting “circuit detector model for the seven-CNOT-layer extraction schedule”, status research-grade: row three, carrying the most-cited number in the literature: 0.8%. BB guide section 6.4 lists its five qualifiers one by one and section 3.3 of the BB decoders guide goes through them again. Section 5 of the QEC-Bank derivation page syndrome-circuits-and-decoding.md lists what 0.8% is not (verbatim):

最后一句是本篇的钥匙：限定项是数字含义的一部分。换电路、换归一化、换译码器，就是另一个实验。而且即便对 BB 码，QEC-Bank 也只是引用了这个数字，没有自己重跑：同一推导页第 2 节说「The QEC-Bank constructor exports only static check incidence. It intentionally does not manufacture a schedule from row support: support does not determine CNOT order, ancilla assignment, reset placement, or detector definitions.」第 7 节列出条目里没有的五样东西：「no ancilla coordinates; no seven-layer gate order; no reset or measurement operations; no detector definitions; no circuit-distance witness.」（BB 译码器教程 4.4 节）人话：知道一条检查碰哪 6 个比特，推不出这 6 个 CNOT 该按什么顺序做。所以哪怕是 QEC-Bank 里研究最充分的 qLDPC 条目，「真实电路」这一栏也只是一个带限定项的引文，不是一份能重放的流程。量子扩展码连引文都没有。

The last sentence is the key to this guide: the qualification is part of the number's meaning. Change the circuit, the normalisation or the decoder and it is another experiment. Moreover, even for the BB code the QEC-Bank only cites the number and has not rerun it: section 2 of the same derivation page says “The QEC-Bank constructor exports only static check incidence. It intentionally does not manufacture a schedule from row support: support does not determine CNOT order, ancilla assignment, reset placement, or detector definitions.” Section 7 lists five things the entry lacks: “no ancilla coordinates; no seven-layer gate order; no reset or measurement operations; no detector definitions; no circuit-distance witness.” (section 4.4 of the BB decoders guide) In plain words: knowing which 6 qubits a check touches does not tell you in which order to do the 6 CNOTs. So even for the most thoroughly studied qLDPC entry in the QEC-Bank, the “real circuit” column is a qualified citation, not a replayable pipeline. The quantum expander code does not even have the citation.

memory：这里指「量子存储实验」——把一个逻辑比特编进码里，跑 T 轮提取电路，最后读出，看逻辑信息还在不在。operational memory performance：在这种实验里每轮丢掉逻辑信息的概率（逻辑错误率），以及它随码变大怎么变（阈值）。HGP memories：超图积码做的存储器；量子扩展码是超图积的一种（qLDPC 教程 2.1 节）。code-capacity algorithms：1.3 节第一行的译码器——第 2 章的三条记录。这句话的前半句承认：理想症状下我们会译；后半句说：这不够，因为存储性能不是 H_X、H_Z 两块矩阵的属性。BB 推导页第一段说得直接：「‘the decoder’ is not a function of H_X, H_Z alone. The static matrices, extraction circuit, detector graph, noise model, and inference algorithm form one experiment.」人话：问「这个码存得好不好」就像问「这辆车省不省油」——不说路况、司机、油的标号，数字没有意义。

memory: here a “quantum memory experiment”: encode one logical qubit, run the extraction circuit for T rounds, read out, and see whether the logical information survived. operational memory performance: the probability of losing the logical information per round in such an experiment (the logical error rate), and how it changes as the code grows (the threshold). HGP memories: memories built from hypergraph-product codes; quantum expander codes are one kind (qLDPC guide section 2.1). code-capacity algorithms: decoders of row one of 1.3, the three records of Chapter 2. The first half of the sentence concedes that we can decode under ideal syndromes; the second half says that is not enough, because memory performance is not a property of the two matrices H_X, H_Z. The BB derivation page's first paragraph says it directly: “‘the decoder’ is not a function of H_X, H_Z alone. The static matrices, extraction circuit, detector graph, noise model, and inference algorithm form one experiment.” In plain words: asking “does this code store well” is like asking “is this car fuel-efficient”: without the road, the driver and the fuel grade the number means nothing.

记号：T = 存储实验跑的轮数；H_X、H_Z = 码的两块检查矩阵（X 型检查和 Z 型检查各一块，每行一条检查、每列一个比特，1 表示这条检查碰这个比特；见 qLDPC 教程 1.2 节）。

Notation: T = number of rounds the memory experiment runs; H_X, H_Z = the code's two check matrices (one for X-type checks, one for Z-type; one row per check, one column per qubit, a 1 where the check touches the qubit; see qLDPC guide section 1.2).

两个形容词。versioned（带版本）：四个零件的每一个都有编号，改一个字就换一个号；QEC-Bank 已经这样对待码距搜索（方法名 rand-lowweight-v1，种子、轮数都登记，码距篇 2.2 节），这里要求对整条译码流水线也这样做。executable（可执行）：合同是一段能运行的程序，不是论文里的一段话。BB 推导页说一篇论文若不写清「操作合同」六项，「‘decoded with BP-OSD’ is incomplete」——因为读者无法重跑。one（一份）：四个零件不能各自分开引用；0.8% 之所以带五个限定项，就是因为它是四个零件的一个特定组合下的数字。人话：合同 = 把电路、探测器、译码器、噪声定义钉在一起、编上号、能一键重跑的那个东西。有了它，「A 码比 B 码存得好」才是一句可以检验的话。

Two adjectives. versioned: each of the four parts carries a number, and changing one character changes the number; the QEC-Bank already treats distance searches this way (method name rand-lowweight-v1, seed and iteration count registered, distance guide section 2.2), and here the same is demanded of the whole decoding pipeline. executable: the contract is a runnable program, not a paragraph in a paper. The BB derivation page says that if a paper does not spell out the six items of the “operational contract”, “‘decoded with BP-OSD’ is incomplete”, because a reader cannot rerun it. one: the four parts cannot be cited separately; 0.8% carries five qualifiers precisely because it is the number of one specific combination of the four parts. In plain words: the contract is the thing that pins circuit, detectors, decoder and noise definition together, gives them a number and reruns with one click. Only with it does “code A stores better than code B” become a testable sentence.

最后一行要小心措辞。答案不会让 0.8% 变成「BB 码的阈值」，也不会让某个新数字变成「量子扩展码的阈值」——每个数字永远属于它的合同。它做到的是：让两个码在同一份合同下各有一个数字，于是比较才是比较。今天一个 QEC-Bank 用户做不到的事：打开两个条目，问「在每个 CNOT 出错概率 0.1% 的机器上，[[125,37,3]] 和 [[72,12,6]] 哪个更适合做存储器」——两张卡片上没有任何一格能回答，连「不知道」都写不出，因为问题本身没有定义好的合同可依附。

The last row needs careful wording. The answer will not turn 0.8% into “the threshold of BB codes”, nor some new number into “the threshold of quantum expander codes”; every number belongs to its contract forever. What it achieves is this: both codes get a number under the same contract, and only then is a comparison a comparison. What a QEC-Bank user cannot do today: open the two entries and ask “on a machine where each CNOT fails with probability 0.1%, is [[125,37,3]] or [[72,12,6]] the better memory?”; no cell on either card can answer, not even with “unknown”, because the question has no well-defined contract to attach to.

从 statement 的字面出发——「a replayable decoder and benchmark adapter for repeated syndrome extraction … including measurement, routing, leakage, and correlated circuit faults」——一个合格的答案至少交付：

Starting from the statement's letter, “a replayable decoder and benchmark adapter for repeated syndrome extraction … including measurement, routing, leakage, and correlated circuit faults”, a qualifying answer delivers at least:

两点澄清。第一，statement 没有要求译码器是 small-set-flip。卡片引的是完美症状下的定理（「the perfect-syndrome small-set-flip theorem」，Leverrier、Tillich、Zémor 2015），但理论不止于此。Fawzi、Grospellier 与 Leverrier 2018 年证明：对扩展足够好的量子扩展码，读数也会出错时 small-set-flip 仍然有效，而且是单发single-shot：每次译码只用一轮带噪读数，不必为了一次译码重复测量多轮；残余错误不为零，但保持很小，下一次译码继续处理。的：只用一轮带噪读数，除了一个概率 e^−Ω(√n) 的例外，修复后的残余错误在相差稳定子的意义下等价于参数为 q^Ω(1) 的局部随机噪声，也就是保持很小、但不保证为零（论文定理 2）。前提是一种「局部随机」噪声：任意指定 |S| 个比特全出错、同时任意指定 |T| 条读数全读反的概率不超过 p^|S|q^|T|，且 p、q 低于两个存在但没有给出数值的常数。他们还把它放进一个完整的容错方案（论文定理 1），在同一类局部随机电路噪声、错误率低于某个同样没给数值的阈值时成立。所以缺的不是「读数出错时能否译码」的理论，而是 QEC-Bank 的基础设施：这一族具体实例（如 [[125,37,3]]）的提取电路、探测器与噪声文件、跑出来的具体数字，以及别人能重跑的合同。况且图 2.1 已经说明，这么小的实例本身不满足定理的扩展假设。一个在探测器图上跑的 BP-OSD，只要五样齐全，同样是合格答案。

Two clarifications. First, the statement does not require the decoder to be small-set-flip. The card cites the perfect-syndrome theorem (“the perfect-syndrome small-set-flip theorem”, Leverrier, Tillich and Zémor 2015), but the theory goes further. Fawzi, Grospellier and Leverrier (2018) proved that for quantum expander codes with good enough expansion, small-set-flip still works when readings can be wrong, and that it is single-shotEach decoding uses one round of noisy readings; no repeated rounds are needed for one decoding. The leftover error is not zero but stays small, and the next decoding keeps handling it.: after one round of noisy readings, except with probability e^−Ω(√n), the residual error is equivalent, up to stabilizers, to local stochastic noise with parameter q^Ω(1), so it stays small but is not guaranteed to vanish (Theorem 2 of the paper). The premise is a “local stochastic” noise: the probability that any chosen |S| qubits are all in error and any chosen |T| readings are all flipped is at most p^|S|q^|T|, with p and q below two constants that exist but are not given numerically. They also put it into a complete fault-tolerant scheme (Theorem 1 of the paper), valid for the same kind of local stochastic circuit noise below a threshold whose value is likewise not given. So what is missing is not the theory of “can one decode when readings fail”, but QEC-Bank infrastructure: extraction circuits for concrete instances of this family (such as [[125,37,3]]), detector and noise files, an actual number, and a contract others can rerun. Besides, Fig. 2.1 already showed that an instance this small does not meet the theorem's expansion hypothesis. A BP-OSD running on the detector graph qualifies just as well, as long as all five deliverables are present.

第二，statement 也没有要求一个「好看」的数字：一份合同下跑出来的逻辑错误率，哪怕很差，只要可重放、可比较，就填上了 none-known 那一格。人话：要的是一台能让别人重跑的机器，不是一个漂亮的百分数。

Second, the statement does not ask for a “nice” number either: a logical error rate obtained under one contract, however poor, fills the none-known cell as long as it is replayable and comparable. In plain words: what is wanted is a machine others can rerun, not a pretty percentage.

Wanted 板用两个词分难度。open-research：没人知道怎么做——比如码距篇里给 144 个比特的码证明下界，QEC-Bank 里至今没有可重放的方法（文献用混合整数规划算出过 gross 的精确码距 12，但那不是 QEC-Bank 能重跑的证书）。hard-engineering：知道怎么做，但要做很多。这道题是后者，理由在第 2 章：表面码已经走完了这条路（探测器错误模型 + 时空匹配），BB 码的文献也走完了一次（深度 7 电路 + 探测器 + BP-OSD，得到 0.8%）。每一步的做法都有先例，没有一步需要新定理。难在三处：

The board grades difficulty with two words. open-research: nobody knows how; for example, proving a lower bound for a 144-qubit code in the distance guide, where the QEC-Bank has no replayable method yet (the literature computed gross's exact distance 12 by mixed-integer programming, but that is not a certificate the QEC-Bank can rerun). hard-engineering: known how, but much to do. This problem is the latter, for the reasons of Chapter 2: the surface code has walked this road (detector error model plus spacetime matching), and the BB literature has walked it once (depth-7 circuit plus detectors plus BP-OSD, giving 0.8%). Every step has a precedent and none needs a new theorem. The difficulty sits in three places:

「做很多」有多多？用 1.1 节的算法数一数。一条碰 w 个比特的检查，一轮提取电路有 1 次制备 + w 个 CNOT + 1 次读出 = w + 2 个可能出错的位置。对 toric_L3 [[18,2,3]]：18 条检查，每条 4 条腿，一轮有 18 × 6 = 108 个位置（不算等待）。对 [[125,37,3]]：qLDPC 教程 3.4 节说它的因子是 K₅ 的 5 × 10 关联矩阵，每行 4 个 1（每个顶点连 4 条边）、每列 2 个 1（每条边两个端点）；超图积的每条检查重量 = 行重 + 列重 = 6，检查数 = 5 × 10 + 10 × 5 = 100；一轮约 100 × 8 = 800 个位置，仍不算等待。存储实验跑多少轮 T 是协议的选择，常见取法是 T = d，这里取 T = 3。BB 教程 6.4 节的电路噪声模型里一个 CNOT 有 15 种不同的出错方式（两个比特上各放 I、X、Y、Z 之一，共 4 × 4 = 16 种组合，去掉「两边都是 I」即什么都没发生的那一种）；每次制备、每次读出各算 1 种。于是三轮里的基本故障有：toric_L3 3 × 18 × (1 + 4 × 15 + 1) = 3,348 种，[[125,37,3]] 3 × 100 × (1 + 6 × 15 + 1) = 27,600 种。「哪种故障点亮哪些探测器、翻不翻逻辑」这张表——文献里叫探测器错误模型detector error model：把每一种基本故障用模拟跑一遍，记下它点亮的探测器集合与是否翻转逻辑；译码器在这张表定义的加权（超）图上工作。——就从这些基本故障逐一模拟得来。点亮同一组探测器、翻转同一组逻辑的故障会合并成一行（两个独立故障 p₁、p₂ 合并后的概率是「恰好一个发生」的 p₁ + p₂ − 2p₁p₂，因为两个都发生就互相抵消），所以表的行数比基本故障少；合并后剩多少行，本篇没有算。「上万」只说 [[125,37,3]] 的基本故障数。这些数字是本篇自己数的（脚本核对），QEC-Bank 页面上没有；它们不吓人，机器一秒就能列完，真正花时间的是把每一行都做对、做成能重放的文件、并在换一个顺序时全部重来。这就是 hard-engineering 的字面意思。

How much is “much”? Count with the procedure of 1.1. A check touching w qubits has, per round of extraction, 1 preparation + w CNOTs + 1 readout = w + 2 locations that can fail. For toric_L3 [[18,2,3]]: 18 checks of 4 legs each, 18 × 6 = 108 locations per round (idling not counted). For [[125,37,3]]: qLDPC guide section 3.4 says its factor is the 5 × 10 incidence matrix of K₅, with 4 ones per row (each vertex meets 4 edges) and 2 per column (each edge has two endpoints); every hypergraph-product check has weight = row weight + column weight = 6, and the number of checks is 5 × 10 + 10 × 5 = 100; about 100 × 8 = 800 locations per round, still without idling. How many rounds T a memory experiment runs is a choice of the protocol; T = d is common, and here T = 3. In the circuit-noise model of BB guide section 6.4 one CNOT has 15 distinct ways to fail (one of I, X, Y, Z on each of its two qubits, 4 × 4 = 16 combinations, minus “I on both”, which is no fault at all); each preparation and each readout counts as 1. So over three rounds the elementary faults number 3 × 18 × (1 + 4 × 15 + 1) = 3,348 for toric_L3 and 3 × 100 × (1 + 6 × 15 + 1) = 27,600 for [[125,37,3]]. The table of “which fault lights which detectors and whether it flips the logical”, called a detector error modelSimulate every elementary fault once and record the set of detectors it lights and whether it flips the logical; the decoder works on the weighted (hyper)graph this table defines. in the literature, is built by simulating these elementary faults one by one. Faults that light the same detectors and flip the same logicals are merged into one row (two independent faults p₁, p₂ merge to p₁ + p₂ − 2p₁p₂, the probability that exactly one happens, since both together cancel), so the table has fewer rows than there are elementary faults; how many remain after merging, this guide did not compute. “Tens of thousands” refers only to the elementary faults of [[125,37,3]]. These counts are this guide's own (checked by script) and appear on no QEC-Bank page; they are not frightening, a machine lists them in a second. What takes the time is getting every row right, making it a replayable file, and redoing all of it when one order changes. That is the literal meaning of hard-engineering.

记号：w = 一条检查触碰的比特数（检查重量）；K₅ = 5 个顶点两两相连的完全图，10 条边；T = 存储实验的轮数，由协议选定；本节取 T = 3，等于码距 d（常见取法，不是必须）。

Notation: w = the number of qubits a check touches (check weight); K₅ = the complete graph on 5 vertices, every pair joined, 10 edges; T = the number of rounds of a memory experiment, chosen by the protocol; here T = 3, equal to the distance d (a common choice, not a requirement).

预设是 toric_L3 [[18,2,3]]。按「新错误」，实验台在 18 个比特上撒下 X 错误，然后直接算出理想的 Z 检查症状（s = H_Ze，一行矩阵乘法），把「亮起的 Z 检查」数给你——注意它大约是错误重量的两倍，第 0 章事实 4。接着让 BP 一轮轮传消息，看每个比特的「相信自己错了」的程度变化，卡住时用 OSD 收尾。这就是 Wanted 板原文里的「BP-OSD … perfect-syndrome」。请特别注意实验台没有的东西：没有帮手比特，没有「轮」，没有读反的可能，没有探测器，没有路由，没有泄漏，没有一次故障翻两个比特。它拿到的症状是纸上的真值。第二条记录的 status 是 research-grade、noise 是 code-capacity，说的就是这台机器。

The preset is toric_L3 [[18,2,3]]. Press “New error” and the bench scatters X errors over the 18 qubits, then directly computes the ideal Z-check syndrome (s = H_Ze, one line of matrix multiplication) and gives you the count of “Z-checks lit”; note it is about twice the error weight, Chapter 0 fact 4. Then let BP pass messages round by round, watch each qubit's degree of “believing itself flipped” change, and finish with OSD when it stalls. This is the “BP-OSD … perfect-syndrome” of the board's wording. Pay special attention to what the bench does not have: no helper qubits, no “rounds”, no possibility of a misread, no detectors, no routing, no leakage, no single fault flipping two qubits. The syndrome it receives is the truth on paper. Record two's status research-grade and noise code-capacity describe exactly this machine.

柱子是什么。每根柱子是一个比特的对数似然比 L = ln[P(0)/P(1)]：P(0) 是 BP 当前估计的「这个比特没被翻」的概率，P(1) 是「被翻了」的概率。L > 0 表示更可能没错，L < 0 表示更可能错了，|L| 越大越有把握。例：BP 开始前每个比特只有先验，滑块 p = 0.06 时 P(0) = 0.94、P(1) = 0.06，L = ln(0.94/0.06) ≈ 2.75；L = −8 表示 P(1) 是 P(0) 的 e⁸ ≈ 3000 倍，几乎肯定错了。纵轴只画到 ±8：超出的柱子截在 ±8，柱端画一个小三角标出。柱子下方的数字是比特编号（比特多时每 5 个或 10 个标一次）。这里的 BP 用归一化最小和规则（消息乘 α = 0.8），是近似算法，L 不是精确概率。滑块上的 p 既是撒错误的概率，也是 BP 的先验；拖动滑块会清掉已经算好的 BP 结果，当前这组错误保留（它是按旧的 p 撒的），下一次运行 BP 按新的 p 重算。

What the bars are. Each bar is one qubit's log-likelihood ratio L = ln[P(0)/P(1)]: P(0) is BP's current estimate of the probability that the qubit was not flipped, P(1) that it was. L > 0 means more likely fine, L < 0 more likely wrong, and the larger |L|, the surer. Example: before BP starts every qubit has only its prior; at slider value p = 0.06, P(0) = 0.94 and P(1) = 0.06, so L = ln(0.94/0.06) ≈ 2.75; L = −8 means P(1) is e⁸ ≈ 3000 times P(0), almost certainly wrong. The vertical axis stops at ±8: taller bars are cut at ±8 and their cut end is marked by a small triangle. The numbers under the bars are qubit indices (every 5th or 10th when there are many qubits). BP here uses the normalized min-sum rule (messages multiplied by α = 0.8), an approximate algorithm, so L is not an exact probability. The slider's p is both the probability used to scatter errors and BP's prior; moving the slider clears any BP result already computed, keeps the current error sample (drawn at the old p), and the next BP run recomputes with the new p.

记号：s = 症状向量（每条 Z 检查一个 0/1）；e = 错误向量（每个比特一个 0/1，1 表示这个比特被 X 翻了）；H_Z = Z 检查矩阵；s = H_Ze 的乘法按模 2 进行；L = ln[P(0)/P(1)] = 一个比特的对数似然比，ln 是自然对数。

Notation: s = the syndrome vector (one 0/1 per Z check); e = the error vector (one 0/1 per qubit, 1 meaning the qubit was flipped by X); H_Z = the Z-check matrix; the product s = H_Ze is taken mod 2; L = ln[P(0)/P(1)] = a qubit's log-likelihood ratio, ln the natural logarithm.

把 1.1 节的玩具加一条检查：c₁ 问 b₁ ⊕ b₂，c₂ 问 b₂ ⊕ b₃。各自用一个帮手，连测 4 轮。画一张 2 列（两条检查）× 4 行（四轮）的格子，在每格写探测器 D。对下面三种情形分别填表，并说出花样：(a) b₂ 在第 2 轮前翻了；(b) c₁ 的帮手在第 2 轮读反；(c) b₁ 在第 2 轮前翻了。答案：(a) 第 2 行两格都亮（横向一对：b₂ 被两条检查盯着）；(b) c₁ 列的第 2、3 行亮（纵向一对）；(c) 只有 c₁ 列第 2 行亮——一个孤零零的点，因为 b₁ 只被一条检查盯着，它是这个玩具的「边界」。第三种情形提醒你：探测器花样是否成对，取决于检查怎么覆盖比特，这正是「detector model」要写清的事。

Add one check to the toy of 1.1: c₁ asks b₁ ⊕ b₂, c₂ asks b₂ ⊕ b₃. Each has its own helper, measured for 4 rounds. Draw a grid of 2 columns (two checks) × 4 rows (four rounds) and write the detector D in each cell. Fill it for three situations and name the pattern: (a) b₂ flips before round 2; (b) c₁'s helper is misread in round 2; (c) b₁ flips before round 2. Answers: (a) both cells of row 2 light (a horizontal pair: b₂ is watched by two checks); (b) rows 2 and 3 of column c₁ light (a vertical pair); (c) only row 2 of column c₁ lights, a lone dot, because b₁ is watched by one check only; it is this toy's “boundary”. The third case reminds you that whether detector patterns come in pairs depends on how checks cover qubits, which is exactly what a “detector model” must spell out.

回到 c₁ 的提取电路：h 置 0，异或 b₁，异或 b₂，读 h。假设在两个异或之间 h 翻了一次。先看经典玩具：h 是两次「异或进来」的目标端，它自己翻了只会弄错这一次读数，传不到 b₂。写出这一轮和下一轮 c₁ 的读数与探测器。答案：本轮 c₁ 读数多翻一次，下一轮恢复正常；于是探测器 D₁ 本轮亮一次、下一轮再亮一次——一个故障、两条记录，这就是「测量错误」那种整齐的时间型一对。经典玩具里一个故障只碰一条读数；量子版的 hook 不同，下面换到真正的码上看。

Back to c₁'s extraction circuit: set h to 0, xor in b₁, xor in b₂, read h. Suppose h flips once between the two xors. First the classical toy: h is the target of both “xor into” operations, so its own flip only spoils this reading and never reaches b₂. Write the readings and detectors of c₁ for this round and the next. Answer: this round c₁'s reading is flipped once extra, next round it is back to normal; so detector D₁ lights this round and again next round — one fault, two entries, the tidy timelike pair of a “measurement error”. In the classical toy one fault touches only one reading; the quantum hook is different, and the rest of this exercise moves to a real code to see it.

标号。取 [[125,37,3]]。它的因子是 K5 的关联矩阵：比特 = 边 e01、e02、…、e34（e01 连顶点 0 与 1），检查 = 顶点 v0 到 v4。超图积的 125 个数据比特分两种：100 个标作 (边, 边)，25 个标作 (顶点, 顶点)。X 检查标作 X(顶点 a, 边 j)，作用在 4 个 (e, j) 上（e 取遍连着 a 的 4 条边）和 2 个 (a, v) 上（v 取遍 j 的两个端点）；Z 检查标作 Z(边 i, 顶点 b)，作用在 4 个 (i, e′) 上（e′ 取遍连着 b 的 4 条边）和 2 个 (u, b) 上（u 取遍 i 的两个端点）。这里看 X 检查 X(v0, e01)，它的 6 个比特记作 q1 到 q6（下图左列）。

Labels. Take [[125,37,3]]. Its factor is the incidence matrix of K5: bits = edges e01, e02, …, e34 (e01 joins vertices 0 and 1), checks = vertices v0 to v4. The 125 data qubits of the hypergraph product come in two kinds: 100 labelled (edge, edge) and 25 labelled (vertex, vertex). An X check is labelled X(vertex a, edge j) and acts on the 4 qubits (e, j), e running over the 4 edges at a, and on the 2 qubits (a, v), v running over the two endpoints of j; a Z check is labelled Z(edge i, vertex b) and acts on the 4 qubits (i, e′), e′ running over the 4 edges at b, and on the 2 qubits (u, b), u running over the two endpoints of i. Here we look at the X check X(v0, e01); its 6 qubits are called q1 to q6 (left column of the figure).

测量顺序（这是合同的一部分，这里定一个）：每一轮先测完全部 X 检查，再测全部 Z 检查。一条 X 检查的做法：帮手 h 制备成 |+⟩，以 h 为控制端依次对 q1、…、q6 做 CNOT，最后测 h 的 X。故障：第 2 个与第 3 个 CNOT 之间，h 上出现一个 X。CNOT 把控制端上的 X 复制到目标端，所以 CNOT 3–6 把它复制到 q3、q4、q5、q6：一个故障，数据上是 4 比特的 X 错误。h 自己的读数不变（|+⟩ 上的 X 不改变 X 测量的结果），X 探测器一个都不亮。

Schedule (part of the contract; here we fix one): every round, all X checks are measured first, then all Z checks. One X check is measured by preparing the helper h in |+⟩, applying CNOTs with h as control onto q1, …, q6 in turn, and measuring h in the X basis. Fault: an X appears on h between the 2nd and the 3rd CNOT. A CNOT copies an X on its control onto its target, so CNOTs 3–6 copy it onto q3, q4, q5, q6: one fault, a 4-qubit X error on the data. The helper's own reading is unchanged (an X on |+⟩ does not change an X measurement), so no X detector lights.

算探测器。这 4 个比特上的 X 在同一轮随后的 Z 检查里被看见：一条 Z 检查亮，当且仅当它与 {q3, q4, q5, q6} 重叠奇数次。逐个数：q3 = (e03, e01) 在 Z(e03,v0)、Z(e03,v1) 里；q4 = (e04, e01) 在 Z(e04,v0)、Z(e04,v1) 里；q5 = (v0, v0) 在 Z(e01,v0)、Z(e02,v0)、Z(e03,v0)、Z(e04,v0) 里；q6 = (v0, v1) 在 Z(e01,v1)、Z(e02,v1)、Z(e03,v1)、Z(e04,v1) 里。Z(e03,·) 与 Z(e04,·) 各被碰两次，抵消；剩下 Z(e01,v0)、Z(e01,v1)、Z(e02,v0)、Z(e02,v1)。探测器 = 本轮读数 ⊕ 上一轮读数，错误留在数据上以后读数不再变，所以表里的这一行是：本轮这 4 个 Z 探测器亮，X 探测器不亮，下一轮什么都不亮。它和 q1、q2 上各一个 X 点亮的完全一样：4 比特的错误乘上这条 X 检查本身（一个稳定子）就是 q1、q2 上的 2 比特错误。所以这一个故障相当于两个数据错误，这就是 hook 可能压低电路码距的原因。

Compute the detectors. The X on these 4 qubits is seen by the Z checks measured next in the same round: a Z check lights exactly when it overlaps {q3, q4, q5, q6} an odd number of times. One by one: q3 = (e03, e01) is in Z(e03,v0) and Z(e03,v1); q4 = (e04, e01) in Z(e04,v0) and Z(e04,v1); q5 = (v0, v0) in Z(e01,v0), Z(e02,v0), Z(e03,v0), Z(e04,v0); q6 = (v0, v1) in Z(e01,v1), Z(e02,v1), Z(e03,v1), Z(e04,v1). Z(e03,·) and Z(e04,·) are each hit twice and cancel; left are Z(e01,v0), Z(e01,v1), Z(e02,v0), Z(e02,v1). A detector is this round's reading ⊕ the previous round's, and once the error sits on the data the readings stop changing, so this fault's row of the table reads: these 4 Z detectors light this round, no X detector lights, nothing lights next round. That is exactly what one X on each of q1 and q2 would light: the 4-qubit error times this X check itself (a stabilizer) is the 2-qubit error on q1, q2. So this single fault acts like two data errors, which is why a hook can lower the circuit distance.

同一个 X 故障放在别的位置，结果如下表（脚本逐个核对）。它不是整齐的一对：从 2 个到 8 个都有，这就是 4.2 节说的超边，也是 4.3 节那张表里的几行。做完你会同意：这些行取决于顺序。把 CNOT 的顺序换掉，同一个位置的故障落到另一组比特上，点亮另一组探测器。顺序是合同。

The same X fault at other positions gives the table below (each checked by script). It is not a tidy pair: anything from 2 to 8, the hyperedges of section 4.2 and a few rows of the table of section 4.3. Having done it you will agree that these rows depend on the order: change the CNOT order and a fault at the same position lands on other qubits and lights other detectors. The order is the contract.

记号：q1…q6 = X(v0, e01) 的 6 个比特，依次是 (e01,e01)、(e02,e01)、(e03,e01)、(e04,e01)、(v0,v0)、(v0,v1)；「乘上这条 X 检查后」= 与第二列差一个稳定子、效果完全相同的错误；两列里较轻的那个，才是这个故障真正「值」几个数据错误。最后一列 = 各比特所在 Z 检查的条数之和，减去被碰偶数次而抵消的那些；(边, 边) 比特各在 2 条 Z 检查里，(顶点, 顶点) 比特各在 4 条里，例如 q5、q6 给 4 + 4 = 8，没有抵消。

Notation: q1…q6 = the 6 qubits of X(v0, e01), namely (e01,e01), (e02,e01), (e03,e01), (e04,e01), (v0,v0), (v0,v1); “times this X check” = the error that differs from the second column by a stabilizer and has exactly the same effect; the lighter of the two columns is how many data errors the fault is really “worth”. Last column = the number of Z checks the qubits sit in, added up, minus those hit an even number of times, which cancel; an (edge, edge) qubit sits in 2 Z checks, a (vertex, vertex) qubit in 4, so q5, q6 give 4 + 4 = 8 with no cancellation.

想看「有答案」长什么样：《译码器 · 高中数学版》第 5 章把表面码从理想症状走到电路级，BB 译码器教程第 4 章把 BB 码的 hook、探测器、三个距离和操作合同讲完。想看这道题的主角：qLDPC 教程 3.4 节的两个实例与6.4 节的 small-set-flip，以及 QEC-Bank 的 quantum_expander 条目与它的译码器页。想看同一块板上相邻的题：「有些码还没有标准译码器」讲的是另一类 none-known——那里缺的是被公认为「标准」的译码器（haah_cubic 在理想症状下有研究级的 RG 与 BP-OSD 记录，缺的是标准件和电路级流程）。这条 Wanted 记录没有 working_repo 字段。

To see what “having the answer” looks like: Chapter 5 of “Decoders · High-School Edition” takes the surface code from ideal syndromes to circuit level, and Chapter 4 of the BB decoders guide covers the BB code's hooks, detectors, three distances and operational contract. To see this problem's protagonist: the two instances of qLDPC guide section 3.4 and the small-set-flip of section 6.4, plus the QEC-Bank quantum_expander entry and its decoders page. For the neighbouring problem on the same board: “Some codes still have no standard decoder” is about the other kind of none-known: codes lacking a decoder accepted as “standard” (haah_cubic has research-grade RG and BP-OSD records under ideal syndromes; what is missing is a standard one and a circuit-level workflow). This Wanted record has no working_repo field.

回放一遍。真实机器里，检查的读数由帮手比特跑一段提取电路收集，帮手自己会错；单看一轮，「数据错了」和「帮手报错了」无法区分，多测一轮、只报变化的探测器把它们分成横向一对与纵向一对。允许哪些东西出错，分出三层噪声模型：码容量、现象学、电路级；三层是三道不同的题，表面码在三层下的阈值 10.3%、2.9%、0.5%–1% 不能互比。QEC-Bank 的 quantum_expander 条目有两个实例 [[125,37,3]]（d verified）与 [[656,170,5]]（d claimed），译码器一栏三条记录：css-pivot 与 bp-osd 都写着完美症状，第三条在完整电路那一格写着 none-known（2026-08-09），small-set-flip 只有讲解没有插头。隔壁 BB 码有一个引文数字 0.8%，带五个限定项，且条目不导出电路。我们想要答案，是因为「存储性能」不是矩阵的属性，而是提取电路、探测器模型、译码器、噪声归一化四个零件捆成一份带版本、可执行的合同之后的属性；有了它，none-known 那格换成一条有 id 的记录，两个码才能在同一份合同下并排比较。合格的答案是五样可交付的东西；难度是 hard-engineering——每一步都有先例，但非几何的连接让每一步都要为这一族重做，而且每个设计选择都要钉进版本号。

Replay. In a real machine a check's reading is collected by a helper qubit running an extraction circuit, and the helper itself can fail; within one round “the data is wrong” and “the helper reported wrongly” cannot be told apart, and one more round plus detectors reporting only change separate them into a horizontal and a vertical pair. What is allowed to fail defines three noise models: code capacity, phenomenological, circuit level; they are three different problems, and the surface code's thresholds of 10.3%, 2.9% and 0.5%–1% cannot be compared across them. The QEC-Bank quantum_expander entry has two instances, [[125,37,3]] (d verified) and [[656,170,5]] (d claimed), and three decoder records: css-pivot and bp-osd both say perfect syndrome, the third says none-known (2026-08-09) in the full-circuit cell, and small-set-flip has an explanation but no plug. Next door the BB code has one cited number, 0.8%, with five qualifiers, and its entry exports no circuit. We want the answer because “memory performance” is not a property of the matrix but of four parts, extraction circuit, detector model, decoder and noise normalisation, bound into one versioned executable contract; with it the none-known cell becomes a record with an id, and two codes can sit side by side under the same contract. A qualifying answer is five deliverables; the difficulty is hard-engineering: every step has a precedent, but the nongeometric connections make every step a redo for this family, and every design choice must be pinned into a version number.