七张推导页，
把 gross 码算到底 Seven derivation pages,
the gross code worked to the bottom

QEC-Bank 索引页开头一句话说明了这七页为什么存在：「低开销 BB 存储器」这个标签把三种性质压在了一起——代数定义、有限块的算术、电路与架构的主张。七页的任务就是把它们拆开。你在《BB 码 · 高中数学版》里已经看见了这七件事，这里要把它们证出来或标清楚：

The QEC-Bank index page opens by saying why these seven pages exist: the label “low-overhead BB memory” compresses three kinds of statement into one, the algebraic definition, the finite-block arithmetic, and the circuit and architecture claims. The seven pages pull them apart. You have already seen these seven things in “BB Codes · High-School Edition”; here they get proved or labelled:

QEC-Bank 对这些页面立了一条规矩：凡是「算出来」的陈述，都必须能在仓库里重跑。脚本 codes/bivariate_bicycle/pedagogy/reproduce.py 先加载随 QEC-Bank 一起发布的构造器 constructor.py，然后自己再造一遍循环移位、x、y、A、B，逐条 assert（断言，即「不等就报错退出」）本教程用到的每个计算结果：gross 的两个多项式、两个周期 x¹² = I 与 y⁶ = I、xy = yx、矩阵形状 72×144、H_XH_Zᵀ = 0、两个秩都是 66、商维数 6、每行每列的重量、第 0 条 X 检查与第 0 条 Z 检查的精确支撑、平移自同构的矩阵恒等式、导出格点与检查的一一对应、种子 5 的重量 12 上界见证，以及 toric_L3 的 [[18,2,3]] 精确码距。任何一条不符，脚本以非零状态退出。普通验证器 make verify-entry 另外独立重算 instances.yaml 里两个实例的全部 claims。

The QEC-Bank imposes one rule on these pages: every statement that was “computed” must rerun inside the repository. The script codes/bivariate_bicycle/pedagogy/reproduce.py loads the constructor shipped with the QEC-Bank, constructor.py, then rebuilds on its own the cyclic shifts, x, y, A and B, and asserts (i.e. “exit with an error if unequal”) every computed result this guide uses: the two gross polynomials, the two periods x¹² = I and y⁶ = I, xy = yx, the 72×144 shapes, H_XH_Zᵀ = 0, both ranks 66, quotient dimension 6, every row and column weight, the exact supports of X check 0 and Z check 0, the matrix identities of the translation automorphisms, the one-to-one match between exported lattice and checks, the seed-5 weight-12 upper witness, and the exact [[18,2,3]] distance of toric_L3. On the first disagreement it exits nonzero. The ordinary verifier, make verify-entry, independently recomputes every claim of both instances in instances.yaml.

契约还有一条更严的细节。条目源文件 card.yaml 的 construction: 块把两个多项式因子钉为 bivariate_polynomial@1、把 CSS 拼装钉为 bivariate_bicycle@1（@1 是版本号）；实例参数只绑定四个东西：两个周期 l、m 和两张指数表 a_terms、b_terms。重放时必须重建出逐元素完全相同的矩阵——行的顺序、比特的顺序、左右两块的顺序都不能差——验证器比较的是整张数组，而不是「秩相等」或「行空间相同」这种弱一些的等价。人话：QEC-Bank 不满足于「你造的码和我的一样好」，它要求「你造的矩阵和我的一个 0/1 都不差」。

The contract has one stricter detail. The construction: block of the entry source card.yaml pins both polynomial factors to bivariate_polynomial@1 and the CSS assembly to bivariate_bicycle@1 (@1 is a version number); the instance parameters bind only four things, the two periods l, m and the two exponent tables a_terms, b_terms. Replay must rebuild a matrix that is identical entry by entry, same row order, same qubit order, same left/right block order; the verifier compares the whole array, not a weaker equivalence such as “equal rank” or “same row space”. In plain words: the QEC-Bank is not satisfied with “your code is as good as mine”; it demands “your matrix differs from mine in not a single 0 or 1”.

出处的划分是刻意的，本教程每一章都会沿用这套标签：

The provenance split is deliberate, and every chapter below reuses these labels:

令 e_i,j 是环面格点 (i, j) 上的标准基向量，i 取 0 到 l−1，j 取 0 到 m−1（数学写法：(i, j) ∈ ℤ_l × ℤ_m，即两个坐标各自绕圈）。构造器的循环移位 S_l 在第 i 行、第 i+1 列（下标模 l）放一个 1——这正是《BB 码 · 高中数学版》2.1 节的那个矩阵，方向约定「第 i 格送到第 i+1 格」。把格点拉平成一维下标 im + j（先数 i、再数 j），定义

Let e_i,j be the standard basis vector at torus site (i, j), with i from 0 to l−1 and j from 0 to m−1 (in symbols (i, j) ∈ ℤ_l × ℤ_m: each coordinate wraps around). The constructor's cyclic shift S_l puts a 1 in row i, column i+1 (indices mod l), exactly the matrix of §2.1 of “BB Codes · High-School Edition”, with the orientation “cell i goes to cell i+1”. Flatten the sites into the one-dimensional index im + j (count i first, then j) and define

人话：x 让格点 (i, j) 变成 (i+1, j)，y 让它变成 (i, j+1)。第一条恒等式 x^l = I 是「往 x 方向走 l 步回到原地」；第二条同理；第三条 xy = yx 是「先右后上和先上后右到的是同一个格子」——它成立的深层原因是 x 和 y 作用在克罗内克积的不同因子上。reproduce.py 对 gross 逐条断言了 x¹² = I、y⁶ = I 与 xy = yx。这些矩阵合起来给出了环 R 的正则表示regular representation：把环里每个元素 r 变成「乘以 r」这个线性映射的矩阵。加法变成矩阵加法，乘法变成矩阵乘法，所以多项式里的等式与矩阵里的等式一一对应。：

In plain words: x sends site (i, j) to (i+1, j), y sends it to (i, j+1). The first identity x^l = I says “l steps in the x direction bring you home”; the second likewise; the third, xy = yx, says “right-then-up and up-then-right reach the same cell”, and the deep reason it holds is that x and y act on different factors of the Kronecker product. reproduce.py asserts x¹² = I, y⁶ = I and xy = yx for gross one by one. Together these matrices give the regular representationThe regular representation turns every ring element r into the matrix of the linear map “multiply by r”. Addition becomes matrix addition and multiplication matrix multiplication, so identities among polynomials and identities among matrices correspond one to one. of the ring R:

每个单项式 x^ay^b 是一个置换矩阵（每行每列恰一个 1）。置换矩阵的转置等于它的逆，所以

Every monomial x^ay^b is a permutation matrix (exactly one 1 per row and per column). A permutation matrix's transpose is its inverse, hence

这就是转置在这里有几何意义的原因：它把一次环面平移倒过来。为什么会这样？置换矩阵 P 在第 i 行第 π(i) 列放 1，「把 i 送到 π(i)」；转置后 1 跑到第 π(i) 行第 i 列，「把 π(i) 送回 i」。人话：往右挪三格的矩阵，转置就是往左挪三格。这一条在 1.4 节要用两次。

That is why the transpose has a geometric meaning here: it reverses a torus translation. Why? A permutation matrix P has its 1 in row i, column π(i), “send i to π(i)”; after transposing, the 1 sits in row π(i), column i, “send π(i) back to i”. In plain words: the matrix that moves everything three cells right has, as its transpose, the matrix that moves everything three cells left. §1.4 will use this twice.

一个二元多项式 A = Σ_{(a,b) ∈ TA} x^ay^b 的矩阵，每一行恰好从每个不同的单项式那里得到一个 1。如果指数表 T_A 里的（约化后的）指数对两两不同，那么每行每列的重量都是 |T_A|——项数。重复的项在 GF(2) 上成对抵消（1 + 1 = 0），所以构造器先把指数对按 l、m 约化、再做 XOR。gross 的输入是

The matrix of a bivariate polynomial A = Σ_{(a,b) ∈ TA} x^ay^b receives, in each row, exactly one 1 from each distinct monomial. If the (reduced) exponent pairs in the table T_A are pairwise distinct, every row and every column has weight |T_A|, the number of terms. Repeated terms cancel in pairs over GF(2) (1 + 1 = 0), so the constructor reduces the exponent pairs mod l, m first and then XORs. The gross inputs are

动手算一条。第 0 条 X 检查坐在格点 (0, 0)（下标 0·6 + 0 = 0）。A 的三项把它连到左层的 (3, 0)、(0, 1)、(0, 2)，拉平后是 18、1、2；B 的三项连到右层的 (0, 3)、(1, 0)、(2, 0)，拉平后 3、6、12，再加右层偏移 72，得 75、78、84。所以第 0 条 X 检查的支撑是 {1, 2, 18, 75, 78, 84}——reproduce.py 第 113 行断言的正是这六个数。第 0 条 Z 检查用反向平移：Bᵀ 的三项把 (0,0) 连到左层的 (0, −3) = (0, 3)、(−1, 0) = (11, 0)、(−2, 0) = (10, 0)，即 3、66、60；Aᵀ 的三项连到右层的 (−3, 0) = (9, 0)、(0, −1) = (0, 5)、(0, −2) = (0, 4)，即 54 + 72 = 126、5 + 72 = 77、4 + 72 = 76。支撑 {3, 60, 66, 76, 77, 126}，与脚本第 114 行一致。注意：没有人另外画一个「方块」——检查图就是这些平移的关联图。到实验台上点一个起点，看三个箭头落在哪里，再看矩阵里这一行的三个 1：

Work one out by hand. X check 0 sits at site (0, 0) (index 0·6 + 0 = 0). The three terms of A join it to left-layer sites (3, 0), (0, 1), (0, 2), flattened 18, 1, 2; the three terms of B join it to right-layer sites (0, 3), (1, 0), (2, 0), flattened 3, 6, 12, plus the right-layer offset 72, giving 75, 78, 84. So the support of X check 0 is {1, 2, 18, 75, 78, 84}, exactly the six numbers asserted on line 113 of reproduce.py. Z check 0 uses the inverse translations: the three terms of Bᵀ join (0,0) to left-layer (0, −3) = (0, 3), (−1, 0) = (11, 0), (−2, 0) = (10, 0), i.e. 3, 66, 60; the three terms of Aᵀ join it to right-layer (−3, 0) = (9, 0), (0, −1) = (0, 5), (0, −2) = (0, 4), i.e. 54 + 72 = 126, 5 + 72 = 77, 4 + 72 = 76. Support {3, 60, 66, 76, 77, 126}, matching line 114 of the script. Note that nobody drew a separate “plaquette”: the check graph is the incidence graph of these translations. Pick a starting cell on the bench, watch where the three arrows land, then find the three 1s of that row in the matrix:

在环面每个格点放两个数据比特：前 lm 列是左块，后 lm 列是右块。定义 H_X = [A | B]。CSS 码要求 H_XH_Zᵀ = 0（碰撞定理的矩阵形式：每对 X、Z 检查重叠偶数个比特）。选 H_Z = [Bᵀ | Aᵀ]，分块相乘：

Place two data qubits at every torus site: the first lm columns are the left block, the second lm the right block. Define H_X = [A | B]. A CSS code needs H_XH_Zᵀ = 0 (the matrix form of the collision theorem: every X check overlaps every Z check on an even number of qubits). Choose H_Z = [Bᵀ | Aᵀ] and multiply blockwise:

AB = BA 来自 1.1 节的 xy = yx：A、B 都是 x、y 的多项式，而两个变量对易，所以它们的任何两个多项式对易——这是「把检查算在阿贝尔（可交换）环上」的全部回报。1 + 1 = 0 是 GF(2)。于是任何 l、m、A、B 都给出一个合法的 CSS 码。

AB = BA comes from xy = yx in §1.1: A and B are polynomials in x, y, and since the two variables commute, so do any two polynomials in them; this is the whole payoff of computing checks over an abelian (commutative) ring. 1 + 1 = 0 is GF(2). Hence any l, m, A, B give a legal CSS code.

转置和交换各有任务。转置把 Z 一侧的关联倒过来，让 Z 检查成为 X 检查的对偶映射——X 检查沿某方向伸出去的腿，Z 检查沿反方向收回来（1.2 节）。交换把重叠项路由成 AB + BA。若只转置不交换，取候选 H_Z' = [Aᵀ | Bᵀ]，乘积变成 A² + B²，没有任何理由为零。对 gross 可以具体算：A² = x⁶ + y² + y⁴，B² = y⁶ + x² + x⁴ = 1 + x² + x⁴（因为 y⁶ = 1），两者相加是一个六项多项式，矩阵每行 6 个 1，秩 48——离零远得很。人话：不交换的话，gross 的每条 X 检查会与 6 条 Z 检查各撞奇数次，码根本不合法。所以「交换 + 转置」不是审美，是让定理成立的唯一便宜路线。

Transpose and swap each have a job. The transpose reverses the Z-side incidence so that the Z check is the dual map of the X check: where an X check reaches out along some direction, the Z check reaches back along the opposite one (§1.2). The swap routes the overlap into AB + BA. Transpose without swapping, with the candidate H_Z' = [Aᵀ | Bᵀ], gives A² + B², which has no reason to vanish. For gross one can compute it: A² = x⁶ + y² + y⁴, B² = y⁶ + x² + x⁴ = 1 + x² + x⁴ (since y⁶ = 1), and their sum is a six-term polynomial whose matrix has 6 ones per row and rank 48, nowhere near zero. In plain words: without the swap, every gross X check would collide an odd number of times with 6 different Z checks, and the code would simply be illegal. So “swap + transpose” is not taste; it is the only cheap route that makes the theorem true.

刚才的计算还有一个名字：「边界的边界为零」。把 ∂₂ ≔ [B ; A]（把一个多项式 r 送到一对 (Br, Ar)）和 ∂₁ ≔ [A | B]（把一对 (u, v) 送到 Au + Bv）串起来：

The calculation just done has another name: “the boundary of a boundary is zero”. Chain ∂₂ ≔ [B ; A] (sending one polynomial r to the pair (Br, Ar)) and ∂₁ ≔ [A | B] (sending a pair (u, v) to Au + Bv):

在二元 CSS 约定里，∂₁ = H_X、∂₂ = H_Zᵀ（即 ∂₂ᵀ = H_Z）。中间那一项 R² 里的一个向量 v 是全部 2lm 个数据比特上的一个 0/1 串；把它读成一个 Z 型 Pauli（v 为 1 的比特上放 Z）。∂₁v = H_Xv = 0 说的是这个 Z 串与每条 X 检查都重叠偶数次，也就是与所有 X 检查对易；这样的 v 叫一个圈cycle：被边界算符送到零的元素。这里指 ∂₁v = H_Xv = 0 的 Z 串 v，即「所有 X 检查都读 +1」。。乘上一个 Z 稳定子相当于加一个边界boundary：某个元素在边界算符下的像。这里指 ∂₂u = H_Zᵀu，即若干条 Z 检查的乘积。。所以非平凡的 Z 逻辑类是「圈模边界」的商 ker H_X / im H_Zᵀ。X 逻辑类住在对偶复形里：把两个箭头都转置、方向倒过来，得到 ker H_Z / im H_Xᵀ（与所有 Z 检查对易的 X 串，模掉 X 检查的乘积）；两个商的维数相同，都是 k。用 1.6 节的 toric_L3 数一遍：rank H_X = 8，所以 ker H_X 有 18 − 8 = 10 维；im H_Zᵀ 的维数是 rank H_Z = 8；商是 10 − 8 = 2 维，正是 k = 2。这个「圈模边界」的商在数学上叫同调（homology；这里是复形中间那一项的同调，所以也叫中间同调）。这正是同调语言赚回本钱的地方：「与检查对易」和「是检查的乘积」是两个不同的子空间，逻辑信息就是两者之间的差。《Surface Code · 高中数学版》用「横跨不横跨」说这件事，这里用「商」说，一个想法两种口音。一般的二元词典在 QEC-Bank 基础页 CSS chain complexes。

In the binary CSS convention, ∂₁ = H_X and ∂₂ = H_Zᵀ (that is, ∂₂ᵀ = H_Z). A vector v in the middle term R² is a 0/1 string on all 2lm data qubits; read it as a Z-type Pauli (a Z on every qubit where v is 1). ∂₁v = H_Xv = 0 says that this Z string overlaps every X check an even number of times, i.e. commutes with all X checks; such a v is a cycleAn element sent to zero by the boundary operator: here a Z string v with ∂₁v = H_Xv = 0, i.e. “every X check reads +1”.. Multiplying by a Z stabilizer adds a boundaryThe image of something under the boundary operator: here ∂₂u = H_Zᵀu, i.e. a product of Z checks.. So the nontrivial Z-logical classes are the quotient “cycles modulo boundaries”, ker H_X / im H_Zᵀ. The X-logical classes live in the dual complex: transpose both arrows and reverse their direction to get ker H_Z / im H_Xᵀ (X strings commuting with every Z check, modulo products of X checks); the two quotients have the same dimension, k. Count it on toric_L3 of §1.6: rank H_X = 8, so ker H_X has 18 − 8 = 10 dimensions; im H_Zᵀ has dimension rank H_Z = 8; the quotient has 10 − 8 = 2 dimensions, exactly k = 2. This quotient “cycles modulo boundaries” is called homology in mathematics (here the homology of the middle term of the complex, hence “middle homology”). This is exactly where the homological language earns its keep: “commutes with the checks” and “is a product of checks” are two different subspaces, and logical information is the gap between them. “The Surface Code · High-School Edition” says it with “spanning or not spanning”; here it is said with “quotient”, one idea in two accents. The general binary dictionary is on the QEC-Bank foundations page CSS chain complexes.

令 l = m = L，取 A = 1 + x、B = 1 + y。每个多项式两项，所以每条检查四腿。得到的复形正是周期方格上的环面码，只是写成 BB 的「按列分块」约定：每个单元格两个比特（左块一个、右块一个）、一条 X 检查、一条 Z 检查。对已发布的 L = 3 实例，构造器给出两张 9×18 矩阵（竖线左边 9 列是左块 q0…q8，右边 9 列是右块 q9…q17）：

Set l = m = L and choose A = 1 + x, B = 1 + y. Each polynomial has two terms, so each check has weight four. The resulting complex is the toric code on the periodic square lattice, written in the BB “column-block” convention: two qubits per unit cell (one in each block), one X check and one Z check per cell. For the shipped L = 3 instance the constructor gives two 9×18 matrices (the 9 columns left of the bar are the left block q0…q8, the 9 on the right the right block q9…q17):

核对一对：X0 = {q0, q3, q9, q10} 与 Z1 = {q0, q1, q10, q16} 重叠 {q0, q10}，两格，偶数，对易；X0 与 Z3 = {q3, q5, q9, q12} 重叠 {q3, q9}，也是两格。两张矩阵的秩都是 8（九行里有一条依赖：全部 X 检查之积是 I，全部 Z 检查之积也是 I——环面码教程的两条恒等式），所以

Check one pair: X0 = {q0, q3, q9, q10} and Z1 = {q0, q1, q10, q16} overlap in {q0, q10}, two cells, even, so they commute; X0 and Z3 = {q3, q5, q9, q12} overlap in {q3, q9}, again two. Both matrices have rank 8 (one dependency among nine rows: the product of all X checks is I, and so is the product of all Z checks, the toric guide's two identities), hence

这个 [[18,2,3]] 锚点检验的是移位方向、交换转置、秩与精确码距这一整条路径：把转置忘掉，检查就不再两两对易；把方向弄反，第 0 条检查的支撑会对不上。在实验台上把预设切到 toric_L3，点一条检查看它的四条腿，再按「搜索低重逻辑算符」穷举出 3：

This [[18,2,3]] anchor tests the whole path, shift orientation, swapped transpose, ranks and the exact-distance routine: forget the transpose and the checks stop commuting; reverse the orientation and check 0's support no longer matches. Switch the bench to the toric_L3 preset, click a check to see its four legs, then press the low-weight search to enumerate 3:

人话：72 个格点，每个格点两个数据比特，共 144；每个格点一条 X 检查、一条 Z 检查，所以两张矩阵各 72 行、144 列。144 只是数据比特数。它不含症状辅助比特、耦合器、路由工作区、备用模块和译码器硬件——这条注脚要一直带到 2.5 节。

In plain words: 72 sites, two data qubits each, 144 in all; one X check and one Z check per site, so both matrices have 72 rows and 144 columns. 144 is the data-qubit count only. It excludes syndrome ancillas, couplers, routing workspace, spare modules and decoder hardware; carry that footnote all the way to §2.5.

gross 的每个多项式含三个互不相同的约化单项式。一个单项式是置换矩阵，每行每列恰贡献一个 1。因此：A 的每行每列重量 3；B 的每行每列重量 3；H_X = [A | B] 的每行重量 6；H_Z = [Bᵀ | Aᵀ] 的每行重量 6；每个数据比特属于 3 条 X 检查和 3 条 Z 检查（H_X 与 H_Z 的每列重量各为 3）。

Each gross polynomial contains three distinct reduced monomials. A monomial is a permutation matrix, contributing exactly one 1 per row and column. Therefore: every row and column of A has weight 3; likewise B; every row of H_X = [A | B] has weight 6; every row of H_Z = [Bᵀ | Aᵀ] has weight 6; every data qubit belongs to 3 X checks and 3 Z checks (each column of H_X and of H_Z has weight 3).

最后一条常被压缩成「度 6」。保留 X/Z 的拆分对电路很要紧：3 条 X 型交互和 3 条 Z 型交互用不同的辅助比特制备方式，故障传播的样子也不同。还有一个方法学细节：reproduce.py 不是「看到三项就推断重量 3」，而是把 A、B、H_X、H_Z 的每一行每一列都数了一遍（脚本第 103–110 行八条断言）——因为如果两个指数对约化后重合，项会抵消，重量会掉到 1。

The last statement is often compressed to “degree 6”. Keeping the X/Z split visible matters for a circuit: three X-type and three Z-type interactions use different ancilla preparations and propagate faults differently. One methodological detail: reproduce.py does not “see three terms and infer weight 3”; it counts every row and every column of A, B, H_X, H_Z (eight assertions, script lines 103–110), because if two exponent pairs coincide after reduction the terms cancel and the weight drops to 1.

GF(2) 上的高斯消元给出 rank H_X = 66，rank H_Z = 66。CSS 码的稳定子秩是两者之和，因为纯 X 行与纯 Z 行在二元辛表示里不可能互相抵消（一行在 X 半边有 1，另一行在 Z 半边有 1，加起来消不掉）：132。于是

Gaussian elimination over GF(2) gives rank H_X = 66 and rank H_Z = 66. The CSS stabilizer rank is their sum, because pure-X and pure-Z rows cannot cancel each other in the binary symplectic form (one row has 1s in the X half, the other in the Z half; their sum cannot vanish): 132. Hence

这里有一个有用的商读法。映射 [A | B]: R² → R，(u, v) ↦ Au + Bv 的像是理想 (A, B)（所有能写成 Au + Bv 的多项式）。它的秩是 66，所以它的余核cokernel：目标空间除以像。这里是 R/(A,B)，即「把所有 Au + Bv 当作零之后剩下的东西」。 R/(A,B) 的维数是 72 − 66 = 6。对偶一侧有同样的 6 维亏损，中间剩下 12 个逻辑维度。人话：72 个格点上的多项式，有 66 维能被 A、B「生成」出来，剩下 6 维生成不到；X 侧、Z 侧各贡献 6，凑成 12 个逻辑比特。

There is a useful quotient reading. The map [A | B]: R² → R, (u, v) ↦ Au + Bv, has image equal to the ideal (A, B) (all polynomials writable as Au + Bv). Its rank is 66, so its cokernelThe target space divided by the image: here R/(A,B), “whatever is left after every Au + Bv is treated as zero”. R/(A,B) has dimension 72 − 66 = 6. The dual side has the same six-dimensional defect, leaving twelve logical dimensions in the middle. In plain words: of the polynomials on 72 sites, 66 dimensions can be “generated” by A and B and 6 cannot; the X side and the Z side each contribute 6, making 12 logical qubits.

这是对这些周期与这些多项式算出来的事实。A 三项、B 三项不决定秩：改指数就改公共理想，k 可以变。直觉教程 5.5 节的活例子是 [[108,8,10]]：同样的 A、B 放到 9 × 6 的环面上，商维数掉到 4、k = 8。商的视角在 postema2025existence 里被发展成一门方法，它把码搜索变成三方拉扯：公共商越大 k 越大；但同样的共享结构可能引入短的非平凡圈、压低码距；而局域性约束又限制允许的单项式偏移。下面的实验台预设为 gross，可以亲手看到这件事：把 l 的滑杆从 12 拉到 9，每条检查仍是 6 条腿，读数里的 k 却从 12 变成 8（就是 bb108）；或者在「A 的项」里只改一个指数：只要改完后 A 的三项在 mod 12、mod 6 之下仍互不相同，项数、腿数就不变，k 却会跳。这个前提不能省：把 y 改成 y²，得到 A = x³ + y² + y²，两个 y² 相加为 0（1 + 1 = 0）而相消，A = x³，每条检查只剩 4 条腿。修订时的脚本在固定的 l = 12、m = 6 上逐个试过（B 不动）：只改 A 的一个指数、三项仍互不相同的 46 种改法里，k 只取 0、8、12；把 A 的一整项换成任意别的项（三项仍互不相同），k 只取 0、4、8、12。这些 k 值只对 12 × 6 这一个环面成立：

This is a computed fact for these periods and these polynomials. Three terms in A and three in B do not determine the rank: change the exponents and the common ideal changes, and k may change with it. The live example in §5.5 of the intuition guide is [[108,8,10]]: the same A, B on a 9 × 6 torus drop the quotient dimension to 4 and k to 8. postema2025existence develops the quotient perspective into a method that turns code search into a three-way tension: a larger common quotient can raise k; the same shared structure may introduce short nontrivial cycles and lower the distance; and locality constraints restrict the permitted monomial offsets. The bench below is preset to gross, so you can see this yourself: drag the l slider from 12 to 9 and every check still has six legs while the readout's k drops from 12 to 8 (that is bb108); or change a single exponent in “A terms”: as long as A's three terms stay distinct modulo 12 and 6, the term count and leg count stay the same, yet k jumps. The condition matters: changing y to y² gives A = x³ + y² + y², the two y² add to 0 (1 + 1 = 0) and cancel, so A = x³ and every check has only 4 legs. A script run during this revision tried every case on the fixed l = 12, m = 6 (B unchanged): over the 46 single-exponent changes of A that keep three distinct terms, k is 0, 8 or 12; replacing one whole term of A by any other term (still three distinct terms), k is 0, 4, 8 or 12. These k values hold for the 12 × 6 torus only:

gross 实例在 instances.yaml 里记的是：

The gross instance records, in instances.yaml:

两侧的意思不同。下界。bravyi2024bicycle（Bravyi、Cross、Gambetta、Maslov、Rall、Yoder，"High-Threshold and Low-Overhead Fault-Tolerant Quantum Memory"，2024）提供「不存在重量低于 12 的非平凡逻辑 Pauli」这一主张。QEC-Bank 目前没有重放这份下界证书。所以这条引用是结构化 origin 的一部分，不是装饰性的参考文献——去掉它，这个 d 就没有下界。上界。仓库自己构造两个中心化子零空间centralizer nullspace：与所有 Z 检查对易的 X 串全体 = ker H_Z（Z 侧同理）。逻辑算符都住在这里，稳定子也住在这里。，用 NumPy 种子 5 采样确定性的二元组合，剔除落在稳定子类里的，再用同类型稳定子贪心地减轻代表元，在 2000 次迭代内找到重量 12。这是一份可重跑的存在性见证：存在一个重量至多 12 的逻辑算符。

The two sides mean different things. Lower bound. bravyi2024bicycle (Bravyi, Cross, Gambetta, Maslov, Rall, Yoder, “High-Threshold and Low-Overhead Fault-Tolerant Quantum Memory”, 2024) 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: remove it and this d has no lower bound. Upper bound. The repository constructs the two centralizer nullspacesAll X strings commuting with every Z check = ker H_Z (and dually for Z). Logical operators live here, and so do stabilizers., samples deterministic binary combinations with NumPy seed 5, rejects stabilizer classes, greedily reduces representatives by same-type stabilizers, and finds weight 12 within 2,000 iterations. That is a rerunnable existence witness: a logical of weight at most 12 exists.

种子不是「调过的数据」。种子 0 跑 2000 次只到重量 14，跑更久才到 12；存下种子 5 是为了有一个紧凑的确定性见证——任何人跑同样的命令都会在同样的步数看到同样的算符。如果将来某个引擎找到更轻的逻辑算符，验证器会报告「改进」或「不符」，然后必须对照文献诊断，而不是悄悄把期望值改掉（QEC-Bank 的铁律：不许为了让验证通过而调数据）。

The seed is not tuned data. A seed-0 run with 2,000 iterations reaches only weight 14; a longer seed-0 run reaches 12. Seed 5 is stored to have a compact deterministic witness: anyone running the same command sees the same operator after the same number of steps. If a future engine finds a lighter logical, the verifier reports an improvement or a mismatch, and the expected value must then be diagnosed against the literature, never adjusted silently (a QEC-Bank iron rule: no tuning data to make verification pass).

这就是 bravyi2024bicycle 强调的十二倍有限块比较。它不是整机比。一个 BB 周期还需要：症状辅助比特（gross 有 144 条检查，原论文每条配一个辅助比特，于是 144 + 144 = 288 个物理比特——直觉教程 5.6 节的那个 288）、度 6 的交互、原提案里 7 层 CNOT 的调度、测量与复位、经典探测器处理、BP-OSD。架构还可能加移动、通信、备用块，或为逻辑操作准备的表面码辅助贴片。人话：比较开销时，分母里放的资源两边必须一样多；只比数据比特，是「同样 144 个数据比特存 12 个 vs 存 1 个」，这句话是对的，但也只有这句话。

This is the twelve-fold finite-block comparison emphasized by bravyi2024bicycle. It is not a full-machine ratio. A BB cycle also needs syndrome ancillas (gross has 144 checks; the original proposal gives each one ancilla, so 144 + 144 = 288 physical qubits, the 288 of §5.6 of the intuition guide), degree-six interactions, the seven-CNOT-layer schedule of the original proposal, measurement and reset, classical detector processing and BP-OSD. Architectures may add movement, communication, spare blocks, or surface-code ancilla patches for logical operations. In plain words: when comparing overhead, both denominators must hold the same resources; comparing data qubits only gives “the same 144 data qubits store 12 versus 1”, which is true, and is also the only thing it says.

12 属于静态稳定子码。一个症状电路定义了一个时空故障码，它的最小不可探测故障集可以更小：CNOT 顺序决定一个辅助比特的错误怎样扩散到几个数据比特上；探测器构造决定哪些历史可以被区分。因此下面四件事没有一件能单从静态账本推出：一个码距 12 的故障电路；一个普适阈值；一个与译码器无关的逻辑错误率；一个逻辑门错误率。它们分别属于第 3 章的电路账本和第 4 章的操作账本。

The value 12 belongs to the static stabilizer code. A syndrome circuit defines a spacetime fault code whose minimum undetectable fault set can be smaller: CNOT ordering controls how an ancilla error spreads across data qubits, and the detector construction controls which histories are distinguishable. Consequently none of the following follows from the static ledger alone: a distance-12 fault circuit; a universal threshold; a decoder-independent logical error rate; a logical-gate error rate. They belong to the circuit ledger of Chapter 3 and the operations ledger of Chapter 4.

测量完美时，一个 X 型错误 e_X 产生 Z 检查症状 H_Ze_X，一个 Z 型错误产生 H_Xe_Z。修复 c_X 首先必须给出同样的症状（H_Zc_X = H_Ze_X），但这只是必要条件。症状相同说明 e_X ⊕ c_X（⊕ = 逐位异或，即错误与修复叠加后剩下的 X 串）与所有 Z 检查对易；修复成功还要求这个剩余串落在 H_X 的行空间里，即它是若干条 X 检查的乘积（一个稳定子）。如果它与所有 Z 检查对易、却不是 X 检查的乘积，它就是一个 X 逻辑算符：症状全部清零了，逻辑比特却被翻转了。Z 型错误同理，换成 H_X 的症状与 H_Z 的行空间。因为 BB 的 Tanner 图有很多短圈、码又是退化degenerate：许多不同的物理错误对应同一个逻辑效果（相差一个稳定子）。译码器该押的是「最可能的逻辑类」，不一定是「最可能的那一个物理错误」。的，最可能的物理错误和最可能的逻辑陪集coset：相差一个稳定子的全部错误组成的类。同一陪集里的错误修复效果相同。未必是同一个对象。人话：一个数据比特有 3 条 Z 检查看着它，所以一个 X 错误点亮 3 盏灯——匹配译码的「每个错误两盏灯」前提在这里不成立（直觉教程 6.3 节）。这个码容量问题（只在数据比特上出错、测量完美，是最简单的噪声设定）是 roffe2020decoding 的 BP-OSD 基线及其广义 bicycle 前身 panteleev2021degenerate 的干净输入。

With perfect measurement, an X-type data error e_X produces the Z-check syndrome H_Ze_X, and a Z-type error produces H_Xe_Z. A correction c_X must first reproduce the syndrome (H_Zc_X = H_Ze_X), but that is only necessary. Equal syndromes mean that e_X ⊕ c_X (⊕ = bitwise XOR: the X string left after error and correction combine) commutes with every Z check; the correction succeeds only if this leftover also lies in the row space of H_X, i.e. is a product of X checks (a stabilizer). If it commutes with every Z check but is not a product of X checks, it is an X logical operator: every syndrome bit is cleared, yet a logical qubit has been flipped. The same holds for Z-type errors with the syndrome of H_X and the row space of H_Z. Because BB Tanner graphs have many short cycles and the code is degenerateMany distinct physical errors have the same logical effect (they differ by a stabilizer). The decoder should bet on the most likely logical class, not necessarily on the single most likely physical error., the most likely physical error and the most likely logical cosetThe class of all errors differing by a stabilizer. Errors in one coset are equivalent as corrections. need not be the same object. In plain words: three Z checks watch every data qubit, so one X error lights 3 lamps; the “two lamps per error” premise of matching fails here (intuition guide, §6.3). This code-capacity problem (errors only on the data qubits, perfect measurements: the simplest noise setting) is the clean input for the BP-OSD baseline of roffe2020decoding and its generalized-bicycle precursor panteleev2021degenerate.

用一个辅助比特测一条稳定子，要经过制备、一串纠缠门、测量、复位。纠缠门的顺序有讲究：一个早期落在辅助比特上的 Pauli 会经后面的门传播到好几个数据比特上；这样的关联「钩子」（hook）错误可能比独立故障更高效地对齐到一个逻辑算符上（《译码器 · 高中数学版》5.4 节讲过表面码上的同一现象）。bravyi2024bicycle 给出的是七层 CNOT 的提取电路，配它的两层连接图。数字 7 是那个调度与那个版图的性质，不是每个六腿矩阵的性质。shaw2025morphing（Shaw、Terhal，"Lowering Connectivity Requirements for Bivariate Bicycle Codes Using Morphing Circuits"，据文献扫描报告，登记待补）走另一条路：在周期中途改变码本身，把连接度从 6 降到 5、用 6 轮 CNOT，对一个相关的家族——这既换了硬件图也换了探测器模型。

Measuring one stabilizer with an ancilla requires preparation, a sequence of entangling gates, measurement and reset. The order of those entangling gates matters: an ancilla Pauli introduced early propagates through later gates to several data qubits, and such a correlated “hook” error may align with a logical operator more efficiently than independent faults (the same phenomenon on the surface code is in “Decoders · High-School Edition”, §5.4). bravyi2024bicycle gives a seven-CNOT-layer extraction for its two-layer connectivity. The number seven is a property of that schedule and layout, not of every weight-six matrix. shaw2025morphing (Shaw, Terhal, “Lowering Connectivity Requirements for Bivariate Bicycle Codes Using Morphing Circuits”, from the scout report, registration pending) takes another road: it changes the code itself during the cycle, reducing the connectivity degree from six to five and using six CNOT rounds for a related family, which changes both the hardware graph and the detector model.

一个钩子错误，逐步追踪。图 3.2 用一个辅助比特测 gross 的 X 检查 0（X 作用在比特 1、2、18、75、78、84 上）。辅助比特 a 先制备成 |+⟩，再依次对六个数据比特各做一次 CNOT（a 是控制端，数据比特是目标端），最后在 X 基下读出 a：六个数据比特上 X 检查的读数就记在 a 的结果里。图里的 CNOT 顺序只是示意，不是 Bravyi 等人实际用的调度。CNOT 有一条传播规则：门之前控制端上的一个 X，门之后变成控制端和目标端各一个 X。假设在第 2 与第 3 个 CNOT 之间，辅助比特上出了一个 X 故障：第 3、4、5、6 个 CNOT 把它依次复制到 q18、q75、q78、q84 上；a 自己的 X 不改变它的 X 基读数，所以这一轮看不出任何异常。结果是一次故障留下了 4 个数据比特上的 X 错误。再乘上 X 检查 0 本身（一个稳定子，不改变逻辑状态），X₁₈X₇₅X₇₈X₈₄ 等价于 X₁X₂：一次故障抵得上两个数据错误。一般地，故障出在第 t 个 CNOT 之后，等价于 min(t, 6 − t) 个数据错误，最多 3 个。如果这些成组出现的错误恰好顺着某个逻辑算符排列，更少的故障就能拼出一个逻辑错误，电路码距就会低于静态码距 12。钩子落在哪几个比特上由 CNOT 的顺序决定，所以调度要和码一起设计。

One hook error, traced step by step. Figure 3.2 measures X check 0 of gross (X on qubits 1, 2, 18, 75, 78, 84) with one ancilla. The ancilla a is prepared in |+⟩, then one CNOT is applied to each of the six data qubits in turn (a is the control, the data qubit the target), and finally a is read out in the X basis: the value of the X check on the six data qubits ends up in a's outcome. The CNOT order in the figure is only illustrative, not the schedule Bravyi et al. actually use. A CNOT has a propagation rule: an X on the control before the gate becomes an X on the control and an X on the target after it. Suppose an X fault hits the ancilla between CNOT 2 and CNOT 3: CNOTs 3, 4, 5, 6 copy it onto q18, q75, q78, q84 in turn; the X on a itself does not change its X-basis outcome, so nothing looks wrong in this round. The result: one fault has left X errors on 4 data qubits. Multiplying by X check 0 itself (a stabilizer, which does not change the logical state), X₁₈X₇₅X₇₈X₈₄ is equivalent to X₁X₂: one fault is worth two data errors. In general a fault after CNOT t is equivalent to min(t, 6 − t) data errors, at most 3. If these grouped errors happen to line up along a logical operator, fewer faults suffice to build a logical error, and the circuit distance falls below the static distance 12. Which qubits a hook lands on is set by the CNOT order, which is why the schedule must be designed together with the code.

测量有噪声、又要反复提取时，不能孤立地译码原始症状比特。探测器detector：把不同时刻的测量结果作比较（通常是同一条检查前后两轮读数的乘积）得到的「变化」事件；一次测量错误会产生一个可辨认的时空图案。事件比较跨时间的测量，让一次测量故障留下可辨认的时空图案。边界轮、初始化、最终的数据测量、以及哪些测量乘积是确定性的，都进入探测器定义（《译码器 · 高中数学版》5.1 节）。

With noisy repeated extraction, one does not decode raw syndrome bits in isolation. DetectorA “change” event obtained by comparing measurements across time (typically the product of one check's readings in consecutive rounds); a single measurement fault then leaves a recognizable spacetime pattern. events compare measurements across time so that a measurement fault produces a recognizable spacetime pattern. The boundary rounds, initialization, final data measurement and which products of results are deterministic all enter the detector definition (“Decoders · High-School Edition”, §5.1).

一段测量历史，算出探测器。盯住一条 Z 检查 c，连续测 4 轮。记 m_t 为第 t 轮的读数（0 表示 +1，1 表示 −1）。若数据比特一开始都制备在 |0⟩，Z 检查在第一轮之前的值是确定的 +1，记 m₀ = 0。探测器是相邻两轮的比较 D_t = m_t ⊕ m_t−1：读数变了记 1，没变记 0。下表比较两种故障：情形 ① 是第 2、3 轮之间 c 看着的一个数据比特出了 X 错误；情形 ② 是第 2 轮的读数本身测错了一次。

A measurement history, turned into detectors. Watch one Z check c for 4 rounds. Let m_t be the outcome in round t (0 for +1, 1 for −1). If the data qubits start in |0⟩, the Z check has the definite value +1 before round 1; write m₀ = 0. A detector compares consecutive rounds, D_t = m_t ⊕ m_t−1: 1 if the outcome changed, 0 if not. The table compares two faults: in case ① a data qubit watched by c gets an X error between rounds 2 and 3; in case ② the round-2 outcome itself is misread once.

情形 ① 的数据错误是持续的：读数从第 3 轮起一直是 1，但探测器只在第 3 轮亮一次；gross 的每个数据比特有 3 条 Z 检查看着，另外两条也在第 3 轮各亮一次，所以一次数据错误点亮同一时刻的 3 个探测器。情形 ② 的测量错误只坏了一次读数：同一条检查在第 2、3 两轮各亮一次，是时间方向上相邻的一对。原始读数里，两种故障是「从某轮起一直是 1」和「孤立的一个 1」；换成探测器后，它们变成时空里形状不同的小图案（空间方向 3 个、时间方向 2 个），译码器读的正是这些图案。有了探测器，就得到三个绝不能混的「码距」：

In case ① the data error persists: the outcome is 1 from round 3 on, yet the detector fires only once, in round 3; every data qubit of gross is watched by 3 Z checks, and the other two also fire once in round 3, so one data error lights 3 detectors at the same time. In case ② the measurement error spoils one outcome only: the same check fires in rounds 2 and 3, a pair adjacent in time. In the raw outcomes the two faults look like “1 from some round on” and “a single isolated 1”; as detectors they become small spacetime patterns of different shapes (3 in the space direction, 2 in the time direction), and these patterns are what the decoder reads. With detectors in hand there are three “distances” that must never be conflated:

静态码距 12 与「坏调度下更小的电路码距」是相容的。反过来，译码器没找到低重量失败，也不证明电路码距。人话：三个数字回答三个不同的问题，「12」只回答第一个。

A static distance of 12 is compatible with a smaller circuit distance under a bad schedule. Conversely, observing no low-weight decoder failures does not prove circuit distance. In plain words: three numbers answer three different questions, and “12” answers only the first.

置信传播belief propagation, BP：在 Tanner 图上让比特与检查反复交换「你有多可能出错」的软消息；在树上精确，在有圈的图上是近似。在错误变量和奇偶约束之间传递软的似然消息。在树上，这些局部消息是精确的。BB 的 Tanner 图不是树：短圈把关联信息喂回它自己，量子退化又让许多错误代表同一个逻辑陪集。于是 BP 可能振荡、收敛到一个不一致的硬判决，或落在错误的陪集上。有序统计译码ordered-statistics decoding, OSD：按 BP 给出的可靠度排序，选一组「信息位」，在其上精确解 H e = s，并测试一小组按可靠度排序的偏离；保证给出一个症状一致的修复。用 BP 的可靠度选一个基集，再在解症状约束的同时测试一组受控的、按可靠度排序的偏离。它的贡献不是另一条症状公式，而是局部消息失败时的全局修复阶段（《qLDPC 码 · 高中数学版》6.3 节）。到实验台上对 [[72,12,6]] 亲手跑一遍：先看消息如何演化、何时卡住，再按 OSD：

Belief propagationBP: qubits and checks on the Tanner graph repeatedly exchange soft “how likely am I to be wrong” messages; exact on a tree, approximate on graphs with cycles. passes soft likelihood messages between error variables and parity constraints. On a tree those local messages are exact. BB Tanner graphs are not trees: short cycles feed correlated information back into itself, and quantum degeneracy means many errors represent the same logical coset. BP may oscillate, converge to an inconsistent hard decision, or settle on the wrong coset. Ordered-statistics decodingOSD: rank qubits by BP reliability, choose a basis set, solve H e = s exactly on it, and test a small reliability-ordered set of deviations; guarantees a syndrome-consistent correction. uses the BP reliabilities to choose a basis set, then solves the syndrome constraints while testing a controlled set of reliability-ordered deviations. Its contribution is not another syndrome formula; it is a global repair stage when local messages fail (“qLDPC Codes · High-School Edition”, §6.3). Run it by hand on [[72,12,6]] below: watch the messages evolve and stall, then press OSD:

一份可操作的译码合同至少要写明：数据、测量、复位、空闲、双比特门各自的故障概率；是否有 X/Z 关联与泄漏；精确的提取调度与探测器构造；BP 的更新规则、迭代上限、阻尼、停止条件；OSD 的阶数与候选选择规则；延迟与吞吐约束。缺任何一项，「用 BP-OSD 译码」这句话就是不完整的。人话：「我用了 BP-OSD」的信息量，和「我用了一台电脑」差不多。

An operational decoding contract must specify at least: the data, measurement, reset, idle and two-qubit fault probabilities; whether X/Z correlations and leakage are present; the exact extraction schedule and detector construction; the BP update rule, iteration limit, damping and stopping condition; the OSD order and candidate-selection rule; latency and throughput constraints. Without these, “decoded with BP-OSD” is incomplete. In plain words: “I used BP-OSD” carries about as much information as “I used a computer”.

卡片记录的是 bravyi2024bicycle 的文献陈述：在论文的标准电路噪声模型、深度 7 的症状提取、论文的探测器构造与 BP-OSD 译码器下，BB 存储器的阈值约为 0.8%（v1 与 v2 摘要的原话是「an error threshold of 0.8% for the standard circuit-based noise model」，说的是「a family of LDPC codes」）。这个数是怎样算出来的：论文对每一块码单独解收支平衡方程 p_L(p) = kp。这里 p 是每个电路位置的物理错误率；p_L 是整块码每个症状测量周期的逻辑错误率（k 个逻辑比特里任一个出错都算）；kp 是 k 个不编码的比特在同一段时间里至少一个出错的概率的近似（各比特独立出错时精确值是 1 − (1 − p)^k，p 小时约等于 kp）。解出的 p₀ 叫这块码的伪阈值pseudo-threshold：一块固定尺寸的码的收支平衡点，即整块码每个症状测量周期的逻辑错误率 p_L（k 个逻辑比特任一个出错）等于 kp 时的 p。kp 近似 k 个不编码比特至少一个出错的概率（各比特独立出错时精确值是 1 − (1 − p)^k）。它是一块码的数，不是一族码的阈值。。p_L 在 p ≥ 10⁻³ 处由模拟算出，更小的 p 用拟合公式外推。各版本的数字不同：v1（2023 年 8 月）表 1 给五块码的 p₀ 印作 0.007(3) 到 0.008(3)（括号里那一位数字的含义论文没有说明），正文说「close to 0.008」；v2（2024 年 2 月）表 1 改为 0.0048 到 0.0069（gross 为 0.0065），正文说 144 与 288 比特两块码「close to 0.007」，摘要仍写 0.8%；Nature 发表版（2024）的摘要写 0.7%。所以 0.8% 不是把几个尺寸的曲线画在一起读出的交叉点（那叫有限尺寸标度finite-size scaling：在几个不同尺寸的码上画「逻辑错误率对 p」的曲线，曲线的交点可用来估计一族码的阈值。这篇论文没有做这一步。，这篇论文没有做），也不是 gross 一块码的数。它还不是：① 码容量阈值；② 对所有 BB 选择都成立的渐近定理；③ BP-OSD 独立于调度的性质；④ 硬件上的逻辑错误率；⑤ 电路码距等于静态码距的证据。换掉码族、电路、噪声归一化或译码器，就定义了另一个阈值实验，数字可能移动。限定项是这个数字含义的一部分——账本规则见《译码器 · 高中数学版》6.1 节，QEC-Bank 译码器侧面对这条记录的标签是 research-grade。

The card records the literature statement from bravyi2024bicycle: under the paper's standard circuit-noise model, depth-seven syndrome extraction, detector construction and BP-OSD decoder, the BB memory has a threshold of about 0.8% (the v1 and v2 abstracts say “an error threshold of 0.8% for the standard circuit-based noise model”, of “a family of LDPC codes”). How the number is obtained: for each code separately the paper solves the break-even equation p_L(p) = kp. Here p is the physical error rate per circuit location; p_L is the whole block's logical error rate per syndrome cycle (a failure of any of its k logical qubits counts); kp approximates the probability that at least one of k unencoded qubits fails in the same time (exactly 1 − (1 − p)^k when they fail independently, about kp for small p). The solution p₀ is that block's pseudo-thresholdThe break-even point of one fixed-size block: the p at which its logical error rate per syndrome cycle, p_L (any of its k logical qubits failing), equals kp. kp approximates the chance that at least one of k unencoded qubits fails (exactly 1 − (1 − p)^k when they fail independently). A number for one block, not the threshold of a family.. p_L is simulated for p ≥ 10⁻³ and extrapolated below with a fitted formula. The versions differ: Table 1 of v1 (August 2023) prints p₀ = 0.007(3) to 0.008(3) for its five codes (the paper does not say what the parenthesized digit means) and the text says “close to 0.008”; Table 1 of v2 (February 2024) gives 0.0048 to 0.0069 (0.0065 for gross) and the text says “close to 0.007” for the 144- and 288-qubit codes, while the abstract still says 0.8%; the abstract of the Nature version (2024) says 0.7%. So 0.8% is not a crossing point read off curves for several sizes (that method is finite-size scalingPlot the logical error rate against p for several code sizes; the p where the curves cross estimates the threshold of the family. This paper does not do that step., which this paper does not use), and it is not a number of the gross block alone. Nor is it: (1) a code-capacity threshold; (2) an asymptotic theorem for all BB choices; (3) a property of BP-OSD independent of schedule; (4) a hardware logical error rate; (5) evidence that circuit distance equals static distance. Changing the code family, circuit, noise normalization or decoder defines a different threshold experiment, and the number can move. The qualification is part of the number's meaning; the ledger rules are in “Decoders · High-School Edition”, §6.1, and the QEC-Bank decoders facet labels this record research-grade.

文献扫描暴露了几条利用更多 BB 结构的路（键后标注的题名与 arXiv 号来自扫描报告；除 koutsioumpas2025autdec 外均为登记待补）。koutsioumpas2025autdec（"Automorphism Ensemble Decoding of Quantum LDPC Codes"，2025，已登记）对码的自同构做一个集成：把一个难的症状搬进不同的对称参考系里各译一次再合并。sahay2026matching（Sahay、Williamson、Brown，"A Matching Decoder for Bivariate Bicycle Codes"，arXiv 2602.22770）把选定的 BB 问题约化到类环面的圆柱上做匹配，再与 BP 和「过匹配」结合——约化的对称性假设是译码器的一部分；QEC-Bank 译码器侧面把它记为 mwpm、噪声设定「BB 对称约化到类环面圆柱」，并注明「不把每个 BB 实例变成独立的环面码」。blue2026machine（"Machine Learning Decoding of Circuit-Level Noise for Bivariate Bicycle Codes"）训练一个 transformer 读电路级历史，训练分布、校准和推断延迟都是主张的一部分。pakhunov2026convergence 预测 BP 何时可能收敛，是一层分流而非完整的纠错规则；pakhunov2026peeling 分析贪心剥皮 / 流式路线，在其基准上报告很大的延迟增益——这些增益不带着同样的信道、实现与精度目标是不能迁移的。卡片只收录共享词表里已登记的译码器标识（bp-osd、mwpm）；更深的推导页记录这些新的复合流水线，但不假装它们是同一个算法。

The literature sweep exposes several ways to use more BB structure (titles and arXiv ids after each key come from the scout report; all but koutsioumpas2025autdec are registration pending). koutsioumpas2025autdec (“Automorphism Ensemble Decoding of Quantum LDPC Codes”, 2025, registered) runs an ensemble over code automorphisms: a hard syndrome is moved into different symmetry frames, decoded in each, and the candidates combined. sahay2026matching (Sahay, Williamson, Brown, “A Matching Decoder for Bivariate Bicycle Codes”, arXiv 2602.22770) reduces selected BB problems to toric-like cylinders, applies matching, and combines it with BP and over-matching; the reduction's symmetry assumptions are part of the decoder, and the QEC-Bank decoders facet records it as mwpm with noise setting “BB symmetry reduction to toric-code-like cylinders” and the note that it “does not turn every BB instance into independent toric codes”. blue2026machine (“Machine Learning Decoding of Circuit-Level Noise for Bivariate Bicycle Codes”) trains a transformer on circuit-level histories; training distribution, calibration and inference latency are part of the claim. pakhunov2026convergence predicts when BP is likely to converge, a triage layer rather than a complete correction rule; pakhunov2026peeling analyzes a greedy peeling/streaming route and reports large latency gains on its benchmarks, gains that do not transfer without the same channel, implementation and accuracy target. The card includes only decoder identifiers already registered in the shared vocabulary (bp-osd, mwpm); the deeper page records the newer compound pipelines without pretending each is the same algorithm.

静态条目作为码的描述是完整的，但它还没有发布一个 BB 的 Schedule（调度对象）：没有辅助比特坐标；没有七层门序；没有复位与测量操作；没有探测器定义；没有电路码距见证。补上这些需要引擎支持的电路 / 调度表示和一份忠于原文的转录。在此之前，通用的格点浏览器只显示它真正拥有的东西——数据坐标与稳定子支撑——一点不多。人话：QEC-Bank 宁愿留白，也不用「看起来像」的调度充数。

The static entry is complete as a code description, but it does not yet ship a BB Schedule object: no ancilla coordinates; no seven-layer gate order; no reset or measurement operations; no detector definitions; no circuit-distance witness. Adding those requires an engine-supported circuit/schedule representation and a source-faithful transcription. Until then, the generic lattice explorer shows exactly what it owns, data coordinates and stabilizer supports, and no more. In plain words: the QEC-Bank would rather leave a blank than fill it with a schedule that merely “looks right”.

取两块相同的 BB 码，记为控制块 c 与目标块 t，对每个 i 从 c 的第 i 个比特到 t 的第 i 个比特做物理 CNOT。物理 CNOT 对 Pauli 的共轭conjugation：门 U 把算符 P 变成 UPU^†；对 Clifford 门，Pauli 串变成 Pauli 串。作用是：

Take two identical BB blocks, labelled control c and target t, and apply physical CNOT from qubit i of c to qubit i of t for every i. Physical CNOT conjugatesConjugation: a gate U turns an operator P into UPU^†; for Clifford gates Pauli strings go to Pauli strings. Paulis as

于是控制块上的一条 X 稳定子获得目标块上对应的 X 稳定子（因为两块相同，同一组下标在目标块上也是一条稳定子）；目标块上的 X 稳定子不变。对偶地，目标块的 Z 稳定子获得控制块对应的 Z 稳定子，控制块的 Z 稳定子不变。每个像都仍在两块的稳定子群里——码空间保持。同样的共轭作用在对应的逻辑 Pauli 对上，给出并行的逻辑 CNOT：12 对逻辑比特同时各做一个 CNOT。这是一条码空间陈述（《格点手术 · 高中数学版》1.2 节对任何两块相同 CSS 码都说过）。硬件实现还需要对齐的比特对、一个覆盖两块关联错误的故障模型、以及能处理它们的译码器。yoder2025tour（Yoder 等，"Tour de Gross: A Modular Quantum Computer Based on Bivariate Bicycle Codes"，arXiv 2506.03094，据文献扫描报告，登记待补）把这条原语嵌进 gross 家族的模块化架构。QEC-Bank 的操作行：「CNOT between identical BB blocks · transversal · claimed · Clifford」，代价「一层对齐的成对物理 CNOT；模块运输、对齐与关联译码是额外的架构成本」。

Therefore an X stabilizer on the control block acquires the matching X stabilizer on the target (the blocks are identical, so the same index set is a stabilizer there too); an X stabilizer on the target stays fixed. Dually, a target Z stabilizer acquires the matching control Z stabilizer, while a control Z stabilizer stays fixed. Every image remains in the two-block stabilizer group: the code space is preserved. The same conjugation applies to corresponding logical Pauli pairs, giving the parallel logical CNOT: 12 pairs of logical qubits each undergo a CNOT at once. This is a code-space statement (“Lattice Surgery · High-School Edition”, §1.2 says it for any two identical CSS blocks). A hardware implementation still needs aligned pairs, a fault model for correlated two-block errors, and a decoder that handles them. yoder2025tour (Yoder et al., “Tour de Gross: A Modular Quantum Computer Based on Bivariate Bicycle Codes”, arXiv 2506.03094, from the scout report, registration pending) embeds this primitive in a gross-family modular architecture. The QEC-Bank operation row: “CNOT between identical BB blocks · transversal · claimed · Clifford”, cost “one aligned pairwise physical-CNOT layer; module transport, alignment, and correlated decoding are additional architecture costs”.

令 P 是环面平移 x 或 y 之一，对两个数据块同时施加 P。因为 A、B、Aᵀ、Bᵀ 都与 P 对易（它们都是 x、y 的多项式，1.1 节），

Let P be either torus translation x or y, and apply P to both data blocks. Since A, B, Aᵀ, Bᵀ all commute with P (they are polynomials in x, y, §1.1),

左乘 P 只是把检查的行重排，所以这个物理比特置换把稳定子群映到自身：它是码的自同构automorphism：把稳定子群映到自身的物理操作。它保证码空间不变，但对逻辑比特的具体作用需要另外计算与认证。。reproduce.py 第 117–120 行对 gross 的 x 与 y 都断言了这两条精确的矩阵恒等式。这证明的是保持码空间，不是有用。要认证一个逻辑门，还要走五步：① 选一组逻辑 Pauli 基（12 个 X̄、12 个 Z̄）；② 用物理置换共轭每个基元；③ 把每个像模稳定子约化；④ 拼出诱导的 24×24 二元辛作用symplectic action：一个 Clifford 操作对 2k 个逻辑 Pauli 生成元的作用矩阵，2k × 2k，保持对易 / 反对易关系。k = 12 时是 24 × 24。；⑤ 判定想要的物理路由是否容错。QEC-Bank 没有做这五步，所以卡片不把平移记成一个已认证的逻辑门。文献里 ①–④ 做过：Bravyi 等人（arXiv:2308.07915 v2，9.2 节「Logical Gates based on Automorphisms」）选了一组以单项式编号的逻辑算符，平移 s 把编号为 n 的 X̄ 换成编号为 s·n 的 X̄（Z̄ 同理），所以每个平移都是一个逻辑 CNOT 电路（最多差一个逻辑 Pauli 修正）；作者还写道除 s = 1 外没有哪个平移作用成逻辑恒等，但对 gross 这一句不成立：逐一检验 72 个平移对全部逻辑 X̄、Z̄ 的作用，x⁶（沿长为 12 的方向平移半圈）作用成逻辑恒等，其余 70 个非恒等平移都作用非平凡（检验方法见直觉教程 6.5 节）；作者也说，还不清楚怎样把这些 CNOT 用于有用的计算。这个结果在文献里，不在 QEC-Bank 的证书里（直觉教程 6.5 节有同样的说法）。

Left multiplication by P merely permutes check rows, so this physical qubit permutation maps the stabilizer group to itself: it is an automorphismA physical operation mapping the stabilizer group to itself. It guarantees the code space is unchanged, but its concrete action on the logical qubits must be computed and certified separately. of the code. Lines 117–120 of reproduce.py assert both exact matrix identities for the gross x and y. This proves code-space preservation, not usefulness. To certify a logical gate, five more steps: (1) choose a logical Pauli basis (12 X̄s and 12 Z̄s); (2) conjugate every basis element by the physical permutation; (3) reduce each image modulo stabilizers; (4) assemble the induced 24×24 binary symplectic actionThe matrix by which a Clifford operation acts on the 2k logical Pauli generators, 2k × 2k, preserving commutation/anticommutation. For k = 12 it is 24 × 24.; (5) determine whether the desired physical routing is fault tolerant. The QEC-Bank has not carried out these five steps, so the card does not record translations as certified logical gates. The literature has done (1)–(4): Bravyi et al. (arXiv:2308.07915 v2, section 9.2 “Logical Gates based on Automorphisms”) choose logical operators labelled by monomials; a translation s turns the X̄ labelled n into the X̄ labelled s·n (likewise for Z̄), so every translation is a logical CNOT circuit (up to a logical Pauli correction); the authors also write that no translation other than s = 1 acts as the logical identity, but for gross that does not hold: checking all 72 translations on every logical X̄ and Z̄ shows that x⁶ (a half-turn along the direction of length 12) acts as the logical identity, while the other 70 nonidentity translations have a nontrivial logical action (the check is described in §6.5 of the intuition guide); the authors add that it is not yet clear how to use these CNOTs for useful computation. That result lives in the literature, not in the QEC-Bank certificates (§6.5 of the intuition guide says the same).

eberhardt2024fold（Eberhardt、Steffan，"Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes"，arXiv 2407.03973，登记待补）为对称的 BB 例子构造了显式的逻辑基和折叠横向 Clifford 门，包括 [[98,6,12]] 与 [[162,8,12]] 两块。一次折叠fold：把由一个反射（或别的对合，即做两次回到原地的变换）配成对的比特两两配对，使常深度的物理 Clifford 实现一个逻辑操作。把由反射或其他对合关联的比特配成对，让常深度的物理 Clifford 实现一个逻辑操作。证书有三部分：物理置换是所选检查空间的自同构；它诱导的逻辑 Pauli 作用被算出；折叠的交互模式可用或可路由。额外的对称性是一个选码条件。这些例子不意味着原来的 [[144,12,12]] gross 块有同样的折叠门。sayginel2025automorphisms（Sayginel、Koutsioumpas、Webster、Rajput、Browne，"Fault-Tolerant Logical Clifford Gates from Code Automorphisms"，2025，已登记）把这套工作流一般化：找 Tanner 图自同构、解出它们的逻辑辛作用、编译出有用的 Clifford。再说一次：只保持图而不算逻辑作用，只是第一步。QEC-Bank 对应两行：「selected logical Clifford generators · fold-transversal · claimed」（代价：一次折叠 / 置换加常深度物理 Clifford 层；只有解出了逻辑作用的对称 BB 实例才算）与「logical Clifford operations through automorphisms · fold-transversal · claimed」（代价：一个已认证的 Tanner 图置换及其物理路由；诱导的辛作用必须对所选块算出）。

eberhardt2024fold (Eberhardt, Steffan, “Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes”, arXiv 2407.03973, registration pending) constructs explicit logical bases and fold-transversal Clifford gates for symmetric BB examples, including [[98,6,12]] and [[162,8,12]] blocks. A foldPair up qubits related by a reflection or another involution (a transformation that returns home when done twice) so that constant-depth physical Cliffords implement a logical operation. pairs qubits related by a reflection or other involution so that constant-depth physical Cliffords implement a logical operation. The certificate has three parts: the physical permutation is an automorphism of the chosen check spaces; its induced logical Pauli action is computed; the folded interaction pattern is available or routable. The extra symmetry is a code-selection condition. These examples do not imply that the original [[144,12,12]] gross block has the same fold gate. sayginel2025automorphisms (Sayginel, Koutsioumpas, Webster, Rajput, Browne, “Fault-Tolerant Logical Clifford Gates from Code Automorphisms”, 2025, registered) generalizes the workflow: find Tanner-graph automorphisms, resolve their logical symplectic actions, compile useful Cliffords. Again, graph preservation without the logical action is only the first step. The two QEC-Bank rows: “selected logical Clifford generators · fold-transversal · claimed” (cost: a fold/permutation plus constant-depth physical Clifford layers; only symmetric BB instances with a resolved logical action qualify) and “logical Clifford operations through automorphisms · fold-transversal · claimed” (cost: a certified Tanner-graph permutation and its physical routing; the induced symplectic action must be computed for the chosen block).

长程手术用一个辅助码或贴片测量数据块之间的联合逻辑 Pauli（机制见《格点手术 · 高中数学版》第 3 章：缝合检查的连乘缩成 Z̄⊗Z̄）。反复的症状提取让这次奇偶测量容错；经典前馈完成传送或 Clifford 小装置。cohen2022longrange（Cohen、Kim、Bartlett、Brown，"Low-Overhead Fault-Tolerant Quantum Computing Using Long-Range Connectivity"，arXiv 2110.10794，登记待补）为高码率码发展了一般的基础设施；yoder2025tour 给出 gross 家族的指令集、编译路径与 d = 12、d = 18 两种模块的资源估计。这些操作不是某一张 BB 检查矩阵的横向性质，它们的代价包括：一个辅助或通信模块；反复的提取轮；边界 / 接口检查或长程耦合器；联合的探测器模型与译码器；被打包的逻辑比特在路由期间的空闲暴露。卡片把机制标为 surgery，正是为了不把这些资源藏在「逻辑 Clifford」四个字下面。QEC-Bank 行：「inter-block Pauli measurements and Clifford surgery · surgery · claimed」，代价「长程辅助贴片、反复测量与联合译码器；资源计数属于所选的模块化协议」。

Long-range surgery uses an ancilla code or patch to measure joint logical Paulis between data blocks (the mechanism is in “Lattice Surgery · High-School Edition”, Chapter 3: the seam checks telescope into Z̄⊗Z̄). Repeated syndrome extraction makes the parity measurement fault tolerant; classical feed-forward completes teleportation or Clifford gadgets. cohen2022longrange (Cohen, Kim, Bartlett, Brown, “Low-Overhead Fault-Tolerant Quantum Computing Using Long-Range Connectivity”, arXiv 2110.10794, registration pending) develops the general infrastructure for high-rate codes; yoder2025tour supplies a gross-family instruction set, compilation path and resource estimates for distance-12 and distance-18 modules. These operations are not transversal properties of one BB check matrix; their cost includes an ancilla or communication module; repeated extraction rounds; boundary/interface checks or long-range couplers; a joint detector model and decoder; idle and routing exposure of the packed logical qubits. The card labels the mechanism surgery precisely so those resources are not hidden under “logical Clifford”. QEC-Bank row: “inter-block Pauli measurements and Clifford surgery · surgery · claimed”, cost “long-range ancilla patches, repeated measurements, and a joint decoder; the resource count belongs to the selected modular protocol”.

liang2025selfdual（Liang、Chen，"Self-Dual Bivariate Bicycle Codes with Transversal Clifford Gates"，arXiv 2510.05211，登记待补）加上额外的自对偶与扭环面条件，得到重量 8 的 BB 块，带横向 CNOT、H、S。这些条件改变了所选的家族。操作行明写「self-dual BB subfamily」，不是 gross 的主张；QEC-Bank 行的代价栏：「在满足额外自对偶与扭环面条件的块上做常深度物理层；不是 gross 码的主张」。breuckmann2026cups（Breuckmann、Davydova、Eberhardt、Tantivasadakarn，"Cups and Gates I"，arXiv 2410.16250，登记待补）用上同调与杯积cup product：上同调类之间的一种乘法。这里它是「常深度逻辑操作存在」的证书：有这个代数不变量的码才有那个门。不变量为具备所需结构的码认证常深度逻辑操作，包括选定的 BB 例子——杯积就是门的证书，一对一般的二元多项式不会自动提供它。xu2025batched（Xu 等，"Batched High-Rate Logical Operations for Quantum LDPC Codes"，arXiv 2510.06159，登记待补）为 CSS qLDPC 架构发展成批的纠错、手术、切换与 Clifford 小装置，并把一个提案特化到自对偶 BB 块；「成批」改变的是资源调度与共享辅助比特的记账，不是静态稳定子参数。后两个键只出现在推导页，不绑定任何操作行。

liang2025selfdual (Liang, Chen, “Self-Dual Bivariate Bicycle Codes with Transversal Clifford Gates”, arXiv 2510.05211, registration pending) imposes extra self-duality and twisted-torus conditions, obtaining weight-eight BB blocks with transversal CNOT, H and S. Those conditions change the selected family. The operation row explicitly says “self-dual BB subfamily” and is not a gross claim; the QEC-Bank cost column reads “constant-depth physical layers on blocks satisfying the extra self-duality and twisted-torus conditions; not a gross-code claim”. breuckmann2026cups (Breuckmann, Davydova, Eberhardt, Tantivasadakarn, “Cups and Gates I”, arXiv 2410.16250, registration pending) uses cohomology and cup-productA multiplication between cohomology classes. Here it is the certificate that a constant-depth logical operation exists: codes with this algebraic invariant have that gate. invariants to certify constant-depth logical operations on codes with the required structure, including selected BB examples; the cup product is the gate certificate, and a generic bivariate polynomial pair does not automatically supply it. xu2025batched (Xu et al., “Batched High-Rate Logical Operations for Quantum LDPC Codes”, arXiv 2510.06159, registration pending) develops batched error correction, surgery, switching and Clifford gadgets for CSS qLDPC architectures and specializes a proposal to self-dual BB blocks; “batched” changes the resource schedule and shared-ancilla accounting, not the static stabilizer parameters. The last two keys appear only on the derivation page and are bound to no operation row.

六镜头文献扫描里没有任何来源为原始 gross 块提供普适的横向门集。诚实的边界是：横向 CNOT 在对齐的相同 CSS 块之间可用；选定的对称或自对偶块获得更多 Clifford；手术与自同构用额外资源扩大指令集；非 Clifford 的补全仍需要魔法态注入 / 培育、码切换、基于测量的资源或另一种已认证的机制（《格点手术 · 高中数学版》1.3 节的 Bravyi–König 天花板说明了为什么二维局域码没有原生 T 门；BB 不局域，但文献里也还没有它的 T）。k = 12 的打包让可寻址性格外重要：对全部 12 个逻辑比特做一次集体操作，不能替代一个「挑出想要的组合、同时保住其他」的编译器。

No source in the six-lens sweep supplies a universal transversal gate set for the original gross block. The honest boundary: transversal CNOT is available between aligned identical CSS blocks; selected symmetric or self-dual blocks gain more Cliffords; surgery and automorphisms enlarge the instruction set with extra resources; non-Clifford completion still needs magic-state injection/cultivation, switching, measurement-based resources, or another certified mechanism (the Bravyi–König ceiling of “Lattice Surgery · High-School Edition”, §1.3 explains why 2D local codes have no native T; BB is not local, but the literature has no T for it either). The packed k = 12 makes addressability especially important: a collective operation on all twelve logical qubits is no substitute for a compiler that selects the desired logical combination while preserving the others.

下表照 QEC-Bank 推导页排列；「状态」一栏是 QEC-Bank 实现侧面的原词（demonstrated / proposed）。除 wang2026lowoverhead、poole2025rydberg、berthusen2025local、shaw2025morphing、yoder2025tour 五个键绑定了实现记录外，wang2026coprime、chen2025fusion、chandra2026distributed 三个只出现在推导页。八个键全部登记待补，题名来自文献扫描报告。

The table follows the QEC-Bank derivation page; the “status” column uses the QEC-Bank realizations facet's own words (demonstrated / proposed). Five keys, wang2026lowoverhead, poole2025rydberg, berthusen2025local, shaw2025morphing, yoder2025tour, are bound to realization records; wang2026coprime, chen2025fusion and chandra2026distributed appear only on the derivation page. All eight keys are registration pending; titles come from the scout report.

wang2026lowoverhead（Ke Wang 等，"Demonstration of Low-Overhead Quantum Error Correction Codes"，arXiv 2505.09684，登记待补）是六镜头扫描认定的唯一一次字面意义的 BB 实验。它的块和硬件数必须一直跟着结果走：码是 [[18,4,4]]，不是 gross；器件是 32 个带长程耦合的 transmon（18 个数据比特；其余 14 个的角色卡片未登记，直觉教程 6.6 节）；操作是全部非局域六腿检查加反复的症状提取；指标是每个逻辑比特每周期的逻辑错误率 (8.91 ± 0.17)%；解释是「低开销、非局域检查的实验可行」。这次运行没有给出家族的交叉：一个「低于阈值」的标度演示需要至少两个码距在可比的电路、噪声与译码下比较，且逻辑错误随码距增大而下降——一个固定的 [[18,4,4]] 点提供不了这个推断。还有，32 这个器件数不能直接与 gross 的数据比特数 144 比：前者是含所选辅助资源的器件实现，后者是静态的数据比特参数（2.5 节）。QEC-Bank 记录原句：「这确立的是可行性，不是低于阈值的标度，也不是 gross 码的运行。」

wang2026lowoverhead (Ke Wang et al., “Demonstration of Low-Overhead Quantum Error Correction Codes”, arXiv 2505.09684, registration pending) is the only literal BB experiment identified by the six-lens sweep. Its block and hardware count must stay attached to the result: the code is [[18,4,4]], not gross; the device is 32 transmons with long-range couplings (18 data qubits; the card assigns no role to the other 14, intuition guide, §6.6); the operation is all nonlocal weight-six checks with repeated syndrome extraction; the metric is logical error per logical qubit per cycle (8.91 ± 0.17)%; the interpretation is “feasibility of a low-overhead, nonlocal-check experiment”. The run does not provide a family crossing: a below-threshold scaling demonstration needs at least two distances under comparable circuits, noise and decoding, with logical error decreasing as distance grows, and one fixed [[18,4,4]] point cannot supply that inference. Nor may the device count 32 be compared directly with the gross data-only n = 144: the former is a device implementation including its chosen ancilla resources, the latter a static data-qubit parameter (§2.5). The QEC-Bank record's own sentence: “this establishes feasibility, not below-threshold scaling and not a gross-code run.”

七个提案攻的是不同的瓶颈。Rydberg 版图：利用可重构或长程的原子相互作用快速实现 Tanner 图。分层局域：付出移动、通信或对昂贵检查的更慢测量，以适应局域基底。Morphing：在提取中途换码，让瞬时图的度更低。互素子类：约束多项式搜索，让码率与冷原子版图一起设计。模块化架构：把块变成带辅助、通信、编译与资源记账的指令集。光子融合：把奇偶检查翻译成发射器产生的资源态和一个考虑损耗的融合网络。分布式模块：把一个 gross 块拆开，暴露出有噪声的非局域 Bell 链路带来的逻辑代价。它们不是同一个探测器模型的可互换实现：移动会增加空闲故障；morphing 改变了哪个稳定子码在起作用；光子损耗信道不同于 transmon 的退极化模型；联网分割引入链路噪声。每一个都需要自己的电路与译码器主张。

The seven proposals attack different bottlenecks. Rydberg layout: exploit reconfigurable or long-range atom interactions to realize the Tanner graph quickly. Layered locality: pay motion, communication, or slower measurement of costly checks to fit a local substrate. Morphing: change the code during extraction so the instantaneous graph has lower degree. Coprime subclass: constrain the polynomial search so rate and cold-atom layout are co-designed. Modular architecture: turn blocks into an instruction set with ancillas, communication, compilation and resource accounting. Photonic fusion: translate parity checks into emitter-generated resource states and a loss-aware fusion network. Distributed modules: split one gross block and expose the logical penalty from noisy nonlocal Bell links. These are not interchangeable implementations of one detector model: movement can add idle faults; morphing changes which stabilizer code is active; a photonic loss channel differs from a transmon depolarizing model; a networked partition introduces link noise. Each needs its own circuit and decoder claim.

截至扫描日期，没有任何被抓取的实验：运行过 [[144,12,12]] gross 存储器；在一个 BB 家族内演示过随码距增大的逻辑压制；实现过容错的 BB 逻辑门；在硬件上验证过完整的 yoder2025tour 指令集。这些「没有」是条目知识状态的一部分：它们阻止一个提案被渲染成一次演示，也标出未来更新的具体目标。将来新增一条实现记录时，必须写明五件事：① 精确的 BB 实例，以及它是普通、自对偶、互素、剪枝还是 morphing；② 数据、辅助、通信与总器件数分开计；③ 提取调度与反复轮数；④ 译码器与噪声 / 归一化约定；⑤ 结果是探测、纠正、标度、态制备还是逻辑操作。没有这五个字段，「BB 已被演示」粗到无法比较。与表面码实现账本的对照（Willow 的 Λ = 2.14 ± 0.02 之类是另一类主张）见Surface Code 用例与实现导读。

As of the sweep date, no fetched experiment had: run the [[144,12,12]] gross memory; demonstrated increasing-distance logical suppression within a BB family; implemented a fault-tolerant BB logical gate; validated the full yoder2025tour instruction set in hardware. These absences are part of the entry's knowledge state: they prevent a proposal from being rendered as a demonstration and mark concrete targets for future updates. A future realization row must state five things: (1) the exact BB instance and whether it is ordinary, self-dual, coprime, pruned or morphing; (2) data, ancilla, communication and total device counts separately; (3) the extraction schedule and number of repeated rounds; (4) the decoder and noise/normalization convention; (5) whether the result is detection, correction, scaling, state preparation or a logical operation. Without those five fields, “BB demonstrated” is too coarse to compare. For the contrast with the surface-code ledger (Willow's Λ = 2.14 ± 0.02 and the like are a different kind of claim), see the surface-code use-cases guide.

先把 QEC-Bank 推导页的「视图地图」原样搬来，再逐条解释。「基本对象」是你在那种说法里操作的东西；「最擅长的问题」是它比别的说法答得更好的那一个；「保住的不变量」是从一种说法翻到另一种说法时一个 0/1 都不会变的数据；「警告」是最常见的越界推理。

First the “view map” of the QEC-Bank derivation page, copied as is, then one explanation per row. The “basic object” is what you manipulate in that view; the “best question” is the one it answers better than any other view; the “preserved invariant” is the data that survives translation into another view without a single 0/1 changing; the “caveat” is the most common overreach.

QEC-Bank 表示阶梯侧面给这些视图分了「梯级」并贴了标签，《BB 码 · 表示阶梯》逐级讲过；本章只用其中与推导有关的部分。八条断言里，两条是机器导出（derived）的：R1 的辛 CSS 检查矩阵（由构造器 bivariate_bicycle.build → CheckMatrix.from_css 生成）与 R2 的三项链复形；一条是canonical（规范的）：R5 的二元多项式说法本身；其余五条——群代数、Tanner 电路图、商模、任意子 / Koszul、环面圆柱——全是 curated（文献整理），各绑一个文献键。人话：QEC-Bank 机器亲手造出来的只有矩阵和复形；其他说法都是「书上这么讲，我们照登」。

The QEC-Bank representation-ladder facet assigns these views to “rungs” and labels them; “BB Codes · Representation Ladder” walks the rungs one by one, and this chapter uses only what bears on the derivations. Of the eight assertions, two are machine-derived (derived): the R1 symplectic CSS check matrix (produced by the constructor bivariate_bicycle.build → CheckMatrix.from_css) and the R2 three-term chain complex; one is canonical: the R5 bivariate-polynomial description itself; the remaining five, group algebra, Tanner-circuit graph, quotient module, anyon/Koszul and toric cylinders, are all curated from the literature, each bound to one bibkey. In plain words: the only things the QEC-Bank machine built with its own hands are the matrix and the complex; every other view is “the books say so, and we record it”.

单项式 x^ay^b 标记群元素 (a, b) ∈ ℤ_l × ℤ_m——就是「往 x 方向走 a 步、往 y 方向走 b 步」这个平移。用1.1 节的正则表示，群的乘法（两个平移先后做）变成矩阵乘法。因为这个群是阿贝尔的（可交换的：先走 a 再走 c，等于先走 c 再走 a），A 和 B 的正则矩阵就对易。人话：BB 码对易定理的全部力量，来自「环面上的平移可以换顺序」这一件事；这在 gross 里是 72 个平移两两可交换。

The monomial x^ay^b labels the group element (a, b) ∈ ℤ_l × ℤ_m, i.e. the translation “a steps along x, b steps along y”. Under the regular representation of §1.1, group multiplication (doing two translations in turn) becomes matrix multiplication. Because the group is abelian (commutative: a steps then c steps equals c steps then a steps), the regular matrices of A and B commute. In plain words: the whole force of the BB commutation theorem comes from the single fact that torus translations can be reordered; in gross that is 72 translations, pairwise interchangeable.

这条链就是 QEC-Bank 关系侧面唯一一条类型化关系的来历：special_case_of → two_bga，原文「BB 码是 ℤ_l × ℤ_m 的阿贝尔两块群代数码；一般的 2BGA 码可以用别的群，不必有二元多项式表示」，引 wang2023abelian（Renyu Wang、Hsiang-Ku Lin、Leonid P. Pryadko，"Abelian and Non-Abelian Quantum Two-Block Codes"，2023，arXiv 2305.06890）与 lin2023twoblock（Hsiang-Ku Lin、Leonid P. Pryadko，"Quantum Two-Block Group Algebra Codes"，2023，arXiv 2306.16400）——两个键据文献扫描报告，登记待补；目标条目 two_bga 本身在 QEC-Bank 里还是 pending（《BB 码 · 关系》讲了这一点）。三个包含为什么都不是等号：① 一个 2BGA 码可以用另一个群（比如非阿贝尔群）；② 非阿贝尔的构造可以让左、右作用对易，而完全没有两个变量的阿贝尔多项式环；③ 一般的提升积可以有比两块 BB 复形大得多的底矩阵（《qLDPC 码 · 高中数学版》4.2 节：BB 是底矩阵 1 × 1 的提升积，底矩阵一大就不是 BB 了）。人话：BB 是一个很特殊的角落；「两块」「群代数」「提升积」每往外走一步都放开一个限制，放开的东西就不再享有本页证过的性质。

This chain is the origin of the single typed relation on the QEC-Bank relations facet, special_case_of → two_bga, whose text reads “Bivariate bicycle codes are the abelian two-block group-algebra codes for ℤ_l × ℤ_m; general 2BGA codes may use other groups and need not have a two-variable polynomial presentation”, citing wang2023abelian (Renyu Wang, Hsiang-Ku Lin, Leonid P. Pryadko, “Abelian and Non-Abelian Quantum Two-Block Codes”, 2023, arXiv 2305.06890) and lin2023twoblock (Hsiang-Ku Lin, Leonid P. Pryadko, “Quantum Two-Block Group Algebra Codes”, 2023, arXiv 2306.16400); both keys are, per the scout report, registration pending, and the target entry two_bga is itself still pending in the QEC-Bank (“BB Codes · Relations” covers this). Why none of the three containments is an equality: (1) a 2BGA code may use another group, for instance a nonabelian one; (2) a nonabelian construction can make left and right actions commute without any two-variable abelian polynomial ring; (3) a general lifted product can have base matrices far larger than the two-block BB complex (“qLDPC Codes · High-School Edition”, §4.2: BB is the lifted product with a 1 × 1 base matrix; enlarge the base and it is no longer BB). In plain words: BB is a very special corner; each step outward to “two-block”, “group algebra” or “lifted product” releases one restriction, and whatever is released no longer enjoys the properties proved on this page.

映射 [B ; A] 与 [A | B] 是 R 线性R-linear：不只对 0/1 系数线性，对「乘以环 R 里任何元素」也线性。比如把输入平移一格（乘 x），输出也恰好平移一格。这正是 4.2 节平移自同构的来源。的，二者复合为零（1.5 节），于是定义了一个长度为二的复形。忘掉 R 线性，只把它们当 GF(2) 上的 0/1 矩阵，就得到二元 CSS 矩阵；保留 R 线性，就看见了平移与商模。两种描述在固定了同一组基与同一个正则表示时，保住的是同一个精确的码。

The maps [B ; A] and [A | B] are R-linearLinear not only over 0/1 coefficients but with respect to “multiply by any ring element of R”: translate the input one cell (multiply by x) and the output translates by exactly one cell too. This is where the translation automorphisms of §4.2 come from., and their composition vanishes (§1.5), so they define a length-two complex. Forget R-linearity, treating them as mere 0/1 matrices over GF(2), and you have the binary CSS matrices; keep it, and translations and quotient modules come into view. With the same bases and the same regular representation fixed, the two descriptions preserve the same exact code.

这里藏着一个对 QEC-Bank 复现契约至关重要的区别。换一组基（比如把比特重新编号、把左右两块对调、把行按别的顺序排）可以保住码空间——同一组量子态、同样的 n、k、d——却改变行序与比特序。而0.2 节的契约要求的是精确矩阵：验证器的匹配器叫 exact-checkmatrix-v1，比较整张 72 × 144 的数组。所以「我的构造和 QEC-Bank 的等价」不满足契约；「我的构造和 QEC-Bank 的逐元素相同」才满足。人话：链复形告诉你「这两个码是同一个码」；复现契约问的是「这两张表是同一张表吗」——问题不同，答案可以不同。

A distinction crucial to the QEC-Bank replay contract hides here. Changing bases (renumbering qubits, swapping the left and right blocks, ordering rows differently) can preserve the code space, the same set of quantum states, the same n, k, d, while changing the row and qubit order. But the contract of §0.2 demands the exact matrix: the verifier's matcher is named exact-checkmatrix-v1 and compares the whole 72 × 144 array. So “my construction is equivalent to the QEC-Bank one” does not satisfy the contract; “my construction agrees with the QEC-Bank one entry by entry” does. In plain words: the chain complex tells you “these two codes are the same code”; the replay contract asks “are these two tables the same table?”, a different question with a possibly different answer.

[A | B] 的像是理想 (A, B)——所有能写成 Au + Bv 的多项式。因此商 R/(A,B) 量的正是一个秩亏损：gross 里像的维数是 66，商的维数是 72 − 66 = 6（2.3 节）。因式分解与公共根的方法能预测这个商何时非零、它的维数怎样随周期或边界变化：postema2025existence（Jasper Johannes Postema、Servaas J. J. M. F. Kokkelmans，"Existence and Characterisation of Bivariate Bicycle Codes"，2025，arXiv 2502.17052，登记待补）用这套结构刻画 BB 码的存在条件与渐近极限；wang2026bbs（Renyu Wang、Leonid P. Pryadko，"Algebra of Bivariate-Bicycle Surface Codes"，2026，arXiv 2606.08771，登记待补）把根 / 边界分析推得更远（第 7 章会用到）。人话：A 和 B「共同碰不到」的那 6 个方向，就是逻辑信息的一半；换周期，比如把同样的 A、B 放到 9 × 6 的环面上，这个商就掉到 54 − 50 = 4，k 变成 8，那正是 bb108（直觉教程 5.5 节）。

The image of [A | B] is the ideal (A, B), every polynomial writable as Au + Bv. The quotient R/(A,B) therefore measures exactly one rank defect: in gross the image has dimension 66 and the quotient dimension 72 − 66 = 6 (§2.3). Factorization and common-root methods predict when this quotient is nonzero and how its dimension changes with periods or boundaries: postema2025existence (Jasper Johannes Postema, Servaas J. J. M. F. Kokkelmans, “Existence and Characterisation of Bivariate Bicycle Codes”, 2025, arXiv 2502.17052, registration pending) uses this structure to characterize existence conditions and asymptotic limitations, and wang2026bbs (Renyu Wang, Leonid P. Pryadko, “Algebra of Bivariate-Bicycle Surface Codes”, 2026, arXiv 2606.08771, registration pending) extends the root/boundary analysis (Chapter 7 uses it). In plain words: the 6 directions that A and B “jointly fail to reach” are half of the logical information; change the periods, say the same A, B on a 9 × 6 torus, and the quotient drops to 54 − 50 = 4, k becomes 8, and that is bb108 (intuition guide, §5.5).

A 或 B 的每一个非零元都是一条边，连一条检查与一个数据比特。gross 的 3 + 3 项给出重量 6 的检查，以及每个数据比特上 3 条 X 关联加 3 条 Z 关联（2.2 节）。数一数：H_X 有 72 行 × 6 个 1 = 432 条 X 型边，H_Z 同样 432 条，Tanner 图共 144 个数据节点、144 个检查节点、864 条边（这三个数是本页按 72 × 6 自己乘出来的，不在 QEC-Bank 页面上）。bravyi2024bicycle 证明这张图能拆成两个平面的度 3 耦合层（直觉教程 6.1 节）——这是硬件工程师最爱的视图，因为它直接说出需要多少根线。

Every nonzero entry of A or B is an edge joining one check to one data qubit. The gross 3 + 3 terms give weight-six checks and three X plus three Z incidences per data qubit (§2.2). Count them: H_X has 72 rows × 6 ones = 432 X-type edges, H_Z another 432, so the Tanner graph has 144 data nodes, 144 check nodes and 864 edges (these three numbers are this page's own 72 × 6 arithmetic, not on any QEC-Bank page). bravyi2024bicycle shows the graph splits into two planar degree-three coupling layers (intuition guide, §6.1), the hardware engineer's favourite view because it says directly how many wires are needed.

但这张图对一个故障电路是必要而不充分的。一条 Tanner 边说「这两个系统有交互」，它不规定：哪个辅助比特负责这条检查；交互发生在哪一时刻；CNOT 的控制端与目标端是谁；复位与测量的时序；哪些测量乘积是探测器。这五条正是3.2 节「QEC-Bank 刻意不做的事」的清单。shaw2025morphing 是最好的反例教材：它在周期中途改变瞬时的码与图，把度 6 降到度 5——所以「降低连接度」不是把同一张静态矩阵重画一遍，而是换了一个码在跑。人话：Tanner 图是电路的骨架照片，不是电路。

Yet the graph is necessary but insufficient for a fault circuit. A Tanner edge says “these two systems interact”; it does not specify which ancilla owns the check; when the interaction happens; which end is CNOT control and which target; reset and measurement timing; which measurement products are detectors. Those five items are exactly the list in §3.2, “what the QEC-Bank deliberately does not do”. shaw2025morphing is the best counter-example: it changes the instantaneous code and graph mid-cycle, lowering degree 6 to degree 5, so “reducing connectivity” is not a redrawing of the same static matrix but a different code running. In plain words: the Tanner graph is an X-ray of the circuit's skeleton, not the circuit.

chen2025anyon（Keyang Chen、Yuanting Liu、Yiming Zhang、Zijian Liang、Yu-An Chen、Ke Liu、Hao Song，"Anyon Theory and Topological Frustration of High-Efficiency Quantum LDPC Codes"，2025，arXiv 2503.04699，登记待补）把多项式 / KoszulKoszul 复形：由一个理想的生成元（这里是 A、B）机械地造出的链复形，1.5 节的 R → R² → R 就是它的两项版本；它把「理想」翻译成「激发的类型与可动性」。数据读成激发excitation：违反某些检查的局部图案，即「被点亮的检查」；在拓扑语言里叫任意子（anyon）。的类型、可动性约束与拓扑挫折。这幅图问的是：哪些缺陷能被局域操作移动，环面的全局约束怎样制造出逻辑扇区。（这里的哈密顿量是把全部检查加起来、前面带负号得到的能量函数，编码态就是它的最低能态；物理学家把码看成一种物质时谈的是它。）它的警告是本质性的：存在 k/2 = 6 个商维度，不等于哈密顿量就是 6 份互不相干的环面码。逻辑简并、局域 Clifford 相等价、译码器分解是三种不同的等价关系。

chen2025anyon (Keyang Chen, Yuanting Liu, Yiming Zhang, Zijian Liang, Yu-An Chen, Ke Liu, Hao Song, “Anyon Theory and Topological Frustration of High-Efficiency Quantum LDPC Codes”, 2025, arXiv 2503.04699, registration pending) reads the polynomial/KoszulThe Koszul complex: a chain complex built mechanically from the generators of an ideal (here A, B); the R → R² → R of §1.5 is its two-term version. It translates “ideal” into “excitation types and mobility”. data as excitationA local pattern violating some checks, i.e. “lit checks”; called anyons in topological language. types, mobility constraints and topological frustration. This picture asks which defects can be moved by local operators and how global torus constraints create logical sectors. (The Hamiltonian here is the energy function obtained by adding up all checks with a minus sign; the code states are its lowest-energy states, and it is what physicists study when they treat the code as a phase of matter.) Its caution is essential: the existence of k/2 = 6 quotient dimensions does not mean the Hamiltonian is 6 decoupled toric codes. Logical degeneracy, local-Clifford phase equivalence and decoder decomposition are three different equivalence relations.

最后一行值得多说一句。sahay2026matching 利用对称性暴露出类环面的圆柱，在上面做最小重量完美匹配，再与 BP 和过匹配结合。它保住的对象是「约化下选定的译码问题」：那部分症状与那部分逻辑歧义。这与拓扑挫折不矛盾：一个译码器可以把推断问题的一部分搬到熟悉的圆柱上，而不必造出一个把整个 BB 哈密顿量拆成环面码副本的有限深度局域酉变换。人话：把一道难题的一部分改写成会做的题，不等于证明这道题本来就是那道会做的题。

The last row deserves one more sentence. sahay2026matching uses symmetries to expose toric-like cylinders, runs minimum-weight perfect matching on them, then combines the result with BP and over-matching. The object it preserves is “the selected decoding problem under the reduction”: that part of the syndrome and that part of the logical ambiguity. This does not contradict topological frustration: a decoder may carry part of an inference problem onto familiar cylinders without producing a finite-depth local unitary that decomposes the whole BB Hamiltonian into toric copies. In plain words: rewriting part of a hard problem as one you can solve is not a proof that the hard problem was that easy one all along.

mackay2004sparse（David J. C. MacKay、Graeme Mitchison、Paul L. McFadden，"Sparse Graph Codes for Quantum Error-Correction"，2004，已登记）引入了用循环矩阵circulant：每一行都是上一行向右挪一格得到的矩阵；一个变量 x 的多项式在 ℤ_n 上的正则表示就是循环矩阵。结构拼出的稀疏量子 bicycle 码——一个变量、一个圈。panteleev2021degenerate（Pavel Panteleev、Gleb Kalachev，"Degenerate Quantum LDPC Codes With Good Finite Length Performance"，2021，已登记）发展了广义 bicycle 码、有限长度搜索与 BP-OSD。现代 BB 码把单一的循环方向换成二维阿贝尔群 ℤ_l × ℤ_m：这就是直觉教程 2.4 节「一个变量的祖先」到两个变量的那一步。精确的父亲是阿贝尔两块群代数码（wang2023abelian、lin2023twoblock，6.2 节）。「广义 bicycle」「2BGA」「提升积」「bivariate bicycle」四个名字在历史上和代数上互相重叠，但不是可以互换的类型名。人话：BB 是阿贝尔 2BGA 的特例，2BGA 又落在提升积之内；「广义 bicycle」是历史上的名字，与这几类互相重叠而不是严格套叠。

mackay2004sparse (David J. C. MacKay, Graeme Mitchison, Paul L. McFadden, “Sparse Graph Codes for Quantum Error-Correction”, 2004, registered) introduced sparse quantum bicycle codes assembled from circulantA matrix in which every row is the previous row shifted one place to the right; the regular representation of a one-variable polynomial over ℤ_n is a circulant. structure: one variable, one cycle. panteleev2021degenerate (Pavel Panteleev, Gleb Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, 2021, registered) developed generalized bicycle codes, finite-length searches and BP-OSD. Modern BB codes replace the single cyclic direction by the two-dimensional abelian group ℤ_l × ℤ_m: this is the step from “the one-variable ancestor” of intuition guide §2.4 to two variables. The exact parent is the abelian two-block group-algebra code (wang2023abelian, lin2023twoblock, §6.2). “Generalized bicycle”, “2BGA”, “lifted product” and “bivariate bicycle” overlap historically and algebraically but are not interchangeable type names. In plain words: BB is a special case of abelian 2BGA, and 2BGA sits inside the lifted products; “generalised bicycle” is a historical name that overlaps these classes rather than nesting strictly among them.

九个键里只有 steffan2025tile（Vincent Steffan、Seung-Hoon Choe、Nikolas P. Breuckmann、Francisco R. F. Pereira、Jens Niklas Eberhardt，"Tile Codes: High-Efficiency Quantum Codes on a Lattice with Boundary"，2025）在注册表里有完整元数据；其余八个据文献扫描报告，登记待补，下文给出的题名与 arXiv 号都来自扫描报告（抓取日期 2026-07-26）。人话：第四列是「换了这一项之后，账本上哪些格子要清零重填」——没有一条轴允许把 gross 的 144、12、12 原样抄过去。

Of the nine keys only steffan2025tile (Vincent Steffan, Seung-Hoon Choe, Nikolas P. Breuckmann, Francisco R. F. Pereira, Jens Niklas Eberhardt, “Tile Codes: High-Efficiency Quantum Codes on a Lattice with Boundary”, 2025) has full metadata in the registry; the other eight are, per the scout report, registration pending, and the titles and arXiv ids below come from that report (fetched 2026-07-26). In plain words: the fourth column is “which cells of the ledger must be wiped and refilled after this change”; no axis lets you copy gross's 144, 12, 12 across unchanged.

多变量。再加一个独立的循环变量 z，得到多变量 bicycle 码（voss2025multivariate：L. Voss、S. J. Xian、T. Haug、K. Bharti，"Multivariate Bicycle Codes"，2025，arXiv 2406.19151）。它仍是阿贝尔群代数的亲戚——群从 ℤ_l × ℤ_m 变成 ℤ_l × ℤ_m × ℤ_p——但不再是二元的。多出的方向可以改变局域维数与硬件嵌入，即使多项式的项数不变。因此一个检查重量同样是 6 的多变量候选，不是「参数表更长的 BB 实例」；构造器与受控关系应当指认它更大的代数。人话：三个变量的码住在三维的环面上，不能塞进本页两维的账本。

Multivariate. Add a third independent cyclic variable z and you have a multivariate bicycle code (voss2025multivariate: L. Voss, S. J. Xian, T. Haug, K. Bharti, “Multivariate Bicycle Codes”, 2025, arXiv 2406.19151). It remains an abelian group-algebra relative, the group growing from ℤ_l × ℤ_m to ℤ_l × ℤ_m × ℤ_p, but it is no longer bivariate. The extra direction can change locality dimension and hardware embedding even when the number of polynomial terms stays fixed. So a multivariate candidate with the same check weight 6 is not “a BB instance with a longer parameter list”; the constructor and the controlled relation should identify its larger algebra. In plain words: a three-variable code lives on a three-dimensional torus and does not fit into this page's two-dimensional ledger.

互素。wang2026coprime（Ming Wang、Frank Mueller，"Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms"，2026，arXiv 2408.10001，3.3 节）的「互素」说的是两个周期：要求 gcd(l, m) = 1l 与 m 的最大公约数为 1，即 l、m 互素（没有大于 1 的公因数），例如 l = 3、m = 2。，而不是对多项式的因子提要求。l、m 互素时，π = xy（向右一步再向上一步）走 lm 步才第一次回到起点，所以 π 的幂 1, π, …, π^lm−1 恰好把 lm 个格点各走一遍；于是 A、B 都能写成 π 的一元多项式 a(π)、b(π)。这样 k 有一条公式：令 g(π) = gcd(a(π), b(π), π^lm + 1)（三个多项式在 GF(2) 上的最大公因式），则 k = 2 deg g。A、B 可以有公共因子，而且正是这个公共因子 g 给出 k。手算一例：l = 3、m = 2，a = 1 + π + π²、b = 1 + π² + π⁴。在 GF(2) 上 b = (1 + π + π²)²、π⁶ + 1 = (1 + π)²(1 + π + π²)²，所以 g = 1 + π + π²，deg g = 2，k = 4。换回 x、y（π^j = x^{j mod 3}y^{j mod 2}）是 A = 1 + xy + x²、B = 1 + x² + x，构造器算出 n = 12、两个秩都是 4、k = 12 − 4 − 4 = 4，与公式一致（修订脚本 notes/bb_pedagogy/verify_review.py）。公式换来的是：不做秩计算就能先挑出想要的 k，再与冷原子版图协同设计。这是一个子类：每个互素 BB 码都是普通 BB 码，反过来不成立。gross 的 l = 12、m = 6，gcd = 6，不在这个子类里，子类的 k 公式不能套到 gross 上（gross 上 π = xy 走 12 步就回到起点，只走到 72 个格点里的 12 个）。

Coprime. In wang2026coprime (Ming Wang, Frank Mueller, “Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms”, 2026, arXiv 2408.10001, section 3.3), “coprime” refers to the two periods: it requires gcd(l, m) = 1The greatest common divisor of l and m is 1, i.e. l and m are coprime (no common factor larger than 1), e.g. l = 3, m = 2.; it puts no condition on the factors of the polynomials. When l and m are coprime, π = xy (one step right and one step up) first returns to its start after lm steps, so the powers 1, π, …, π^lm−1 visit each of the lm sites exactly once, and A and B can both be written as one-variable polynomials a(π), b(π). Then k has a formula: with g(π) = gcd(a(π), b(π), π^lm + 1) (the greatest common divisor of the three polynomials over GF(2)), k = 2 deg g. A and B may share factors, and that shared factor g is exactly what produces k. A worked example: l = 3, m = 2, a = 1 + π + π², b = 1 + π² + π⁴. Over GF(2), b = (1 + π + π²)² and π⁶ + 1 = (1 + π)²(1 + π + π²)², so g = 1 + π + π², deg g = 2, k = 4. Back in x, y (π^j = x^{j mod 3}y^{j mod 2}) this is A = 1 + xy + x², B = 1 + x² + x, and the constructor gives n = 12, both ranks 4, k = 12 − 4 − 4 = 4, matching the formula (revision script notes/bb_pedagogy/verify_review.py). What the formula buys: the desired k can be chosen before any rank computation, and the layout on cold atoms co-designed with it. This is a subclass: every coprime BB code is an ordinary BB code, not the other way round. Gross has l = 12, m = 6, gcd 6, so it is outside the subclass, and the subclass's k formula does not apply to gross (on gross, π = xy returns to its start after 12 steps and reaches only 12 of the 72 sites).

周期 BB 码把环面相对的两条边粘在一起（x^l = 1、y^m = 1 就是「粘」的代数写法）。删掉一些比特 / 检查，或者把一块反复出现的瓷砖在某处截断，就撕开了这些粘合。于是出现三个互相独立的问题：

A periodic BB code glues opposite torus edges together (x^l = 1, y^m = 1 are the algebraic form of the gluing). Removing qubits/checks, or terminating a repeated tile somewhere, tears those identifications open. Three separate questions then appear:

eberhardt2024pruning（Jens Niklas Eberhardt、Francisco Revson F. Pereira、Vincent Steffan，"Pruning qLDPC Codes: Towards Bivariate Bicycle Codes with Open Boundary Conditions"，2024，arXiv 2412.04181）的剪枝过程对循环 HGP 子情形有保证，在此之外只是演示了例子——那不是对任意 BB 多项式的定理。1.6 节的环面锚点 A = 1 + x、B = 1 + y 恰好具有 A 只含 x、B 只含 y 的形状，也就是两个长度 3 循环重复码的超图积（《qLDPC 码 · 高中数学版》3.1 节）；gross 的 A = x³ + y + y² 同时含 x 和 y，不具有这种形状。liang2025planar（Zijian Liang、Jens Niklas Eberhardt、Yu-An Chen，"Planar Quantum Low-Density Parity-Check Codes with Open Boundaries"，2025，arXiv 2504.08887）与 steffan2025tile 造出与 BB 体内图案相关的平面 / 开边界码，但共享一块瓷砖不决定共享一个逻辑空间。wang2026bbs 进一步说明为什么多项式在零或无穷处的根需要特殊的边界生成元，并区分了矩形、对角、倾斜三种边界方案——所以「开边界 BB」不是一种边界条件，而是一族。

eberhardt2024pruning (Jens Niklas Eberhardt, Francisco Revson F. Pereira, Vincent Steffan, “Pruning qLDPC Codes: Towards Bivariate Bicycle Codes with Open Boundary Conditions”, 2024, arXiv 2412.04181) guarantees its pruning procedure for the cyclic-HGP subcase and, beyond it, demonstrates examples; that is not a theorem for arbitrary BB polynomials. The toric anchor A = 1 + x, B = 1 + y of §1.6 has exactly the shape “A involves only x, B only y”, i.e. it is the hypergraph product of two length-3 cyclic repetition codes (“qLDPC Codes · High-School Edition”, §3.1); gross's A = x³ + y + y² involves both x and y and does not have that shape. liang2025planar (Zijian Liang, Jens Niklas Eberhardt, Yu-An Chen, “Planar Quantum Low-Density Parity-Check Codes with Open Boundaries”, 2025, arXiv 2504.08887) and steffan2025tile build planar/open codes related to BB bulk patterns, but sharing a tile does not determine sharing a logical space. wang2026bbs further shows why polynomial roots at zero or infinity require special boundary generators and distinguishes rectangular, diagonal and tilted prescriptions; “open BB” is therefore not one boundary condition but a family of them.

自对偶。普通 BB 已经有一个联系 X 与 Z 的「交换转置」形式，但那不自动给出一个能实现 H 或 S 的物理自对偶。liang2025selfdual（Zijian Liang、Yu-An Chen，"Self-Dual Bivariate Bicycle Codes with Transversal Clifford Gates"，2025，arXiv 2510.05211）在扭环面twisted torus：粘合时带一个错位（绕 x 方向一圈回来时 y 坐标平移若干格）的环面；改变了 x、y 的周期关系。上加额外约束，报告了带横向 Clifford 门的重量 8 例子（扫描报告列出 [[16,4,4]] 与更大的有限块）。两个推论必须分开：① CSS 的转置对称帮助证明对易——这是已发布构造的通性（1.4 节）；② 物理自对偶加上解出的逻辑作用可以认证横向 H/S——这只属于被引的子族（4.5 节）。人话：「H_Z 长得像 H_X 的转置」是所有 BB 的性质；「对每个比特做 H 是一个逻辑门」只是某些 BB 的性质。

Self-duality. Ordinary BB already has a swapped-transpose form relating X and Z, but that does not automatically give a physical self-duality implementing H or S. liang2025selfdual (Zijian Liang, Yu-An Chen, “Self-Dual Bivariate Bicycle Codes with Transversal Clifford Gates”, 2025, arXiv 2510.05211) imposes extra constraints on a twisted torusA torus glued with an offset (going once around x shifts the y coordinate by some cells); it changes the period relations of x and y. and reports weight-eight examples with transversal Clifford gates (the scout report lists [[16,4,4]] and larger finite blocks). Two implications must stay separate: (1) the CSS transpose symmetry helps prove commutation, generic to the shipped construction (§1.4); (2) physical self-duality plus a resolved logical action can certify transversal H/S, belonging only to the cited subfamily (§4.5). In plain words: “H_Z looks like the transpose of H_X” is a property of every BB code; “H on every qubit is a logical gate” is a property of some.

qudit。liang2026qudit（Zijian Liang、Yu-An Chen，"Generalized Z_p Toric Codes as Qudit Low-Density Parity-Check Codes"，2026，arXiv 2602.20158）把扭环面的想法推到素数维 qudit。qudit 稳定子用的是 GF(p) 上的辛形式symplectic form over GF(p)：两个 qudit Pauli 对易与否由 Σ(a_ib'_i − a'_ib_i) mod p 是否为零决定；p = 2 时退化为本页一直在用的二元 XOR 规则。，不是二元的 XOR；它的 k 数的是逻辑 p 能级系统，硬件错误信道有不同的 Pauli 类型与归一化。不写明 p 就跨 qubit / qudit 比较 kd²/n，是把资源单位混在一起。将来的 qudit QEC-Bank 条目应当用一个真正验证 GF(p) 对易的引擎，而不是把数据硬塞进二元构造器。人话：一个 p = 3 的 qudit 存的是「三选一」，一个 qubit 存的是「二选一」，两边的 n、k、d 不是同一种数。

Qudits. liang2026qudit (Zijian Liang, Yu-An Chen, “Generalized Z_p Toric Codes as Qudit Low-Density Parity-Check Codes”, 2026, arXiv 2602.20158) extends the twisted-torus ideas to prime-dimensional qudits. A qudit stabilizer uses a symplectic formOver GF(p): two qudit Paulis commute iff Σ(a_ib'_i − a'_ib_i) mod p vanishes; at p = 2 it reduces to the binary XOR rule used throughout this page. over GF(p), not binary XOR; its k counts logical p-level systems, and its hardware error channel has different Pauli types and normalization. Comparing kd²/n across qubit and qudit examples without stating p mixes resource units. A future qudit QEC-Bank entry should use an engine that actually validates GF(p) commutation rather than forcing the data through the binary constructor. In plain words: a p = 3 qudit stores “one of three”, a qubit “one of two”; their n, k, d are not the same kind of number.

搜索产生的扰动。cruzbenito2026evolutionary（Juan Cruz-Benito、Andrew W. Cross、David Kremer、Ismael Faro，"Evolutionary Discovery of Bivariate Bicycle Codes with LLM-Guided Search"，2026，arXiv 2606.02418）把程序变异、GF(2) 秩、码距认证、图去重与局域 Clifford 检查串成一条流水线；一些候选离开了严格的 CSS BB 拟设。一旦 X 与 Z 混进同一行稳定子，定义对象就变成一张完整的辛检查矩阵，本页第 1 章的 CSS 多项式对易证明不再适用——对易要逐对按辛内积检查。Tanner 图相似、或者出身于一次 BB 搜索，都不足以保留严格的家族标签。人话：从 BB 出发搜到的码，未必还是 BB。

Search-generated perturbations. cruzbenito2026evolutionary (Juan Cruz-Benito, Andrew W. Cross, David Kremer, Ismael Faro, “Evolutionary Discovery of Bivariate Bicycle Codes with LLM-Guided Search”, 2026, arXiv 2606.02418) chains program mutation, GF(2) rank, distance certification, graph deduplication and local-Clifford checks into one pipeline; some candidates leave the strict CSS BB ansatz. Once X and Z are mixed within a stabilizer row, the defining object becomes a full symplectic check matrix and the CSS polynomial commutation proof of Chapter 1 no longer applies; commutation must be checked pair by pair via the symplectic inner product. Similarity of a Tanner graph, or ancestry in a BB search, is not enough to keep the strict family label. In plain words: a code found by searching outward from BB is not necessarily still BB.

postema2025existence 对所研究的 BB 子类证明了渐近的坏：固定这套构造约束，不能同时得到非零的渐近码率与非零的相对码距（《qLDPC 码 · 高中数学版》4.4 节「阿贝尔的坏，非阿贝尔的好」；直觉教程 1.1 节的 BPT 定理是它的近亲）。这个结果与 gross 这个强有力的有限点并存：有限搜索能找到有价值的高码率、中等码距的存储器；而固定的构造约束不必在 n → ∞ 时保持这种好。所以正确的产品说法是「在特定连接模型下强有力的有限常数」，而不是「来自二维局域环面的渐近好 qLDPC」。人话：gross 是一个好的点，不是一条好的曲线；k = 12、d = 12 是 n = 144 处的成绩，不是一个随 n 增长的承诺。

postema2025existence proves asymptotic badness for the studied BB subclass: under fixed construction constraints one cannot have nonzero asymptotic rate and nonzero relative distance simultaneously (“qLDPC Codes · High-School Edition”, §4.4, “abelian is bad, nonabelian is good”; the BPT theorem of intuition guide §1.1 is its close relative). That result coexists with the strong finite point gross: finite search can find a valuable high-rate, moderate-distance memory, while the fixed construction constraints need not keep that goodness as n → ∞. The correct product claim is therefore “strong finite constants under a specific connectivity model”, not “asymptotically good qLDPC from a two-dimensional local torus”. In plain words: gross is a good point, not a good curve; k = 12, d = 12 is a score at n = 144, not a promise that grows with n.

任何一问答「否」，新对象就需要自己的可执行实例和自己的出处账本——也就是 QEC-Bank 对 bb72、bb90、bb108、bb288、bb5、bicycle 各开一个条目的原因（第 0 章的范围说明）。人话：名字像不能当证据，账要一块一块地算。

If any answer is “no”, the new object needs its own executable instance and its own provenance ledger, which is why the QEC-Bank gives bb72, bb90, bb108, bb288, bb5 and bicycle entries of their own (the scope note of Chapter 0). In plain words: a similar name is not evidence; the ledger is settled block by block.

回放整条链。循环移位 S_l 与克罗内克积给出 x、y，x^l = I、y^m = I、xy = yx；置换矩阵的转置是它的逆，所以 (x^ay^b)ᵀ 是反向平移；H_X = [A | B]、H_Z = [Bᵀ | Aᵀ] 的乘积是 AB + BA = 0，转置负责对偶、交换负责配对，不交换则得到 A² + B²（gross 里每行 6 个 1、秩 48）；同一算式读成 R → R² → R 的复形；A = 1 + x、B = 1 + y 在 l = m = 3 上给出 [[18,2,3]] verified（第 1 章）。四个输入 l = 12、m = 6、A = x³ + y + y²、B = y³ + x + x² 算到 72 × 144、重量 6、度 3 + 3、秩 66 与 66、商维数 6、k = 12；d = 12 由文献下界与种子 5 的机器上界夹出，标 bounded；144 是数据比特数，加 144 个辅助比特是 288；静态 12 不是电路码距（第 2 章）。静态症状 s = H_Ze_X 由构造器重算；六腿测量是一段电路，探测器是跨时间的比较，三个「码距」不能混；BP-OSD 是一个带十来个参数的算法；0.8% 是一个协议阈值，带五个「不是」；QEC-Bank 不导出任何调度（第 3 章）。横向 CNOT 在纸上证到码空间；平移是自同构，它的 24 × 24 辛作用 QEC-Bank 没有算（Bravyi 等人算出是逻辑 CNOT 电路）；折叠、自同构、手术、自对偶各有证书形状，五行全是 claimed，没有普适门集（第 4 章）。唯一一次真机实验是昆仑的 [[18,4,4]]、32 个 transmon、(8.91 ± 0.17)%，标 demonstrated，确立可行性而非低于阈值；七个提案各攻不同瓶颈（第 5 章）。七种表示各保住一个不变量、各禁止一种推理；八条视图断言里两条 derived、一条 canonical、五条 curated（第 6 章）。七条变体轴、三个开边界问题、一条安全工作流、六问清单；gross 是一个好点而不是一条好曲线（第 7 章）。

Replay the whole chain. The cyclic shift S_l and the Kronecker product give x, y with x^l = I, y^m = I, xy = yx; a permutation matrix's transpose is its inverse, so (x^ay^b)ᵀ is the inverse translation; the product of H_X = [A | B] and H_Z = [Bᵀ | Aᵀ] is AB + BA = 0, the transpose supplying duality and the swap the pairing, and without the swap one gets A² + B² (6 ones per row and rank 48 in gross); the same calculation reads as the complex R → R² → R; A = 1 + x, B = 1 + y at l = m = 3 gives [[18,2,3]] verified (Chapter 1). Four inputs, l = 12, m = 6, A = x³ + y + y², B = y³ + x + x², compute to 72 × 144, weight 6, degree 3 + 3, ranks 66 and 66, quotient dimension 6, k = 12; d = 12 is pinned between a literature lower bound and the seed-5 machine upper bound, hence bounded; 144 counts data qubits, 288 with 144 ancillas; the static 12 is not a circuit distance (Chapter 2). The static syndrome s = H_Ze_X is recomputed by the constructor; a weight-six measurement is a circuit, detectors are comparisons across time, three “distances” must not be conflated; BP-OSD is an algorithm with a dozen parameters; 0.8% is a protocol threshold with five “is nots”; the QEC-Bank exports no schedule (Chapter 3). Transversal CNOT is proved on paper to the code space; a translation is an automorphism whose 24 × 24 symplectic action the QEC-Bank has not computed (Bravyi et al. computed logical CNOT circuits); fold, automorphism, surgery and self-dual gates each have a certificate shape, all five rows are claimed, and there is no universal gate set (Chapter 4). The only hardware run is Kunlun's [[18,4,4]] on 32 transmons at (8.91 ± 0.17)%, labelled demonstrated, establishing feasibility rather than below-threshold operation; seven proposals attack different bottlenecks (Chapter 5). Seven representations each preserve one invariant and forbid one inference; of eight view assertions two are derived, one canonical, five curated (Chapter 6). Seven variant axes, three open-boundary questions, one safe workflow, a six-question checklist; gross is a good point, not a good curve (Chapter 7).

QEC-Bank 的直觉短文 intuition.md 以五道问题收尾，用来检验读者是否真的掌握了这个条目。先自己答，再展开对照；每个答案都指回本页证明它的那一节。

The QEC-Bank intuition essay intuition.md closes with five questions meant to test whether a reader has really mastered this entry. Answer first, then unfold and compare; each answer points back to the section of this page that proves it.