BB 码 · 条目总览BB Codes · Entry Overview
QEC-Bank 条目导读 · BB 码（双变量自行车码）QEC-Bank entry guide · Bivariate bicycle codes

BB 码：
两个多项式的十一个侧面 BB codes:
two polynomials, eleven facets

码距 12 的表面码（12 × 12 的旋转版）用 144 个数据比特只存 1 个逻辑比特；BB 码家族里的 gross 码用同样 144 个比特存 12 个，码距也是 12。它的全部检查由两个多项式 A、B 写出，多项式的每一项是 12 × 6 环面上的一次平移（例如 x³ = 向右走 3 格，y = 向上走 1 格）；怎样从这些平移读出每条检查碰哪六个比特（每格两个比特分左右两层、环面首尾相接、Z 检查倒着走），直觉篇 3.1 节用带标注的图一步步讲。QEC-Bank（本系列依托的纠错码知识库）的 bivariate_bicycle 条目记录这一族码，发布了两个实例：gross 码 [[144,12,12]]，以及 3 × 3 环面上的小码 toric_L3 [[18,2,3]]，它小到可以手算，用来核对 QEC-Bank 的编号约定和论文一致。这一页是该条目的目录：十一个侧面各一篇互动教程，从直觉一直到参考文献，每个数字旁都写着它的证据词（说明它是机器重算的、有上下界的，还是文献声称的，见下方第 1 章）。 A surface code of distance 12 (the rotated 12 × 12 version) spends 144 data qubits on a single logical qubit; the gross code, one member of the BB family, stores 12 in the same 144 qubits, also at distance 12. All of its checks are written with two polynomials A and B, each term of which is one shift on a 12 × 6 torus (for example x³ = three cells to the right, y = one cell up); how the six qubits of each check are read off from these shifts (two qubits per cell in a left and a right layer, the torus wrapping around, Z checks stepping backwards) is shown step by step, with labelled figures, in §3.1 of the intuition guide. The bivariate_bicycle entry of the QEC-Bank (the knowledge base of codes this series is built on) records this family and publishes two instances: the gross code [[144,12,12]], and toric_L3 [[18,2,3]] on a 3 × 3 torus, small enough to work by hand, which checks that the QEC-Bank's numbering conventions agree with the paper's. This page is the entry's table of contents: one interactive tutorial per facet, from intuition to references, with an evidence word next to every number (saying whether a machine recomputed it, only bounds are known, or a paper claims it; see chapter 1 below).

从哪里开始：先读直觉篇（高中数学版，需要先读过 Surface Code）；查数字、查引用直接去对应侧面。QEC-Bank 原始条目页：在 QEC-Bank 中查看 → Where to start: the intuition guide (high-school edition, which assumes the surface-code guide); to check a number or a citation, go straight to that facet. The QEC-Bank entry itself: Open in the QEC-Bank →
选择侧面 ↓Choose a facet ↓ 系列目录：全部码与教程 →　对照阅读：Surface Code 条目总览 →Series index: all codes and tutorials →　Compare: surface-code entry overview →
目录Contents
十一个侧面Eleven facets

探索这个码Explore this code

选你想回答的问题。高中数学版用矩阵、秩、多项式讲；条目导读假定你已读过直觉篇，直接对照 QEC-Bank 的数据、证书和文献键（指向一篇登记过的论文的短字符串）来讲，每个数字都带证据词：verified = 机器独立重算、结果相符；bounded = 只登记了上界和下界（对 gross 的码距：上界由机器找到的例子给出，下界引自论文）；claimed = 只有文献声称，QEC-Bank 没有重算；demonstrated = 有实验论文在真实硬件上报告过；proposed = 设计具体到可以模拟或估算成本，但还没有硬件执行过。 Choose the question you want answered. The high-school guide uses matrices, rank and polynomials; the entry guides assume it and walk the QEC-Bank's data, certificates and reference keys (short strings that point to a registered paper) directly, with an evidence word on every number: verified = recomputed independently by a machine, and it matched; bounded = only an upper and a lower bound are recorded (for the gross distance: the upper bound from an example a machine found, the lower bound cited from the paper); claimed = only a paper says so, the QEC-Bank has not recomputed it; demonstrated = an experimental paper has reported it on real hardware; proposed = the design is concrete enough to simulate or cost, but no hardware has run it.

  • 直觉Intuition→ 为什么要 BB：Bravyi、Poulin、Terhal 的定理（arXiv:0909.5200）说，比特在平面上分布的密度有上限、每条检查只碰一定距离内的比特时，k·d² ≤ c·n，其中 c 是只取决于这两个上限的常数；表面码在常数因子以内达到这个上限（k 块独立的表面码约用 k·d² 个比特）。然后从零推导：单项式是环面上的一步平移，多项式是几步平移盖在一起的「印章」，由它直接读出每条检查碰哪几个比特；一行代数证明所有检查对易；逻辑比特数从哪来（gross 的 72 条 X 检查里只有 66 条互相独立，72 条 Z 检查也是，所以 k = 144 − 66 − 66 = 12，也就是 2 × (72 − 66)）；八个重要的码逐个讲；最后是从矩阵到机器要付的账：长程连线、深度 7 的测量电路、译码、0.8% 这个数是什么不是什么（见下方说明）、逻辑门。Why BB: the theorem of Bravyi, Poulin and Terhal (arXiv:0909.5200) says that when the qubits sit in the plane with bounded density and every check touches only qubits within a bounded distance, k·d² ≤ c·n, where c is a constant that depends only on those two bounds; the surface code reaches this limit up to a constant factor (k independent surface-code patches use about k·d² qubits). Then a derivation from zero: a monomial is one shift step on the torus, a polynomial is a “stamp” of several shifts, and it tells you directly which qubits each check touches; one line of algebra shows that all checks commute; where the logical qubits come from (only 66 of gross's 72 X checks are independent, and likewise 66 of its 72 Z checks, so k = 144 − 66 − 66 = 12, which is 2 × (72 − 66)); eight important codes one by one; and last the bill between a matrix and a machine: long-range wiring, the depth-7 measurement circuit, decoding, what the 0.8% number is and is not (see the note below), logical gates. 高中数学high-school
  • 导览Guided tour→ 导览是 QEC-Bank 给每个条目配的固定路线：不讲新理论，只把程序导出的真实数据一站站指给你看。八站：144 个比特的坐标由哪条公式算出；第 0 条 X 检查碰到的六个比特 {1, 2, 18, 75, 78, 84} 各由哪一步平移得到；Z 检查为什么要「倒着走、换位置」才和所有 X 检查对易；缩到 3 × 3 环面上的 [[18,2,3]] 用手核对；72 条 X 检查为什么只有 66 条独立；以及导览不讲的东西（电路和门）。A guided tour is the fixed route the QEC-Bank attaches to each entry: no new theory, just the real data the program exported, pointed out stop by stop. Eight stops: which formula gives the coordinates of the 144 qubits; which shift step reaches each of the six qubits {1, 2, 18, 75, 78, 84} touched by X check 0; why the Z checks must “walk backwards and swap places” to commute with every X check; shrinking to [[18,2,3]] on a 3 × 3 torus to check by hand; why only 66 of the 72 X checks are independent; and what the tour leaves out (circuits and gates). 条目导读entry guide
  • 表示阶梯Representation ladder→ 同一对多项式的八种写法：两张 72 × 144 的 0/1 检查矩阵；一条链复形（把比特和两族检查用矩阵串起来的代数写法）；一张 864 条边的 Tanner 图把比特和检查都画成点，某条检查碰到某个比特，就在两点之间连一条线。；以及二元多项式、群代数、商模、任意子四本代数「词典」。QEC-Bank 按机器能多直接地执行，把它们排成 R1 到 R5 五级，并标明每条是机器算出来的、照公认定义写的，还是从论文整理来的。R3 这一级（按时间顺序排好的测量）目前是空的，页面照实说明为什么。Eight ways of writing the same pair of polynomials: two 72 × 144 0/1 check matrices; a chain complex (an algebraic form that links the qubits and the two check families with matrices); a Tanner graphDraw every qubit and every check as a dot, and join a check to each qubit it touches with a line. with 864 edges; and four algebraic “dictionaries”: bivariate polynomials, the group algebra, a quotient module, anyons. The QEC-Bank sorts them into five levels, R1 to R5, by how directly a machine can execute them, and marks each as computed by a machine, written from the accepted definition, or collected from papers. Level R3 (measurements laid out in time order) is empty for now, and the page says why. 条目导读entry guide
  • 参数与实例Parameters & instances→ 四个旋钮：环面的宽 l、高 m，以及多项式 A、B 各有哪几项。n = 2lm（每个格子两个比特）；k = 2(lm − r)，r 是 X 检查矩阵的秩，即其中互相独立的检查条数（Z 检查的秩也是 r，所以 k = 2lm − r − r；gross 为 144 − 66 − 66 = 12）；gross [[144,12,12]] 与 toric_L3 [[18,2,3]] 每个数字旁的证据词；为什么 gross 的码距是 bounded（上界 12 由机器的随机搜索每次重现，下界 12 引自论文）；算上测量用的帮手比特共 288 个；以及「十二倍」这个说法比的究竟是什么。Four knobs: the torus width l and height m, and which terms the polynomials A and B contain. n = 2lm (two qubits per cell); k = 2(lm − r), where r is the rank of the X-check matrix, the number of independent checks in it (the Z checks have rank r too, so k = 2lm − r − r; for gross 144 − 66 − 66 = 12); the evidence word beside every number for gross [[144,12,12]] and toric_L3 [[18,2,3]]; why the gross distance is bounded (the upper bound 12 is reproduced every time by the machine's random search, the lower bound 12 is cited from the paper); 288 qubits once the helper qubits for measurement are counted; and what exactly the “twelve times” claim compares. 条目导读entry guide
  • 形式化 IR 与证书Formal IR & certificates→ 「verified」是哪台机器、拿什么、对哪个对象核对的：gross 与 toric_L3 各自的身份编号（由矩阵算出的 64 位十六进制哈希，改一个比特就会变）；QEC-Bank 怎样只凭四个变量和两个版本号把两张矩阵一比特不差地重新拼出来；为什么 toric_L3 有精确码距证书而 gross 没有（一个算术原因，一个策略原因）；证书不覆盖哪些东西；以及怎样自己重跑。证书、哈希、证明检查程序 Lean 都在第 1 章从零讲起。Which machine “verified” something, with what, and against which object: the identity numbers of gross and toric_L3 (a 64-hex-digit hash computed from the matrix, which changes if a single bit changes); how the QEC-Bank rebuilds both matrices bit for bit from just four variables and two version numbers; why toric_L3 has an exact-distance certificate and gross does not (one arithmetic reason, one policy reason); what the certificates do not cover; and how to rerun it all yourself. Certificates, hashes and the proof checker Lean are all taught from zero in chapter 1. 条目导读entry guide
  • 译码器Decoders→ 测量本身不出错时，表面码内部的一个 X 或 Z 数据比特错误点亮两个警报（贴着边界的比特可能只点亮一个），把警报两两配对（或连到边界）就是修复；gross 码里一个 X 或 Z 错误点亮三个（Y 错误是 X、Z 同时发生，点亮六个），「配对」失去意义。本篇讲置信传播（BP：比特和检查之间来回传「我错了吗」的消息）怎样工作；它为什么会在短圈和退化许多个不同的错误只差一些检查的乘积，对信息的效果完全相同；译码器不必分清它们，但 BP 会因此在几个答案之间摇摆。上卡住；有序统计译码（OSD）怎样按可靠度排序、解方程收尾；QEC-Bank 登记的三条译码记录各用什么噪声模型，包括 0.8% 这个数的来历（见下方说明）；最后说明为什么电路里的码距不是静态的 12。With error-free measurements, a single X or Z error on an interior data qubit of the surface code lights two alarms (a qubit on the boundary may light only one), and pairing the alarms (or joining them to the boundary) is the repair; in the gross code a single X or Z error lights three (a Y error, X and Z at once, lights six), and “pairing” stops making sense. This guide covers how belief propagation (BP: bits and checks passing “am I wrong?” messages back and forth) works; why it gets stuck on short cycles and on degeneracyMany different errors differ only by a product of checks and have exactly the same effect on the information; a decoder need not tell them apart, but BP can swing between several answers because of them.; how ordered-statistics decoding (OSD) finishes by sorting bits by reliability and solving equations; which noise model each of the three decoding records in the QEC-Bank uses, including where the 0.8% number comes from (see the note below); and last, why the distance inside a circuit is not the static 12. 条目导读entry guide
  • 逻辑操作Logical operations→ 12 个逻辑比特挤在一块里，门从哪里来。QEC-Bank 登记了五条逻辑操作，证据词都是 claimed：两块相同 BB 码之间的横向 CNOT（两块的对应比特一对一各做一次 CNOT）、对称实例上的折叠横向 Clifford 门一类容易容错实现的门，包括 H、S、CNOT。只用它们不能做任意的量子计算，还需要 T 这样的门。、由自同构（保持所有检查不变的比特重排）导出的 Clifford 门、块间 Pauli 测量与手术、一个自对偶子族的 H/S/CNOT。本篇逐条读它们：环面平移保持全部检查，但光凭这一点说不出它对 12 个逻辑比特做了什么：只给坐标改名什么也没做，真在芯片上把每个比特的状态按平移换到另一处才是一个操作；Bravyi 等人（arXiv:2308.07915 v2 附录）算出，平移 s 把编号为 n 的逻辑 X̄ 送到编号为 s·n 的逻辑 X̄（Z̄ 同理），所以除了原地不动，每个平移都是一段非平凡的逻辑 CNOT 电路（差一个逻辑 Pauli 修正），作者也说还不清楚怎样用它们做有用的计算，主要拿它们把选中的逻辑比特挪到能被测量的位置；为什么「自对偶子族有横向 H」不能推到 gross 头上；以及 T 门为什么仍要靠魔法态事先制备并提纯的一种特殊量子态，用一段只含 Clifford 门和测量的电路把它「注入」，就实现了一个 T 门。。Twelve logical qubits in one block: where do the gates come from? The QEC-Bank records five logical operations, all with the evidence word claimed: transversal CNOT between two identical BB blocks (one CNOT on each pair of corresponding qubits), fold-transversal Clifford gatesA class of gates that are easy to implement fault-tolerantly, including H, S and CNOT. They alone cannot perform arbitrary quantum computation; a gate such as T is also needed. on symmetric instances, Clifford gates derived from automorphisms (reorderings of the qubits that keep every check), inter-block Pauli measurement and surgery, and H/S/CNOT in one self-dual subfamily. This guide reads them one by one: a torus shift keeps every check, but that alone does not say what it does to the 12 logical qubits: renaming the coordinates does nothing, while actually moving each qubit's state along the shift on the chip is an operation; Bravyi et al. (arXiv:2308.07915 v2, supplement) work out that a shift s sends the logical X̄ labelled n to the X̄ labelled s·n (likewise Z̄), so every shift other than staying put is a nontrivial logical CNOT circuit (up to a logical Pauli correction); the authors add that it is not clear how to use these for useful computation and use them mainly to move a chosen logical qubit to where it can be measured; why “the self-dual subfamily has a transversal H” cannot be carried over to gross; and why the T gate still needs magic statesA special quantum state prepared and purified in advance; “injecting” it with a circuit of only Clifford gates and measurements performs a T gate.. 条目导读entry guide
  • 用例与实现Use cases & realizations→ BB 的检查要伸到几格之外，所以芯片得有能跨格连线的超导耦合器、能被光镊搬动的中性原子，或者把码分到几台用量子链路相连的处理器上。这份 QEC-Bank 快照只登记了一次 demonstrated 记录：昆仑超导处理器上的 [[18,4,4]]，每个逻辑比特每轮的逻辑错误率 (8.91 ± 0.17)%。这个数是 X 基与 Z 基两种存储实验各自拟合出的值（8.67% 与 9.15%）的平均，译码器是按电路级去极化噪声设置的 BP-OSD，± 是统计与拟合不确定度合起来的一个标准差（Wang 等人，arXiv:2505.09684）。快照之后，2026 年 6 月 IonQ 的团队在离子阱上演示了三个 BB5 码（每条检查碰 5 个比特）[[18,4,3]]、[[24,4,4]]、[[30,4,5]]（Tham 等人，arXiv:2606.06455），QEC-Bank 还没收录（其中 [[30,4,5]] 与 QEC-Bank 的 bb5 条目参数相同，本页没有核对两者是不是同一个码）；另有四条 proposed（设计具体到可以模拟，但还没有硬件执行过）的方案，本篇说清各在哪一层成立。中间一章教你用五个字段读任何一条 BB 实验记录。BB checks reach several cells away, so the chip needs superconducting couplers that can span cells, neutral atoms that optical tweezers can move, or the code split across several processors joined by quantum links. This QEC-Bank snapshot records one demonstration: [[18,4,4]] on the Kunlun superconducting processor, with a logical error rate of (8.91 ± 0.17)% per logical qubit per round. That number averages the rates fitted separately for the X-basis and Z-basis memory experiments (8.67% and 9.15%), decoded with BP-OSD set up for a circuit-level depolarizing noise model; the ± is one standard deviation of statistical and fit uncertainty (Wang et al., arXiv:2505.09684). After the snapshot, in June 2026, an IonQ team demonstrated three BB5 codes (every check touches 5 qubits), [[18,4,3]], [[24,4,4]] and [[30,4,5]], on a trapped-ion machine (Tham et al., arXiv:2606.06455); the QEC-Bank has not added them yet (their [[30,4,5]] has the same parameters as the QEC-Bank bb5 entry; this page has not checked whether the two are the same code); there are also four proposed designs (concrete enough to simulate, not yet run on hardware), and this guide says at which layer each one holds. One chapter teaches you to read any BB experiment record with five fields. 条目导读entry guide
  • 关系Relations→ BB 码在 QEC-Bank 的地图上和谁相连。QEC-Bank 只登记了一条关系：BB 码是「两块群代数码」在可交换群上的特例，而那一端的条目还没建（pending）。然后是家谱：2004 年一个变量的 bicycle 码 → 广义 bicycle → 两个变量的 BB → 群代数码与提升积，每一代都用 QEC-Bank 里真有的实例 [[6,2,2]]、[[18,2,3]]、[[144,12,12]] 算一遍；QEC-Bank 里一张家族卡和七张成员卡是什么关系；最后七根「变体轴」（多变量、剪开环面加边界、自对偶、qudit 等），每改一根，哪些数字要重算。Who BB codes are connected to on the QEC-Bank map. The QEC-Bank records exactly one relation: BB codes are the commutative-group special case of “two-block group-algebra codes”, and the entry at the other end has not been built yet (pending). Then the family tree: the one-variable bicycle code of 2004 → generalised bicycle → the two-variable BB → group-algebra codes and lifted products, each generation worked out on an instance the QEC-Bank really has, [[6,2,2]], [[18,2,3]], [[144,12,12]]; how one family card relates to seven member cards in the QEC-Bank; and last, seven “variant axes” (more variables, cutting the torus open to add boundaries, self-duality, qudits and others): change one, and which numbers must be recomputed. 条目导读entry guide
  • 推导与指南Derivations & guides→ QEC-Bank 给每个码写两层文字：直觉短文，和「推导页」（Level-2：每页只回答一个问题、把证明做完，并逐句标明哪句是代数、哪句是机器重算、哪句是文献）。BB 条目有七张推导页，本篇一页一章：多项式怎样变成一对对易的检查矩阵；gross 的 72 × 144、秩 66、k = 12、d = 12 各从哪来；症状电路与译码为什么是五个不同的对象；逻辑门各自的证书；真机做到了哪一步；七种表示各保住什么；变体地图上哪条线不能跨。外加复现契约与自测题。The QEC-Bank writes two layers of text for each code: a short intuition piece, and “derivation pages” (Level-2: each page answers one question, finishes the proof, and labels every sentence as algebra, machine recomputation or literature). The BB entry has seven derivation pages, and this guide gives each a chapter: how the polynomials become a commuting pair of check matrices; where gross's 72 × 144, rank 66, k = 12 and d = 12 each come from; why syndrome circuits and decoding are five different objects; the certificate behind each logical gate; how far hardware has got; what each of seven representations preserves; which lines on the variant map cannot be crossed. Plus the reproduction contract and a self-test. 条目导读entry guide
  • 参考文献References→ QEC-Bank 里凡不是机器重算的数字（0.8% 这个伪阈值、昆仑实验的 8.91%、五条逻辑门……）都挂在一个文献键上：一个指向登记过的论文的短字符串。BB 条目有 32 个键，只有 7 个登记了标题、作者、年份和 arXiv 链接，其余 25 个还显示 metadata pending（待登记）。本篇讲怎样读一个键；7 条各替哪条说法作证；25 条按引用它的侧面分组，并给出文献扫描报告里的标题与 arXiv 号；以及怎样顺着一个键找到它在 card.yaml 里支撑的那句话。Every number in the QEC-Bank that a machine did not recompute (the 0.8% pseudo-threshold, the 8.91% from the Kunlun experiment, the five logical gates…) hangs on a reference key: a short string pointing to a registered paper. The BB entry has 32 keys; only 7 have a title, authors, year and arXiv link on file, and the other 25 still show metadata pending. This guide covers how to read a key; which statement each of the 7 vouches for; the 25 grouped by the facet that cites them, with the title and arXiv number from the literature-scan report; and how to follow a key to the sentence it supports in card.yaml. 条目导读entry guide
范围Scope

一个条目，八个亲戚One entry, eight relatives

这里的十一篇讲的是 QEC-Bank 的家族条目 bivariate_bicycle，它交付两个可执行实例：主实例 gross 码 [[144,12,12]]（l = 12、m = 6，A = x³ + y + y²，B = y³ + x + x²）和用来核对编号约定的环面锚点 toric_L3 [[18,2,3]]。Bravyi 等人 2024 年 Table 3 里的其他四个码 [[72,12,6]]、[[90,8,10]]、[[108,8,10]]、[[288,12,18]]，以及小码 bb5 [[30,4,5]] 和一变量祖先 bicycle [[6,2,2]]，在 QEC-Bank 里各有自己的独立条目（等级较低的 T0 / T1 卡：构造器重算了 n、k，其余侧面还没有整理）。它们的码距证据词不一样：bicycle 和 bb5 小到能穷举，d 是 verified；Table 3 的四个大码 d 是 claimed，照抄论文。直觉篇第 5 章把八个条目逐个讲过；关系篇说明它们与家族条目的关系。

These eleven guides cover the QEC-Bank family entry bivariate_bicycle, which ships two executable instances: the primary gross code [[144,12,12]] (l = 12, m = 6, A = x³ + y + y², B = y³ + x + x²) and the toric anchor toric_L3 [[18,2,3]] used to check the numbering conventions. The other four codes of Bravyi et al.'s 2024 Table 3 — [[72,12,6]], [[90,8,10]], [[108,8,10]], [[288,12,18]] — together with the small bb5 [[30,4,5]] and the one-variable ancestor bicycle [[6,2,2]] each have their own separate entry in the QEC-Bank (lower-tier T0 / T1 cards: the constructor recomputes n and k, the other facets have not been written up yet). Their distances carry different evidence words: bicycle and bb5 are small enough to search exhaustively, so d is verified; for the four large Table 3 codes d is claimed, copied from the paper. Chapter 5 of the intuition guide walks all eight; the relations guide explains how they relate to the family entry.

证据词Evidence words

gross 的 n = 144、k = 12 是机器重算的（verified）；d = 12 是 bounded：下界引自 bravyi2024bicycle，上界由 QEC-Bank 用随机低重搜索（种子 5，2000 次）复现出一个重量 12 的逻辑算符（改写信息却不触发任何检查的操作，碰到 12 个比特），所以码距至多 12。0.8%、深度 7 电路、全部逻辑门和昆仑实验都是文献声称，不在任何证书里。各篇都会在数字旁写明这些词。

0.8% 要带着来历读。它出自 Bravyi 等人（arXiv:2308.07915）：在电路级噪声下（测量电路里每个操作以同一个概率 p 出错，彼此独立），用深度 7 的测量电路和 BP-OSD 译码器模拟，再对每个码单独解盈亏平衡方程 pL(p) = kp。pL 是整块码每轮失败的概率（k 个逻辑比特里任何一个出错都算）；kp 近似于 k 个不编码的裸比特里至少一个出错的概率；解出的 p 叫这块码的伪阈值。pL 只在 p ≥ 10−3 处模拟，更低处用拟合公式外推。v2（2024 年 2 月）的摘要写 0.8%，而它的 Table 1 里 gross 的伪阈值是 0.0065（五个码从 0.0048 到 0.0069），正文说 144 与 288 比特两个码「接近 0.007」。所以它不是几种码长的曲线交出来的阈值，也不是 gross 一个码独有的性质；译码器篇讲细节。

For gross, n = 144 and k = 12 are recomputed by machine (verified); d = 12 is bounded: the lower bound is cited to bravyi2024bicycle and the QEC-Bank reruns a seeded low-weight search (seed 5, 2000 iterations) that finds a weight-12 logical operator (an operation that rewrites the information without setting off any check, touching 12 qubits) as the upper witness, so the distance is at most 12. The 0.8%, the depth-7 circuit, every logical gate and the Kunlun experiment are literature claims outside any certificate. Every guide writes these words next to its numbers.

Read the 0.8% with its history. It comes from Bravyi et al. (arXiv:2308.07915): under circuit-level noise (every operation of the measurement circuit fails independently with the same probability p), with the depth-7 measurement circuit and the BP-OSD decoder, they solve for each code separately the break-even equation pL(p) = kp. Here pL is the per-round failure rate of the whole block (any of its k logical qubits failing counts), and kp approximates the chance that at least one of k unencoded qubits fails; the solution is that block's pseudo-threshold. pL is simulated only for p ≥ 10−3 and extrapolated below with a fitted formula. The v2 abstract (February 2024) says 0.8%, while its Table 1 gives gross a pseudo-threshold of 0.0065 (0.0048 to 0.0069 across five codes), and the text calls the 144- and 288-qubit codes “close to 0.007”. So it is not a threshold read off a crossing of curves for several sizes, and not a property of gross alone; the decoders guide has the details.

尾声Epilogue

关于这套教程About these guides

每一篇都是单个 HTML 文件，共用同一套页面样式与交互部件：右上角切换中英文，左侧目录随滚动高亮，橙色虚线引用点开即弹出定义卡片，浅灰点线的符号点一下告诉你怎么读。BB 构造器、多项式平移、置信传播等实验台都在浏览器本地重算，不上传任何数据。

Each guide is a single HTML file sharing the same page style and interactive components: switch language top right, the left rail follows your scroll, amber dashed citations pop up definition cards, and faint dotted symbols tell you how they are read. The BB constructor, polynomial-shift and belief-propagation benches all recompute locally in your browser; nothing is uploaded.