四个输入，三个数字，
和每个数字旁边的那个词 Four inputs, three numbers,
and the word beside each number

量子码的名片是[[n, k, d]]双方括号是量子码的记法（单方括号 [n,k,d] 留给经典码）：n 个物理量子比特，编码 k 个逻辑量子比特，码距 d。：n 个物理比特一起保护 k 个逻辑比特，d 是码距能通过全部检查、却改写逻辑信息的最轻算符（逻辑算符）的重量，即它触碰的量子比特数。——能骗过全部检查、却改写逻辑信息的最轻算符触碰的比特数。n 与 k 的记账规则在《从高考数学到 Toric Code》的 4.4 节（注意那里用 k 记独立检查的条数；从 BB 教程起 k 直接记逻辑比特数，见BB 教程第 0 章的记号说明），d 的定义在同一教程的 5.8 节。BB 教程里你已经算过 gross 的 k：两个秩都是 66，k = 2 × 6 = 12。

A quantum code's calling card is [[n, k, d]]Double brackets denote a quantum code (single brackets [n,k,d] are reserved for classical codes): n physical qubits encoding k logical qubits with distance d.: n physical qubits jointly protecting k logical qubits, with d the distanceThe weight, i.e. the number of qubits touched, of the lightest operator that passes every check yet rewrites the logical information (a logical operator).: the number of qubits touched by the lightest operator that fools every check yet rewrites logical data. The bookkeeping rule for n and k is in section 4.4 of “From Exam Maths to the Toric Code” (note that k there counts independent checks; from the BB guide onward k counts logical qubits directly, see the notation note in Chapter 0 of the BB guide), and d is defined in section 5.8 of the same guide. In the BB guide you have already computed k for gross: both ranks are 66, so k = 2 × 6 = 12.

QEC-Bank 比一般的参数表多做一件事：每个数字旁边站着一个证据词QEC-Bank 给每条声明标注来源类别的固定词汇（origin kind）：verified（机器独立重算并相符）、bounded（只有上下界，未穷举）、claimed（引自文献、未重算）等。。说人话：verified 是「QEC-Bank 的机器自己重算过，结果一样」；bounded 是「机器只算出了一半（上界），另一半（下界）靠引文」；claimed 是「照抄文献，机器没算」。同一个 d = 12，写成 verified、bounded 或 claimed，是三句强度不同的话。本篇每张表都会把这个词原样保留。

The QEC-Bank does one thing more than an ordinary parameter table: beside every number stands an evidence wordThe fixed vocabulary (origin kind) the QEC-Bank attaches to every claim's source: verified (independently recomputed by machine and matching), bounded (only bounds, no exhaustive check), claimed (cited from the literature, not recomputed), and others.. In plain words: verified means “the QEC-Bank machine recomputed it itself and got the same answer”; bounded means “the machine computed only half (the upper bound), the other half (the lower bound) rests on a citation”; claimed means “copied from a paper, the machine did not compute it”. The same d = 12 written as verified, bounded or claimed is three statements of different strength. Every table in this guide keeps that word verbatim.

表面码的参数页只登记一个家族变量生成一族码中某一个成员所需的输入参数。BB 条目有四个：l、m、a_terms、b_terms。 L。BB 条目的参数页登记了四个，原文如下（左列是 instances.yaml 里的字段名）：

The surface-code parameters page registers a single family variableAn input parameter that selects one member of a code family. The BB entry has four: l, m, a_terms, b_terms., L. The BB entry's page registers four, in these words (the left column is the field name in instances.yaml):

读这张表时有两个细节值得停一下。第一，[3,0] 是「x 的指数 3、y 的指数 0」，所以是 x³，不是 y³；[0,3] 才是 y³。第二，「reduced modulo (l,m)」的意思是：写 [15,0] 和写 [3,0] 在 l = 12 时是同一项，因为往右挪 15 格等于挪 3 格。constructor.py 里对应的一行是 (term[0] % l, term[1] % m)。四个变量取定之后，就得到一个实例家族变量取定具体值后得到的那一个有限码对象，有确定的检查矩阵和确定的 [[n,k,d]]。QEC-Bank 的 BB 条目发布了两个：gross 与 toric_L3。。QEC-Bank 为 BB 条目发布了两个实例，标签叫 gross 和 toric_L3；第 3 章逐字转录它们的记录。

Two details in this table deserve a pause. First, [3,0] means “exponent 3 on x, exponent 0 on y”, hence x³, not y³; [0,3] is y³. Second, “reduced modulo (l,m)” means that writing [15,0] and writing [3,0] give the same term when l = 12, because moving 15 sites right is the same as moving 3. The matching line in constructor.py is (term[0] % l, term[1] % m). Once the four variables are fixed you have an instanceThe one finite code object obtained once the family variables are fixed: a definite check matrix and a definite [[n,k,d]]. The QEC-Bank BB entry ships two: gross and toric_L3.. The QEC-Bank ships two instances for the BB entry, labelled gross and toric_L3; Chapter 3 transcribes their records word for word.

codes/bivariate_bicycle/constructor.py 只有几十行，做的正是 BB 教程 2.3 节与3.1 节的翻译：① 造一个 l × l 的循环移位 S（第 i 行的 1 在第 (i+1) mod l 列，这是「正向」约定）；② 令 x = S_l ⊗ I_m、y = I_l ⊗ S_m；③ 对 a_terms 的每一对 (a, b) 算出置换矩阵 x^ay^b，全部按 mod 2 相加，得到 lm × lm 的矩阵 A，B 同理；④ 拼出 H_X = [A | B]，H_Z = [Bᵀ | Aᵀ]；⑤ 交给 CheckMatrix.from_css 并调用 validate()，即当场核对 CSS 条件。格子 (i, j) 的编号是 i·m + j（先数行再数列），左层比特占第 0 到 lm−1 列，右层占第 lm 到 2lm−1 列。

codes/bivariate_bicycle/constructor.py is a few dozen lines and performs exactly the translation of BB guide section 2.3 and section 3.1: (1) build an l × l cyclic shift S (row i has its 1 in column (i+1) mod l, the “forward” convention); (2) set x = S_l ⊗ I_m, y = I_l ⊗ S_m; (3) for each pair (a, b) in a_terms form the permutation matrix x^ay^b and add them all mod 2, giving the lm × lm matrix A, and likewise B; (4) assemble H_X = [A | B], H_Z = [Bᵀ | Aᵀ]; (5) hand them to CheckMatrix.from_css and call validate(), which checks the CSS condition on the spot. Site (i, j) is numbered i·m + j (rows first, then columns); left-layer qubits occupy columns 0 to lm−1, right-layer qubits columns lm to 2lm−1.

这套约定不是抽象的。reproduce.py 断言 gross 第 0 条 X 检查（格子 (0,0)）恰好碰这六个比特：[1, 2, 18, 75, 78, 84]。自己数一遍：A 的三项 x³、y、y² 把 (0,0) 送到 (3,0)、(0,1)、(0,2)，编号 3·6+0 = 18、1、2，这是左层的三腿；B 的三项 y³、x、x² 送到 (0,3)、(1,0)、(2,0)，编号 3、6、12，加上右层的偏移 72 得 75、78、84。第 0 条 Z 检查是 [3, 60, 66, 76, 77, 126]：Bᵀ 把三次平移倒过来走，(0,−3) = (0,3) → 3、(−1,0) = (11,0) → 66、(−2,0) = (10,0) → 60；Aᵀ 在右层给 (−3,0) = (9,0) → 54+72 = 126、(0,−1) = (0,5) → 77、(0,−2) = (0,4) → 76。下面的实验台把预设固定在 gross，点任何一条检查看它的六条腿，把读数里的比特编号和这两行对一对。图上横轴是 i（x 方向，向右），纵轴是 j（y 方向，向上）；实心点是左层比特，空心点是右层比特；高亮的腿旁边标着它的编号，格子 (0,0) 在左下角。

None of this is abstract. reproduce.py asserts that gross X check 0 (site (0,0)) touches exactly these six qubits: [1, 2, 18, 75, 78, 84]. Count them yourself: the three terms x³, y, y² of A send (0,0) to (3,0), (0,1), (0,2), numbered 3·6+0 = 18, 1, 2, the three left-layer legs; the three terms y³, x, x² of B send it to (0,3), (1,0), (2,0), numbered 3, 6, 12, plus the right-layer offset 72 gives 75, 78, 84. Z check 0 is [3, 60, 66, 76, 77, 126]: Bᵀ walks the three translations backwards, (0,−3) = (0,3) → 3, (−1,0) = (11,0) → 66, (−2,0) = (10,0) → 60; Aᵀ on the right layer gives (−3,0) = (9,0) → 54+72 = 126, (0,−1) = (0,5) → 77, (0,−2) = (0,4) → 76. The bench below is preset to gross; click any check to see its six legs and compare the qubit indices in the readout with these two lines. In the picture the horizontal axis is i (the x direction, rightwards) and the vertical axis is j (the y direction, upwards); filled dots are left-layer qubits, hollow dots right-layer qubits; each highlighted leg carries its index, and site (0,0) is at the bottom left.

必须说清一次：QEC-Bank 的这个条目对应 eczoo（Error Correction Zoo，一个公开的纠错码百科网站）的集合 qcga（正式身份 eczoo-entity:qcga#collection），构造器覆盖的是平移不变、写在有限二维环面上、按 H_X = [A | B]、H_Z = [Bᵀ | Aᵀ] 拼装的那一族。文献里同样冠以 BB 之名的开边界平面版、剪枝版、自对偶版、多变量版、qudit 版，都不是这个构造器生成的对象，它们的 n、k、d 不在本篇的表里；QEC-Bank 的「关系」侧面和 pedagogy/variant-boundary-map.md 专门处理它们。反过来，环面码本身在这一族之内：A = 1 + x、B = 1 + y 就是它（BB 教程 3.1 节）——这正是 QEC-Bank 把 toric_L3 收作第二个实例的原因。

This must be said once, plainly: this QEC-Bank entry corresponds to the eczoo (the Error Correction Zoo, a public encyclopaedia website of error-correcting codes) collection qcga (formal identity eczoo-entity:qcga#collection), and the constructor covers the family that is translation-invariant, written on a finite two-dimensional torus, and assembled as H_X = [A | B], H_Z = [Bᵀ | Aᵀ]. The open-boundary planar, pruned, self-dual, multivariate and qudit variants that the literature also files under “BB” are not objects this constructor produces, and their n, k, d are not in this guide's tables; the QEC-Bank “Relations” facet and pedagogy/variant-boundary-map.md handle them. Conversely, the toric code itself is inside the family: A = 1 + x, B = 1 + y is exactly it (BB guide section 3.1), which is why the QEC-Bank keeps toric_L3 as its second instance.

环面有 N = lm 个格子，每个格子放两个数据比特（左层一个、右层一个），所以 n = 2lm。矩阵语言里：A、B 都是 N × N，[A | B] 有 2N 列，列数就是比特数。gross：N = 12 × 6 = 72，n = 144；toric_L3：N = 3 × 3 = 9，n = 18。人话：n 数的是数据比特，账本原文特意补了一句，它「excludes syndrome ancillas, couplers, routing workspace, spare modules, and decoder hardware」——不含帮手比特、耦合器、布线、备用模块和译码硬件。第 5 章再把帮手加回来。

The torus has N = lm sites and each site carries two data qubits (one in the left layer, one in the right), so n = 2lm. In matrix language: A and B are both N × N, [A | B] has 2N columns, and the column count is the qubit count. Gross: N = 12 × 6 = 72, n = 144; toric_L3: N = 3 × 3 = 9, n = 18. In plain words: n counts data qubits; the ledger adds deliberately that it “excludes syndrome ancillas, couplers, routing workspace, spare modules, and decoder hardware”. Chapter 5 puts the helpers back.

一个单项式是置换矩阵，每行每列恰有一个 1；一个有 |A| 个不同约化项的多项式，矩阵每行每列恰有 |A| 个 1（BB 教程 2.3 节）。于是 H_X = [A | B] 的每一行有 |A| + |B| 个 1：每条 X 检查碰 |A| + |B| 个比特；H_Z 同理。反过来看列：左层比特所在的列在 A 里有 |A| 个 1、在 Bᵀ 里有 |B| 个 1，所以每个左层比特被 |A| 条 X 检查和 |B| 条 Z 检查碰到；右层比特的列在 B 里、在 Aᵀ 里，所以反过来，被 |B| 条 X 检查和 |A| 条 Z 检查碰到。|A| = |B| 时两层一样；不相等时两层不同，例如 bb5（|A| = 2，|B| = 3）：左层比特 2 X + 3 Z，右层 3 X + 2 Z。gross：|A| = |B| = 3，检查重量 6，比特度 3 + 3。账本特意说，「degree six」这个压缩说法会丢掉 Pauli 类型的划分，而电路里三条 X 型和三条 Z 型相互作用「have different ancilla preparations and fault propagation」——帮手比特的制备方式与错误传播路径都不同，所以要写成 3 + 3。reproduce.py 不是从「三项」推出重量，而是逐行逐列真的数了一遍：A、B 的每行每列都是 3，H_X、H_Z 的每行都是 6、每列都是 3。

A monomial is a permutation matrix, exactly one 1 per row and per column; a polynomial with |A| distinct reduced terms is a matrix with exactly |A| ones per row and per column (BB guide section 2.3). So every row of H_X = [A | B] has |A| + |B| ones: every X check touches |A| + |B| qubits, and likewise H_Z. Reading columns instead: a left-layer qubit's column has |A| ones in A and |B| ones in Bᵀ, so every left-layer qubit sits in |A| X checks and |B| Z checks; a right-layer qubit's column lies in B and in Aᵀ, so it is the other way round, |B| X checks and |A| Z checks. When |A| = |B| the two layers agree; otherwise they differ, e.g. bb5 (|A| = 2, |B| = 3): left-layer qubits 2 X + 3 Z, right-layer 3 X + 2 Z. Gross: |A| = |B| = 3, check weight 6, qubit degree 3 + 3. The ledger points out that the compressed phrase “degree six” loses the Pauli split, while in a circuit the three X-type and three Z-type interactions “have different ancilla preparations and fault propagation”, so it is written 3 + 3. reproduce.py does not infer the weight from “three terms”; it actually counts every row and column: every row and column of A and B is 3, every row of H_X and H_Z is 6, every column is 3.

稳定子计数给 k = n − rank H_X − rank H_Z。BB 教程 4.1 节证明两个秩相等（记作 r），所以 k = 2lm − 2r = 2(lm − r)。账本对这一步补了一句为什么可以直接相加：CSS 码的稳定子秩是两个秩之和，「because pure-X and pure-Z binary symplectic rows cannot cancel each other」——纯 X 行和纯 Z 行写在辛表示的不同半边，互相抵消不了。gross：GF(2) 上高斯消元得 rank H_X = rank H_Z = 66，稳定子秩 132，k = 144 − 132 = 12。toric_L3：两个秩各 8，k = 18 − 16 = 2。

The stabilizer count gives k = n − rank H_X − rank H_Z. BB guide section 4.1 proves the two ranks are equal (call the value r), so k = 2lm − 2r = 2(lm − r). The ledger adds why the two may simply be added: the CSS stabilizer rank is their sum “because pure-X and pure-Z binary symplectic rows cannot cancel each other”; pure-X and pure-Z rows live in different halves of the symplectic representation and cannot cancel. Gross: Gaussian elimination over GF(2) gives rank H_X = rank H_Z = 66, stabilizer rank 132, k = 144 − 132 = 12. toric_L3: both ranks 8, k = 18 − 16 = 2.

亏损 lm − r 有一个更有解释力的读法（BB 教程 4.2 节）：把一对多项式 (u, v) 送到 uA + vB，所有这样的 uA + vB 组成理想 (A, B)（补课 2.A 在 toric_L3 上数了一遍）。按「系数 ↔ 格子」的编码（xⁱy^j ↔ 格子 (i, j)，编号 i·m + j）写成矩阵，这个映射是 [Aᵀ | Bᵀ] 乘列向量 (u; v)，不是 [A | B]：构造器约定下，A 的第 0 列落在编号 4、5、54 上，那是 Aᵀ 的图案；A 的第 0 行（1、2、18）才是多项式 A 的图案，所以 Aᵀ 的第 s 列正是把 A 的形状平移到格子 s。[Aᵀ | Bᵀ] 与 H_Z = [Bᵀ | Aᵀ] 只差左右两块的次序，秩相同，都是 r = 66，这就是理想的维数；剩下的 72 − 66 = 6 维是商空间 R/(A,B)。账本原话：「the dual side has the same six-dimensional defect, leaving twelve logical dimensions in the middle」。两个重要的诚实限定紧跟其后。第一，这是「a computed fact for these periods and polynomials」——对这组 l、m、A、B 算出来的事实，不是所有三项 BB 多项式都编码 12 个比特的承诺。第二，「Three terms in A and three in B do not determine the rank: changing their exponents changes the common ideal and can change k」——三项不决定 k，改指数就改理想，k 随之变。最直接的证据你在 BB 教程见过：同一对 A、B 放到 9 × 6 的环面上，k 从 12 掉到 8。

The deficit lm − r has a more explanatory reading (BB guide section 4.2): send a pair of polynomials (u, v) to uA + vB; all such uA + vB form the ideal (A, B) (catch-up 2.A counts it on toric_L3). Written as a matrix under the coefficient-to-site encoding (xⁱy^j ↔ site (i, j), index i·m + j), this map is [Aᵀ | Bᵀ] acting on the column vector (u; v), not [A | B]: under the constructor's convention column 0 of A sits on indices 4, 5, 54, which is the pattern of Aᵀ; row 0 of A (1, 2, 18) is the pattern of the polynomial A, so column s of Aᵀ is the shape of A translated to site s. [Aᵀ | Bᵀ] differs from H_Z = [Bᵀ | Aᵀ] only in the order of its two blocks, so both have rank r = 66, and that is the dimension of the ideal; the remaining 72 − 66 = 6 dimensions are the quotient R/(A,B). In the ledger's words: “the dual side has the same six-dimensional defect, leaving twelve logical dimensions in the middle”. Two honest qualifiers follow at once. First, this is “a computed fact for these periods and polynomials”, not a promise that every three-term BB polynomial pair encodes twelve qubits. Second, “Three terms in A and three in B do not determine the rank: changing their exponents changes the common ideal and can change k”. The most direct evidence you have already seen in the BB guide: the same A, B on a 9 × 6 torus drops k from 12 to 8.

n、重量、k 都是一次高斯消元的事；d 不是。账本这一节的标题就叫「Distance: two certificates, not one label」。说 d = 12 要同时交两份东西：下界——证明不存在重量小于 12 的非平凡逻辑算符（要排除的对象太多，不能靠找）；上界——拿出一个重量恰为 12 的逻辑算符（一个具体的见证witness：一个具体的、可机器复核的例子。这里指一个与所有检查对易、却不是检查乘积、重量恰为某个数的 Pauli 算符；它只证明 d 不超过那个数。，找到就算）。BB 教程 4.4 节讲过这个分工；第 3 章把 gross 的两份证书各自是谁写的、写在哪里，逐字抄下来——那正是「bounded」这个词的全部内容。

n, weight and k are one Gaussian elimination each; d is not. The ledger's section is titled precisely “Distance: two certificates, not one label”. To say d = 12 you must hand over two things at once: a lower bound, a proof that no non-trivial logical operator of weight below 12 exists (too many objects to exclude by searching); and an upper bound, an actual logical operator of weight exactly 12 (a concrete witnessA concrete, machine-checkable example: here a Pauli operator that commutes with every check, is not a product of checks, and has weight exactly some number; it proves only that d does not exceed that number., which counts as soon as it is found). BB guide section 4.4 introduced this division of labour; Chapter 3 copies out who wrote each of gross's two certificates and where, which is the entire content of the word “bounded”.

下表逐字转录 codes/bivariate_bicycle/instances.yaml 与 QEC-Bank 参数页。斜体是证据词（QEC-Bank 称 origin kind）。

The table transcribes codes/bivariate_bicycle/instances.yaml and the QEC-Bank parameters page verbatim. Italics are the evidence words (the QEC-Bank calls them origin kinds).

注意两行的 d 写法不同。toric_L3 是 {property: d, exact: 3}——一个数、一个词 verified。gross 是三行：lower、upper、origin——同一个 12 出现两次，各带自己的来源。这个结构差异不是排版偶然，它就是本章的主题。

Notice that the two rows write d differently. toric_L3 is {property: d, exact: 3}: one number, one word, verified. gross is three lines, lower, upper, origin: the same 12 appears twice, each with its own source. This structural difference is no accident of layout; it is the subject of this chapter.

n、k、css 的 verified 直白：验证器（make verify-entry E=codes/bivariate_bicycle）重新运行构造器，读矩阵宽度得 n，做两次 GF(2) 高斯消元得 k，检查 H_XH_Z^T 是否全零得 css，再与 yaml 里的声明比对。两个实例三项全部相符，所以六个 verified；加上 toric_L3 的 d，共七个。toric_L3 的 d 也是 verified，方法 exact-css-enum：逐一枚举所有重量小于 3 的候选，确认每一个要么触发某条检查、要么本身是检查的乘积，没有一个是逻辑算符；再出示一个重量 3 的逻辑算符。18 个比特的枚举量很小（BB 教程 5.1 节说过它是「检验约定的锚点」：把转置忘掉，检查立刻不再两两对易），所以 toric_L3 三个数全是 verified。

The verified on n, k and css is straightforward: the verifier (make verify-entry E=codes/bivariate_bicycle) reruns the constructor, reads n off the matrix width, does two Gaussian eliminations over GF(2) for k, checks that H_XH_Z^T is all zero for css, and compares with the yaml claims. All three items agree for both instances, hence six verifieds; with toric_L3's d that makes seven. The d of toric_L3 is also verified, by exact-css-enum: enumerate one by one every candidate of weight below 3, confirm that each either trips some check or is itself a product of checks, so none is a logical; then exhibit one logical of weight 3. For 18 qubits the enumeration is tiny (BB guide section 5.1 called it “the anchor that checks conventions”: forget the transpose and the checks stop commuting at once), so all three numbers of toric_L3 are verified.

gross 有 144 个比特。要穷举重量 1 到 11 的全部候选，数量级远超验证器的例行预算（下一篇形式化证书导读会算这个数）。于是 QEC-Bank 把 d 的声明拆成两半分别记账，instances.yaml 原文：

Gross has 144 qubits. Exhausting every candidate of weight 1 to 11 is far beyond the verifier's routine budget by many orders of magnitude (the formal-certificates guide computes that number). So the QEC-Bank splits the distance claim into two separately book-kept halves; the instances.yaml text reads:

下界 d ≥ 12：引文。账本原话：「bravyi2024bicycle supplies the claim that no nontrivial logical Pauli has weight below 12. The QEC-Bank does not currently replay that lower-bound certificate. The citation is therefore part of the structured origin, not decorative bibliography.」人话：「没有比 12 更轻的逻辑算符」这句话是 Bravyi、Cross、Gambetta、Maslov、Rall、Yoder 的论文（“High-Threshold and Low-Overhead Fault-Tolerant Quantum Memory”，2024）说的，QEC-Bank 的机器没有自己重算这一半；所以 ref: bravyi2024bicycle 写在 origin 里，是这条声明的结构性组成部分，不是文末装饰。

Lower bound d ≥ 12: a citation. In the ledger's words: “bravyi2024bicycle supplies the claim that no nontrivial logical Pauli has weight below 12. The QEC-Bank does not currently replay that lower-bound certificate. The citation is therefore part of the structured origin, not decorative bibliography.” Plainly: the statement “nothing lighter than 12 is a logical” comes from the paper by Bravyi, Cross, Gambetta, Maslov, Rall and Yoder (“High-Threshold and Low-Overhead Fault-Tolerant Quantum Memory”, 2024), and the QEC-Bank machine did not recompute that half; so ref: bravyi2024bicycle sits inside origin as a structural part of the claim, not as an ornament at the end.

上界 d ≤ 12：种子搜索。验证器每次都重跑一个确定性的低重量搜索 rand-lowweight-v1。账本描述了它的步骤：构造中心化子零空间与全部同类检查对易的 Pauli 算符构成的空间：X 型即 H_Z v = 0 的解 v。逻辑算符就住在这个空间里、稳定子行空间之外。（与所有 Z 检查对易的 X 串，即 H_Zv = 0 的解），用 NumPy 种子 5 抽取确定性的二元组合，排除落在稳定子类里的，再用同类稳定子贪心地减轻代表的重量，在 2000 次迭代内找到重量 12。这是「a rerunnable existence witness for a logical of weight at most 12」——一个可重跑的存在性见证：d 至多 12。

Upper bound d ≤ 12: a seeded search. On every run the verifier reruns a deterministic low-weight search, rand-lowweight-v1. The ledger describes its steps: build the centralizer nullspaceThe space of Pauli operators commuting with every check of one type: for X-type, the solutions v of H_Z v = 0. Logical operators live in this space, outside the stabilizer row space. (X strings commuting with every Z check, i.e. solutions of H_Zv = 0), sample deterministic binary combinations with NumPy seed 5, reject those that fall in a stabilizer class, greedily reduce each representative's weight by same-type stabilizers, and find weight 12 within 2,000 iterations. This is “a rerunnable existence witness for a logical of weight at most 12”: d is at most 12.

为什么两个 12 相等还叫 bounded。账本：「Because the cited lower bound and machine upper bound meet, d = 12. The origin is bounded, not verified: the machine has not exhaustively excluded weights one through eleven at n = 144.」规则很干脆：只要下界那一半不是 QEC-Bank 自己穷举出来的，整条声明就叫 bounded。bounded 不是「可能不对」，它是一句精确的话：「上界机器可重跑，下界靠引文，两者没有被同一台穷举机器合拢」。

Why two equal twelves are still bounded. The ledger: “Because the cited lower bound and machine upper bound meet, d = 12. The origin is bounded, not verified: the machine has not exhaustively excluded weights one through eleven at n = 144.” The rule is blunt: whenever the lower half was not exhaustively produced by the QEC-Bank itself, the whole claim is called bounded. Bounded does not mean “possibly wrong”; it is the precise sentence “the upper bound is machine-rerunnable, the lower bound rests on a citation, and the two were not closed by one exhaustive machine”.

先让浏览器把七个 BB 码的 n、rank、k 现场重算一遍（下表第一行的 bicycle 是单变量的祖先，不在这个构造器里，只出现在静态表中；点「重算 n 与 k」；再点「再加低重搜索」可以看到本页几秒钟的搜索能把 d 压到哪里，那只是上界）。表的「证据」一列就是 QEC-Bank 各条目上 d 旁边的词。

First let the browser recompute n, rank and k for the seven BB codes live (bicycle, the first row of the table below, is the univariate ancestor and is not in this constructor; it appears only in the static table; press “Recompute n and k”; “Also run the low-weight search” shows how far a few seconds of search on this page can push d, an upper bound only). The “evidence” column is the word beside d on each QEC-Bank entry.

下表把每个码送回它在 QEC-Bank 里的家。Table 3 指 bravyi2024bicycle 论文（arXiv 2308.07915）里列出 BB 码的那张表，共七个：[[72,12,6]]、[[90,8,10]]、[[108,8,10]]、[[144,12,12]]、[[288,12,18]]，以及两个只知道码距上界的更大例子 [[360,12,≤24]] 与 [[756,16,≤34]]（「≤」表示论文写作时只有上界）。本篇与 BB 教程只讲前五个，下表也只列这五个；bb72、bb90、bb108、bb288 是它们在 QEC-Bank 里的单独条目（T1 级别的「object」卡：构造器存着精确的多项式约定，重算 n、k，d 照抄论文，其他侧面尚未调研）。

The table below sends each code back to its QEC-Bank home. Table 3 is the table of BB codes in the bravyi2024bicycle paper (arXiv 2308.07915), seven in all: [[72,12,6]], [[90,8,10]], [[108,8,10]], [[144,12,12]], [[288,12,18]], and two larger examples known only up to a distance upper bound, [[360,12,≤24]] and [[756,16,≤34]] (“≤” means only an upper bound was known when the paper was written). This guide and the BB guide cover only the first five, and so does the table below; bb72, bb90, bb108, bb288 are their separate QEC-Bank entries (T1 “object” cards: the constructor stores the exact polynomial convention, recomputes n and k, copies d from the paper, and the other facets are not yet surveyed).

三个词的分布不是随机的。能穷举的小码（bicycle 6 比特、toric_L3 18 比特、bb5 30 比特）d 全是 verified；Table 3 里本篇讲到的另外四个码 d 全是 claimed——QEC-Bank 重算了它们的 n、k，但 d = 6、10、10、18 是照抄论文，连上界见证也没有登记；只有 gross 是 bounded，因为它是 bivariate_bicycle 这张 T3 级别条目的主实例（T0、T1、T3 是 QEC-Bank 的条目等级；这里只需知道：上面说的 T1 卡只有构造器与 n、k，而 bivariate_bicycle 这张卡的各个侧面都已调研），条目为它跑了种子搜索。人话：同一张 Table 3 里，「d = 12」比「d = 6」在 QEC-Bank 里多了半份机器证书，不是因为 12 更可信，而是因为 QEC-Bank 对 gross 多做了工作。

The distribution of the three words is not random. The small codes that can be exhausted (bicycle at 6 qubits, toric_L3 at 18, bb5 at 30) all have d verified; the other four Table 3 codes covered here all have d claimed: the QEC-Bank recomputed their n and k, but d = 6, 10, 10, 18 are copied from the paper, with not even an upper-bound witness recorded; only gross is bounded, because it is the primary instance of the T3-level bivariate_bicycle entry (T0, T1, T3 are QEC-Bank entry grades; all that matters here is that the T1 cards above hold only a constructor with n and k, while every facet of the bivariate_bicycle card has been surveyed), which ran the seeded search for it. Plainly: within the same Table 3, “d = 12” carries half a machine certificate more than “d = 6” in the QEC-Bank, not because 12 is more believable but because the QEC-Bank did more work on gross.

表里出现了两次 [[144, 12, ·]]。一张是 bivariate_bicycle 条目的实例 gross——本篇的主角，d = 12 bounded。另一张是 QEC-Bank 里单独的 gross 条目，主体是 eczoo 的集合 eczoo-entity:gross#collection，级别 T1，状态「admitted」；它的构造器同样重算了 n = 144、k = 12（都 verified），但参数表的 d 一栏是空的，页面直接显示 d=?，且译码、操作、关系、实现四个侧面「remain unsurveyed」。两张卡描述的是同一个 144 比特的矩阵，但 QEC-Bank 的原则是「Source, page, and formal object are separate identities」——来源、页面与形式对象是三个不同的身份，eczoo 里有一个叫 gross code 的条目，QEC-Bank 就为它保留一张卡。读到「gross code」时先看是哪张卡：本篇引用的所有数字与证据词都来自 bivariate_bicycle 那张。

[[144, 12, ·]] appears twice in the table. One is the instance gross of the bivariate_bicycle entry, the subject of this guide, d = 12 bounded. The other is the QEC-Bank's separate gross entry, whose subject is the eczoo collection eczoo-entity:gross#collection, tier T1, status “admitted”; its constructor likewise recomputes n = 144 and k = 12 (both verified), but the d column of its parameters table is empty, the page shows d=?, and its decoder, operation, relation and realization facets “remain unsurveyed”. Both cards describe the same 144-qubit matrix, but the QEC-Bank principle is “Source, page, and formal object are separate identities”: eczoo has an entry called gross code, so the QEC-Bank keeps a card for it. When you read “gross code”, check which card it is; every number and evidence word quoted in this guide comes from the bivariate_bicycle card.

账本的算法很简单：gross 每个逻辑比特平均占 n/k = 144/12 = 12 个数据比特，码距 12。文献里的旋转表面码要达到同样的码距 12，一块贴片用 d² = 144 个数据比特存 1 个逻辑比特（旋转贴片 [[d², 1, d]] 的数字见表面码「参数与实例」导读 4.2 节，那是本侧面的表面码版本）。144 ÷ 12 = 12，这就是 bravyi2024bicycle 强调的「twelve-fold finite-block comparison」：码距相同时每个逻辑比特占的数据比特数之比。同一篇论文的标题数字是另一种比法，来自电路级噪声下的数值模拟（第 3、8 节）：每个操作（CNOT、制备、测量、空等）以概率 p = 0.1% 独立出错的标准去极化电路噪声；gross 一侧用 MIN-SUM 置信传播加 combination-sweep 版本的 OSD 译码，288 个物理比特（含帮手），估计每轮逻辑错误率约 2 × 10⁻⁷，论文由此估计能把 12 个逻辑比特保存近一百万轮症状测量；对照是 12 块独立的旋转表面码贴片，要把每轮错误率从 10⁻³ 压到 10⁻⁶ 需要近 3000 个物理比特，论文说节省超过 10 倍。两个数字比的东西不同：一个固定码距、只数数据比特；另一个在同一噪声模型下、逻辑错误率大致相当时比物理比特总数（约 2 × 10⁻⁷ 对 10⁻⁶，并不相等，所以「10 倍」是近似的），而且依赖电路、噪声模型与译码器。BB 教程的1.2 节已经画过这张图，这里再放一次，把鼠标停在点上看 k·d²/n：表面码约为 1，gross 是 12。

The ledger's arithmetic is simple: gross spends n/k = 144/12 = 12 data qubits per logical qubit at distance 12. To reach the same distance 12, the rotated surface code of the literature uses one patch of d² = 144 data qubits for 1 logical qubit (the rotated-patch numbers [[d², 1, d]] are in section 4.2 of the surface-code “Parameters & instances” guide, the surface-code version of this facet). 144 ÷ 12 = 12: this is the “twelve-fold finite-block comparison” that bravyi2024bicycle emphasises, the ratio of data qubits per logical qubit at equal distance. The headline number of the same paper compares something else and comes from numerical simulation under circuit-level noise (sections 3 and 8): the standard depolarizing circuit noise in which every operation (CNOT, preparation, measurement, idling) fails independently with probability p = 0.1%; on the gross side, MIN-SUM belief propagation with the combination-sweep version of OSD as decoder, 288 physical qubits (helpers included), an estimated logical error rate of about 2 × 10⁻⁷ per cycle, from which the paper estimates that 12 logical qubits survive nearly one million syndrome cycles; the comparator is 12 separate rotated surface-code patches, which need nearly 3000 physical qubits to push the per-cycle error rate from 10⁻³ down to 10⁻⁶, and the paper claims a saving of more than 10 times. The two numbers compare different things: one at equal distance counting data qubits only; the other counts all physical qubits at roughly comparable logical error rates under one noise model (about 2 × 10⁻⁷ versus 10⁻⁶, not equal, so “10 times” is approximate), and it depends on the circuit, the noise model and the decoder. Section 1.2 of the BB guide drew this picture already; here it is again. Hover a point for k·d²/n: about 1 for surface codes, 12 for gross.

[[n,k,d]] 的 n 只数数据比特。真实芯片上每条检查要一个帮手比特辅助比特（ancilla）：专职测一条检查的量子比特，依次和该检查的数据比特做 CNOT 再被读出。表面码教程里打开过这个黑盒。去测它。gross 有 72 条 X 检查加 72 条 Z 检查共 144 条，按每条一个帮手，就是 144 个帮手比特；数据 144 加帮手 144，共 288 个物理比特存 12 个逻辑比特。BB 教程 5.6 节给过这个 288。对照组：旋转表面码 d = 12 是 144 个数据比特加 143 个帮手，共 287 个物理比特存 1 个（表面码参数导读的同一节，来自表面码条目的 reproduce.py 断言）。288 对 287，几乎相同的物理比特数，12 个逻辑比特对 1 个——加上帮手后，倍数仍然接近十二。

The n in [[n,k,d]] counts data qubits only. On a real chip every check needs a helper qubitAn ancilla: a qubit dedicated to measuring one check, which performs CNOTs with that check's data qubits in turn and is then read out. The surface-code guide opened this box. to measure it. Gross has 72 X checks plus 72 Z checks, 144 in all, so one helper each gives 144 helper qubits; 144 data plus 144 helpers is 288 physical qubits storing 12 logical qubits. BB guide section 5.6 gave this 288. The comparison: the rotated surface code at d = 12 is 144 data qubits plus 143 helpers, 287 physical qubits storing 1 (the same section of the surface-code parameters guide, from the surface entry's reproduce.py assertions). 288 against 287, nearly identical physical-qubit counts, 12 logical qubits against 1: with helpers included the factor stays close to twelve.

但账本立刻划出边界：「It is not a full-machine ratio.」一个 BB 周期还需要「syndrome ancillas, degree-six interactions, a seven-CNOT-layer schedule in the original proposal, measurement/reset, classical detector processing, and BP-OSD」——六腿的相互作用要两层耦合的芯片、原论文深度 7 的 CNOT 调度、测量与复位、经典探测器处理和 BP-OSD 译码器；架构还可能加上「movement, communication, spare blocks, or surface-code ancillas for logical operations」。结论是一条方法论：「Comparisons must put the same resources in both denominators」——比较时两边的分母必须数同样的东西。

But the ledger draws the line at once: “It is not a full-machine ratio.” A BB cycle also needs “syndrome ancillas, degree-six interactions, a seven-CNOT-layer schedule in the original proposal, measurement/reset, classical detector processing, and BP-OSD”: the six-legged interactions need a two-layer coupling chip, the original paper's depth-7 CNOT schedule, measurement and reset, classical detector processing and the BP-OSD decoder; architectures may further add “movement, communication, spare blocks, or surface-code ancillas for logical operations”. The conclusion is a rule of method: “Comparisons must put the same resources in both denominators”.

账本还有一节「Static versus circuit distance」，说的是：12 属于静态稳定子码。一旦用电路去测症状，就定义了一个时空故障码spacetime fault code：把「电路里每个位置可能出的每种错」当作比特、「探测器」当作检查得到的更大的码；电路码距是能在其中造成一个不可探测的非平凡逻辑错误的最少故障数，可以小于静态码距。，其「minimum undetectable fault set can be smaller」。这里要读准：电路码距数的是最少几个电路故障能既骗过所有探测器、又真的改变逻辑信息（造成一个非平凡的逻辑错误），这个数可能少于 12。只骗过探测器还不够：有些故障组合彼此抵消，或者合起来只等于一个稳定子，逃过了探测也无害，不算。Bravyi 等（arXiv 2308.07915 第 6 节）的定义是：Z 型电路码距是「the minimum number of faulty operations in the SM circuit that can generate an undetectable Z-type logical error」，X 型同理，电路码距取两者中较小的；他们对 gross 的深度 7 电路只算出上界 d_circ ≤ 10（表 1）。原因在 BB 教程 6.2 节讲过：帮手比特中途出错会沿后续 CNOT 传播到多个数据比特，CNOT 的顺序决定它传到哪里。于是账本列出四件不能从静态账本推出的事：「a distance-12 fault circuit; a universal threshold; a decoder-independent logical error rate; a logical-gate error rate」。这四件事分别归译码器与逻辑操作两个侧面管。人话：本篇的 12 是矩阵的性质；芯片上能不能拿到 12，要看电路。

The ledger has a further section, “Static versus circuit distance”: the 12 belongs to the static stabilizer code. As soon as a circuit measures the syndrome, it defines a spacetime fault codeThe larger code obtained by treating “each possible fault at each circuit location” as a bit and “detectors” as checks; its circuit distance, the fewest faults that can cause an undetectable non-trivial logical error, can be smaller than the static distance. whose “minimum undetectable fault set can be smaller”. Read it carefully: the circuit distance counts the fewest circuit faults that both fool every detector and actually change the logical information (cause a non-trivial logical error), and that number may be below 12. Fooling the detectors is not enough: some fault combinations cancel each other, or add up to a mere stabilizer; they escape detection and do no harm, and they do not count. The definition of Bravyi et al. (arXiv 2308.07915, section 6): the Z-type circuit distance is “the minimum number of faulty operations in the SM circuit that can generate an undetectable Z-type logical error”, likewise for X, and the circuit distance is the smaller of the two; for the depth-7 circuit of gross they obtain only the upper bound d_circ ≤ 10 (Table 1). The reason was given in BB guide section 6.2: a helper qubit that fails midway spreads through later CNOTs onto several data qubits, and the CNOT order decides where. Hence the ledger lists four things that do not follow from the static ledger: “a distance-12 fault circuit; a universal threshold; a decoder-independent logical error rate; a logical-gate error rate”. They belong to the Decoders and Logical operations facets. Plainly: this guide's 12 is a property of a matrix; whether a chip gets 12 depends on the circuit.

gross 的 k·d²/n = 12 远高于表面码的约 1，很容易让人以为「把环面放大，这个数会一直涨」。QEC-Bank 的 intuition.md 把这列为失败模式之一：「Finite constants do not imply asymptotic goodness. The gross point is a strong finite block. It is not evidence that fixed sparse polynomials evade two-dimensional tradeoffs as the torus grows.」人话：gross 是一个很强的有限块；固定两个多项式、只放大环面，并不能逃出 BPT 界那种二维权衡。BPT 界（Bravyi、Poulin、Terhal，arXiv 0909.5200）的前提有两条：比特排在二维格子上，单位面积里的比特数有上限；每条检查只碰相距不超过某个固定距离的比特。满足这两条的码必须有 k·d² ≤ c·n，常数 c 只取决于检查的作用范围和每个格点的维数，与码有多大无关。固定两个多项式、只放大环面时，每条检查的腿仍是那几个固定的平移，作用范围不变，两条前提都成立，界直接管用，k·d²/n 不能无限上涨。若只固定项数、让指数随环面一起变大，检查可以伸得越来越远，第二条前提不再成立，BPT 不再直接适用；那种情形的限制来自下面这条文献声明。card.yaml 的表示阶梯里有一条 R5 层的「quotient-module」断言，证据词 curated、出处 postema2025existence：商空间 R/(A,B) 控制 k，多项式的因子与公共根解释了「why finite blocks can have high rate, and why the studied subclass is not an asymptotically good family」——为什么有限块可以有高码率，以及为什么所研究的子类不是渐近好的一族。BB 教程 6.6 节用一句话转述过：固定项数、无限放大环面时，码率与相对距离不能同时保持非零。

Gross's k·d²/n = 12 is far above the surface code's roughly 1, and it is tempting to think “enlarge the torus and this number keeps rising”. The QEC-Bank file intuition.md lists this among the failure modes: “Finite constants do not imply asymptotic goodness. The gross point is a strong finite block. It is not evidence that fixed sparse polynomials evade two-dimensional tradeoffs as the torus grows.” Plainly: gross is a strong finite block; fixing the two polynomials and only enlarging the torus does not escape the two-dimensional trade-off of the BPT bound. The BPT bound (Bravyi, Poulin, Terhal, arXiv 0909.5200) has two premises: the qubits sit on a two-dimensional lattice with a bounded number per unit area, and every check touches only qubits within some fixed distance of each other. Codes meeting both must satisfy k·d² ≤ c·n, where the constant c depends only on the check range and the dimension per lattice site, not on the size of the code. With the two polynomials fixed and only the torus growing, the legs of every check are still the same few translations, the range does not change, both premises hold, and the bound applies directly: k·d²/n cannot rise without limit. If only the number of terms is fixed and the exponents grow with the torus, checks can reach further and further, the second premise fails and BPT no longer applies directly; the limit in that case comes from the literature claim below. In card.yaml's representation ladder there is an R5 “quotient-module” assertion, evidence word curated, source postema2025existence: the quotient R/(A,B) controls k, and polynomial factors and common roots explain “why finite blocks can have high rate, and why the studied subclass is not an asymptotically good family”. BB guide section 6.6 paraphrased it in one sentence: with a fixed number of terms and an ever-larger torus, rate and relative distance cannot both stay non-zero.

证据词要读准：这条是 curated（QEC-Bank 整理的文献结论），在 derivations.md 的 provenance split 里归入「Literature claims」的「asymptotic limitation」一项，QEC-Bank 的机器没有、也不可能对「渐近」做任何重算。而且 postema2025existence 在 QEC-Bank 的引文登记里目前是「metadata pending」——文献扫描报告给出的题目是 Postema 与 Kokkelmans 的 “Existence and Characterisation of Bivariate Bicycle Codes”（2025，arXiv 2502.17052），登记待补。真正渐近好的码要换掉阿贝尔群 Z_l × Z_m，那是qLDPC 教程 4.4 节的主题。

Read the evidence word precisely: this assertion is curated (a literature conclusion organised by the QEC-Bank), filed under “Literature claims” as the “asymptotic limitation” item in the provenance split of derivations.md; the QEC-Bank machine has not, and could not, recompute anything about “asymptotics”. Moreover, postema2025existence is currently “metadata pending” in the QEC-Bank citation registry; the scout report gives the title as Postema and Kokkelmans, “Existence and Characterisation of Bivariate Bicycle Codes” (2025, arXiv 2502.17052), registration pending. Truly asymptotically good codes replace the abelian group Z_l × Z_m, the subject of section 4.4 of the qLDPC guide.

回放一遍：BB 条目的家族变量是 l、m、a_terms、b_terms；构造器按正向移位、先行后列、左层在前、交换转置的固定约定拼出 H_X = [A | B]、H_Z = [Bᵀ | Aᵀ]。n = 2lm 与检查重量 |A| + |B| 是数数，k = 2(lm − r) 是一次 GF(2) 高斯消元，d 要下界证明加上界见证。QEC-Bank 发布 gross 与 toric_L3 两个实例：n、k、css 全部 verified；toric_L3 的 d = 3 由穷举 verified；gross 的 d = 12 由 bravyi2024bicycle 的下界引文与 rand-lowweight-v1（seed 5，iters 2000）的上界见证合成 bounded。本篇讲到的 Table 3 另外四个码住在 bb72、bb90、bb108、bb288 各自的条目里（Table 3 还有 [[360,12,≤24]]、[[756,16,≤34]] 两个，本篇未涉及），d 均为 claimed；bicycle 与 bb5 由穷举 verified；另有一张 T1 的 gross 卡，d 未登记。「十二倍」只比数据比特，加帮手是 288 对 287，静态 12 不是电路 12，渐近坏是一条 curated 的文献声明。

Replay: the BB entry's family variables are l, m, a_terms, b_terms; the constructor assembles H_X = [A | B], H_Z = [Bᵀ | Aᵀ] under the fixed conventions of forward shift, rows before columns, left layer first, swapped transposes. n = 2lm and the check weight |A| + |B| are counting, k = 2(lm − r) is one Gaussian elimination over GF(2), d needs a lower-bound proof plus an upper-bound witness. The QEC-Bank ships two instances, gross and toric_L3: n, k, css verified throughout; toric_L3's d = 3 verified by enumeration; gross's d = 12 bounded, from the cited lower bound bravyi2024bicycle and the upper witness of rand-lowweight-v1 (seed 5, iters 2000). The other four Table 3 codes covered here live in their own entries bb72, bb90, bb108, bb288 (Table 3 also has [[360,12,≤24]] and [[756,16,≤34]], not covered here) with d claimed; bicycle and bb5 are verified by enumeration; a separate T1 gross card records no d. “Twelve-fold” compares data qubits only, becomes 288 against 287 with helpers, the static 12 is not the circuit 12, and asymptotic badness is a curated literature claim.